View source for 18.17

{{DISPLAYTITLE:Orthogonal Polynomials - 18.17 Integrals}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/18.17.E1 18.17.E1] || <math qid="Q5742">2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{y}\diff{y} = \JacobipolyP{\alpha+1}{\beta+1}{n-1}@{0}-(1-x)^{\alpha+1}(1+x)^{\beta+1}\JacobipolyP{\alpha+1}{\beta+1}{n-1}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{y}\diff{y} = \JacobipolyP{\alpha+1}{\beta+1}{n-1}@{0}-(1-x)^{\alpha+1}(1+x)^{\beta+1}\JacobipolyP{\alpha+1}{\beta+1}{n-1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*n*int((1 - y)^(alpha)*(1 + y)^(beta)* JacobiP(n, alpha, beta, y), y = 0..x) = JacobiP(n - 1, alpha + 1, beta + 1, 0)-(1 - x)^(alpha + 1)*(1 + x)^(beta + 1)* JacobiP(n - 1, alpha + 1, beta + 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*n*Integrate[(1 - y)^\[Alpha]*(1 + y)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], y], {y, 0, x}, GenerateConditions->None] == JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, 0]-(1 - x)^(\[Alpha]+ 1)*(1 + x)^(\[Beta]+ 1)* JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x]</syntaxhighlight> || Failure || Successful || Manual Skip! || Successful [Tested: 81]
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| [https://dlmf.nist.gov/18.17.E2 18.17.E2] || <math qid="Q5743">\int_{0}^{x}\LaguerrepolyL[]{m}@{y}\LaguerrepolyL[]{n}@{x-y}\diff{y} = \int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\LaguerrepolyL[]{m}@{y}\LaguerrepolyL[]{n}@{x-y}\diff{y} = \int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(LaguerreL(m, y)*LaguerreL(n, x - y), y = 0..x) = int(LaguerreL(m + n, y), y = 0..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LaguerreL[m, y]*LaguerreL[n, x - y], {y, 0, x}, GenerateConditions->None] == Integrate[LaguerreL[m + n, y], {y, 0, x}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 27] || Successful [Tested: 27]
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| [https://dlmf.nist.gov/18.17.E2 18.17.E2] || <math qid="Q5743">\int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y} = \LaguerrepolyL[]{m+n}@{x}-\LaguerrepolyL[]{m+n+1}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y} = \LaguerrepolyL[]{m+n}@{x}-\LaguerrepolyL[]{m+n+1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(LaguerreL(m + n, y), y = 0..x) = LaguerreL(m + n, x)- LaguerreL(m + n + 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LaguerreL[m + n, y], {y, 0, x}, GenerateConditions->None] == LaguerreL[m + n, x]- LaguerreL[m + n + 1, x]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 27]
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| [https://dlmf.nist.gov/18.17.E3 18.17.E3] || <math qid="Q5744">\int_{0}^{x}\HermitepolyH{n}@{y}\diff{y} = \frac{1}{2(n+1)}(\HermitepolyH{n+1}@{x}-\HermitepolyH{n+1}@{0})</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}\HermitepolyH{n}@{y}\diff{y} = \frac{1}{2(n+1)}(\HermitepolyH{n+1}@{x}-\HermitepolyH{n+1}@{0})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(HermiteH(n, y), y = 0..x) = (1)/(2*(n + 1))*(HermiteH(n + 1, x)- HermiteH(n + 1, 0))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[HermiteH[n, y], {y, 0, x}, GenerateConditions->None] == Divide[1,2*(n + 1)]*(HermiteH[n + 1, x]- HermiteH[n + 1, 0])</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
Test Values: {x = 3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.500000000+0.*I
Test Values: {x = 3/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 9]
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| [https://dlmf.nist.gov/18.17.E4 18.17.E4] || <math qid="Q5745">\int_{0}^{x}e^{-y^{2}}\HermitepolyH{n}@{y}\diff{y} = \HermitepolyH{n-1}@{0}-e^{-x^{2}}\HermitepolyH{n-1}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}e^{-y^{2}}\HermitepolyH{n}@{y}\diff{y} = \HermitepolyH{n-1}@{0}-e^{-x^{2}}\HermitepolyH{n-1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(exp(- (y)^(2))*HermiteH(n, y), y = 0..x) = HermiteH(n - 1, 0)- exp(- (x)^(2))*HermiteH(n - 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- (y)^(2)]*HermiteH[n, y], {y, 0, x}, GenerateConditions->None] == HermiteH[n - 1, 0]- Exp[- (x)^(2)]*HermiteH[n - 1, x]</syntaxhighlight> || Failure || Successful || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/18.17.E5 18.17.E5] || <math qid="Q5746">\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}}}{\ultrasphpoly{\lambda}{n}@{1}}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{2}}}}{\ultrasphpoly{\lambda}{n}@{1}} = \frac{\EulerGamma@{\lambda+\frac{1}{2}}}{\pi^{\frac{1}{2}}\EulerGamma@{\lambda}}\*\int_{0}^{\pi}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}}{\ultrasphpoly{\lambda}{n}@{1}}(\sin@@{\phi})^{2\lambda-1}\diff{\phi}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}}}{\ultrasphpoly{\lambda}{n}@{1}}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{2}}}}{\ultrasphpoly{\lambda}{n}@{1}} = \frac{\EulerGamma@{\lambda+\frac{1}{2}}}{\pi^{\frac{1}{2}}\EulerGamma@{\lambda}}\*\int_{0}^{\pi}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}}{\ultrasphpoly{\lambda}{n}@{1}}(\sin@@{\phi})^{2\lambda-1}\diff{\phi}</syntaxhighlight> || <math>\lambda > 0, \realpart@@{(\lambda+\frac{1}{2})} > 0, \realpart@@{(\lambda)} > 0</math> || <syntaxhighlight lang=mathematica>(GegenbauerC(n, lambda, cos(theta[1])))/(GegenbauerC(n, lambda, 1))*(GegenbauerC(n, lambda, cos(theta[2])))/(GegenbauerC(n, lambda, 1)) = (GAMMA(lambda +(1)/(2)))/((Pi)^((1)/(2))* GAMMA(lambda))* int((GegenbauerC(n, lambda, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)))/(GegenbauerC(n, lambda, 1))*(sin(phi))^(2*lambda - 1), phi = 0..Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 1]]],GegenbauerC[n, \[Lambda], 1]]*Divide[GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 2]]],GegenbauerC[n, \[Lambda], 1]] == Divide[Gamma[\[Lambda]+Divide[1,2]],(Pi)^(Divide[1,2])* Gamma[\[Lambda]]]* Integrate[Divide[GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]],GegenbauerC[n, \[Lambda], 1]]*(Sin[\[Phi]])^(2*\[Lambda]- 1), {\[Phi], 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E6 18.17.E6] || <math qid="Q5747">\LegendrepolyP{n}@{\cos@@{\theta_{1}}}\LegendrepolyP{n}@{\cos@@{\theta_{2}}} = \frac{1}{\pi}\int_{0}^{\pi}\LegendrepolyP{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}\diff{\phi}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LegendrepolyP{n}@{\cos@@{\theta_{1}}}\LegendrepolyP{n}@{\cos@@{\theta_{2}}} = \frac{1}{\pi}\int_{0}^{\pi}\LegendrepolyP{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}\diff{\phi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LegendreP(n, cos(theta[1]))*LegendreP(n, cos(theta[2])) = (1)/(Pi)*int(LegendreP(n, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)), phi = 0..Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[n, Cos[Subscript[\[Theta], 1]]]*LegendreP[n, Cos[Subscript[\[Theta], 2]]] == Divide[1,Pi]*Integrate[LegendreP[n, Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]], {\[Phi], 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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| [https://dlmf.nist.gov/18.17.E7 18.17.E7] || <math qid="Q5748">\left(\LegendrepolyP{n}@{x}\right)^{2}+4\pi^{-2}\left(\FerrersQ[]{n}@{x}\right)^{2} = 4\pi^{-2}\*\int_{1}^{\infty}\assLegendreQ[]{n}@{x^{2}+(1-x^{2})t}(t^{2}-1)^{-\frac{1}{2}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\LegendrepolyP{n}@{x}\right)^{2}+4\pi^{-2}\left(\FerrersQ[]{n}@{x}\right)^{2} = 4\pi^{-2}\*\int_{1}^{\infty}\assLegendreQ[]{n}@{x^{2}+(1-x^{2})t}(t^{2}-1)^{-\frac{1}{2}}\diff{t}</syntaxhighlight> || <math>-1 < x, x < 1</math> || <syntaxhighlight lang=mathematica>(LegendreP(n, x))^(2)+ 4*(Pi)^(- 2)*(LegendreQ(n, x))^(2) = 4*(Pi)^(- 2)* int(LegendreQ(n, (x)^(2)+(1 - (x)^(2))*t)*((t)^(2)- 1)^(-(1)/(2)), t = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(LegendreP[n, x])^(2)+ 4*(Pi)^(- 2)*(LegendreQ[n, x])^(2) == 4*(Pi)^(- 2)* Integrate[LegendreQ[n, 0, 3, (x)^(2)+(1 - (x)^(2))*t]*((t)^(2)- 1)^(-Divide[1,2]), {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0.+Float(infinity)*I
Test Values: {x = 1/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0.+Float(infinity)*I
Test Values: {x = 1/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 3]
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| [https://dlmf.nist.gov/18.17.E8 18.17.E8] || <math qid="Q5749">\left(\HermitepolyH{n}@{x}\right)^{2}+2^{n}(n!)^{2}e^{x^{2}}\left(\paraV@{-n-\tfrac{1}{2}}{2^{\frac{1}{2}}x}\right)^{2} = \frac{2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}\tanh@@{t}}}{(\sinh@@{2t})^{\frac{1}{2}}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\HermitepolyH{n}@{x}\right)^{2}+2^{n}(n!)^{2}e^{x^{2}}\left(\paraV@{-n-\tfrac{1}{2}}{2^{\frac{1}{2}}x}\right)^{2} = \frac{2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}\tanh@@{t}}}{(\sinh@@{2t})^{\frac{1}{2}}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(HermiteH(n, x))^(2)+ (2)^(n)*(factorial(n))^(2)* exp((x)^(2))*(CylinderV(- n -(1)/(2), (2)^((1)/(2))* x))^(2) = ((2)^(n +(3)/(2))* factorial(n)*exp((x)^(2)))/(Pi)*int((exp(-(2*n + 1)*t + (x)^(2)* tanh(t)))/((sinh(2*t))^((1)/(2))), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(HermiteH[n, x])^(2)+ (2)^(n)*((n)!)^(2)* Exp[(x)^(2)]*(Divide[GAMMA[1/2 + - n -Divide[1,2]], Pi]*(Sin[Pi*(- n -Divide[1,2])] * ParabolicCylinderD[-(- n -Divide[1,2]) - 1/2, (2)^(Divide[1,2])* x] + ParabolicCylinderD[-(- n -Divide[1,2]) - 1/2, -((2)^(Divide[1,2])* x)]))^(2) == Divide[(2)^(n +Divide[3,2])* (n)!*Exp[(x)^(2)],Pi]*Integrate[Divide[Exp[-(2*n + 1)*t + (x)^(2)* Tanh[t]],(Sinh[2*t])^(Divide[1,2])], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 9] || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E9 18.17.E9] || <math qid="Q5750">\frac{(1-x)^{\alpha+\mu}\JacobipolyP{\alpha+\mu}{\beta-\mu}{n}@{x}}{\EulerGamma@{\alpha+\mu+n+1}} = \int_{x}^{1}\frac{(1-y)^{\alpha}\JacobipolyP{\alpha}{\beta}{n}@{y}}{\EulerGamma@{\alpha+n+1}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{(1-x)^{\alpha+\mu}\JacobipolyP{\alpha+\mu}{\beta-\mu}{n}@{x}}{\EulerGamma@{\alpha+\mu+n+1}} = \int_{x}^{1}\frac{(1-y)^{\alpha}\JacobipolyP{\alpha}{\beta}{n}@{y}}{\EulerGamma@{\alpha+n+1}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</syntaxhighlight> || <math>\mu > 0, -1 < x, x < 1, \realpart@@{(\alpha+\mu+n+1)} > 0, \realpart@@{(\alpha+n+1)} > 0, \realpart@@{(\mu)} > 0</math> || <syntaxhighlight lang=mathematica>((1 - x)^(alpha + mu)* JacobiP(n, alpha + mu, beta - mu, x))/(GAMMA(alpha + mu + n + 1)) = int(((1 - y)^(alpha)* JacobiP(n, alpha, beta, y))/(GAMMA(alpha + n + 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(1 - x)^(\[Alpha]+ \[Mu])* JacobiP[n, \[Alpha]+ \[Mu], \[Beta]- \[Mu], x],Gamma[\[Alpha]+ \[Mu]+ n + 1]] == Integrate[Divide[(1 - y)^\[Alpha]* JacobiP[n, \[Alpha], \[Beta], y],Gamma[\[Alpha]+ n + 1]]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E10 18.17.E10] || <math qid="Q5751">\frac{x^{\beta+\mu}(x+1)^{n}}{\EulerGamma@{\beta+\mu+n+1}}\JacobipolyP{\alpha}{\beta+\mu}{n}@{\frac{x-1}{x+1}} = \int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\EulerGamma@{\beta+n+1}}\JacobipolyP{\alpha}{\beta}{n}@{\frac{y-1}{y+1}}\*\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x^{\beta+\mu}(x+1)^{n}}{\EulerGamma@{\beta+\mu+n+1}}\JacobipolyP{\alpha}{\beta+\mu}{n}@{\frac{x-1}{x+1}} = \int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\EulerGamma@{\beta+n+1}}\JacobipolyP{\alpha}{\beta}{n}@{\frac{y-1}{y+1}}\*\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</syntaxhighlight> || <math>\mu > 0, x > 0, \realpart@@{(\beta+\mu+n+1)} > 0, \realpart@@{(\beta+n+1)} > 0, \realpart@@{(\mu)} > 0</math> || <syntaxhighlight lang=mathematica>((x)^(beta + mu)*(x + 1)^(n))/(GAMMA(beta + mu + n + 1))*JacobiP(n, alpha, beta + mu, (x - 1)/(x + 1)) = int(((y)^(beta)*(y + 1)^(n))/(GAMMA(beta + n + 1))*JacobiP(n, alpha, beta, (y - 1)/(y + 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 0..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(x)^(\[Beta]+ \[Mu])*(x + 1)^(n),Gamma[\[Beta]+ \[Mu]+ n + 1]]*JacobiP[n, \[Alpha], \[Beta]+ \[Mu], Divide[x - 1,x + 1]] == Integrate[Divide[(y)^\[Beta]*(y + 1)^(n),Gamma[\[Beta]+ n + 1]]*JacobiP[n, \[Alpha], \[Beta], Divide[y - 1,y + 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 0, x}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E11 18.17.E11] || <math qid="Q5752">\frac{\EulerGamma@{n+\alpha+\beta-\mu+1}}{x^{n+\alpha+\beta-\mu+1}}\JacobipolyP{\alpha}{\beta-\mu}{n}@{1-2x^{-1}} = \int_{x}^{\infty}\frac{\EulerGamma@{n+\alpha+\beta+1}}{y^{n+\alpha+\beta+1}}\JacobipolyP{\alpha}{\beta}{n}@{1-2y^{-1}}\*\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\EulerGamma@{n+\alpha+\beta-\mu+1}}{x^{n+\alpha+\beta-\mu+1}}\JacobipolyP{\alpha}{\beta-\mu}{n}@{1-2x^{-1}} = \int_{x}^{\infty}\frac{\EulerGamma@{n+\alpha+\beta+1}}{y^{n+\alpha+\beta+1}}\JacobipolyP{\alpha}{\beta}{n}@{1-2y^{-1}}\*\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</syntaxhighlight> || <math>\alpha+\beta+1 > \mu, \mu > 0, x > 1, \realpart@@{(n+\alpha+\beta-\mu+1)} > 0, \realpart@@{(n+\alpha+\beta+1)} > 0, \realpart@@{(\mu)} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(n + alpha + beta - mu + 1))/((x)^(n + alpha + beta - mu + 1))*JacobiP(n, alpha, beta - mu, 1 - 2*(x)^(- 1)) = int((GAMMA(n + alpha + beta + 1))/((y)^(n + alpha + beta + 1))*JacobiP(n, alpha, beta, 1 - 2*(y)^(- 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Gamma[n + \[Alpha]+ \[Beta]- \[Mu]+ 1],(x)^(n + \[Alpha]+ \[Beta]- \[Mu]+ 1)]*JacobiP[n, \[Alpha], \[Beta]- \[Mu], 1 - 2*(x)^(- 1)] == Integrate[Divide[Gamma[n + \[Alpha]+ \[Beta]+ 1],(y)^(n + \[Alpha]+ \[Beta]+ 1)]*JacobiP[n, \[Alpha], \[Beta], 1 - 2*(y)^(- 1)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E12 18.17.E12] || <math qid="Q5753">\frac{\EulerGamma@{\lambda-\mu}\ultrasphpoly{\lambda-\mu}{n}@{x^{-\frac{1}{2}}}}{x^{\lambda-\mu+\frac{1}{2}n}} = \int_{x}^{\infty}\frac{\EulerGamma@{\lambda}\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\EulerGamma@{\lambda-\mu}\ultrasphpoly{\lambda-\mu}{n}@{x^{-\frac{1}{2}}}}{x^{\lambda-\mu+\frac{1}{2}n}} = \int_{x}^{\infty}\frac{\EulerGamma@{\lambda}\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</syntaxhighlight> || <math>\lambda > \mu, \mu > 0, x > 0, \realpart@@{(\lambda-\mu)} > 0, \realpart@@{(\lambda)} > 0, \realpart@@{(\mu)} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(lambda - mu)*GegenbauerC(n, lambda - mu, (x)^(-(1)/(2))))/((x)^(lambda - mu +(1)/(2)*n)) = int((GAMMA(lambda)*GegenbauerC(n, lambda, (y)^(-(1)/(2))))/((y)^(lambda +(1)/(2)*n))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Gamma[\[Lambda]- \[Mu]]*GegenbauerC[n, \[Lambda]- \[Mu], (x)^(-Divide[1,2])],(x)^(\[Lambda]- \[Mu]+Divide[1,2]*n)] == Integrate[Divide[Gamma[\[Lambda]]*GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],(y)^(\[Lambda]+Divide[1,2]*n)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E13 18.17.E13] || <math qid="Q5754">\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\EulerGamma@{\lambda+\mu+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda+\mu}{n}@{x^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda+\mu}{n}@{1}} = \int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{\EulerGamma@{\lambda+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda}{n}@{1}}\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\EulerGamma@{\lambda+\mu+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda+\mu}{n}@{x^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda+\mu}{n}@{1}} = \int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{\EulerGamma@{\lambda+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda}{n}@{1}}\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</syntaxhighlight> || <math>\mu > 0, x > 1, \realpart@@{(\lambda+\mu+\tfrac{1}{2})} > 0, \realpart@@{(\lambda+\tfrac{1}{2})} > 0, \realpart@@{(\mu)} > 0</math> || <syntaxhighlight lang=mathematica>((x)^((1)/(2)*n)*(x - 1)^(lambda + mu -(1)/(2)))/(GAMMA(lambda + mu +(1)/(2)))*(GegenbauerC(n, lambda + mu, (x)^(-(1)/(2))))/(GegenbauerC(n, lambda + mu, 1)) = int(((y)^((1)/(2)*n)*(y - 1)^(lambda -(1)/(2)))/(GAMMA(lambda +(1)/(2)))*(GegenbauerC(n, lambda, (y)^(-(1)/(2))))/(GegenbauerC(n, lambda, 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 1..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(x)^(Divide[1,2]*n)*(x - 1)^(\[Lambda]+ \[Mu]-Divide[1,2]),Gamma[\[Lambda]+ \[Mu]+Divide[1,2]]]*Divide[GegenbauerC[n, \[Lambda]+ \[Mu], (x)^(-Divide[1,2])],GegenbauerC[n, \[Lambda]+ \[Mu], 1]] == Integrate[Divide[(y)^(Divide[1,2]*n)*(y - 1)^(\[Lambda]-Divide[1,2]),Gamma[\[Lambda]+Divide[1,2]]]*Divide[GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],GegenbauerC[n, \[Lambda], 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 1, x}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E14 18.17.E14] || <math qid="Q5755">\frac{x^{\alpha+\mu}\LaguerrepolyL[\alpha+\mu]{n}@{x}}{\EulerGamma@{\alpha+\mu+n+1}} = \int_{0}^{x}\frac{y^{\alpha}\LaguerrepolyL[\alpha]{n}@{y}}{\EulerGamma@{\alpha+n+1}}\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x^{\alpha+\mu}\LaguerrepolyL[\alpha+\mu]{n}@{x}}{\EulerGamma@{\alpha+\mu+n+1}} = \int_{0}^{x}\frac{y^{\alpha}\LaguerrepolyL[\alpha]{n}@{y}}{\EulerGamma@{\alpha+n+1}}\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</syntaxhighlight> || <math>\mu > 0, x > 0, \realpart@@{(\alpha+\mu+n+1)} > 0, \realpart@@{(\alpha+n+1)} > 0, \realpart@@{(\mu)} > 0</math> || <syntaxhighlight lang=mathematica>((x)^(alpha + mu)* LaguerreL(n, alpha + mu, x))/(GAMMA(alpha + mu + n + 1)) = int(((y)^(alpha)* LaguerreL(n, alpha, y))/(GAMMA(alpha + n + 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 0..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(x)^(\[Alpha]+ \[Mu])* LaguerreL[n, \[Alpha]+ \[Mu], x],Gamma[\[Alpha]+ \[Mu]+ n + 1]] == Integrate[Divide[(y)^\[Alpha]* LaguerreL[n, \[Alpha], y],Gamma[\[Alpha]+ n + 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 0, x}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Manual Skip!
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| [https://dlmf.nist.gov/18.17.E15 18.17.E15] || <math qid="Q5756">e^{-x}\LaguerrepolyL[\alpha]{n}@{x} = \int_{x}^{\infty}e^{-y}\LaguerrepolyL[\alpha+\mu]{n}@{y}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-x}\LaguerrepolyL[\alpha]{n}@{x} = \int_{x}^{\infty}e^{-y}\LaguerrepolyL[\alpha+\mu]{n}@{y}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}</syntaxhighlight> || <math>\mu > 0, \realpart@@{(\mu)} > 0</math> || <syntaxhighlight lang=mathematica>exp(- x)*LaguerreL(n, alpha, x) = int(exp(- y)*LaguerreL(n, alpha + mu, y)*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- x]*LaguerreL[n, \[Alpha], x] == Integrate[Exp[- y]*LaguerreL[n, \[Alpha]+ \[Mu], y]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E16 18.17.E16] || <math qid="Q5757">\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}e^{ixy}\diff{x} = \frac{(iy)^{n}e^{iy}}{n!}2^{n+\alpha+\beta+1}\EulerBeta@{n+\alpha+1}{n+\beta+1}\genhyperF{1}{1}@{n+\alpha+1}{2n+\alpha+\beta+2}{-2iy}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}e^{ixy}\diff{x} = \frac{(iy)^{n}e^{iy}}{n!}2^{n+\alpha+\beta+1}\EulerBeta@{n+\alpha+1}{n+\beta+1}\genhyperF{1}{1}@{n+\alpha+1}{2n+\alpha+\beta+2}{-2iy}</syntaxhighlight> || <math>\realpart@@{(n+\alpha+1)} > 0, \realpart@@{(n+\beta+1)} > 0, \realpart@@{((n+\alpha+1)+b)} > 0, \realpart@@{(a+(n+\beta+1))} > 0</math> || <syntaxhighlight lang=mathematica>int((1 - x)^(alpha)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x)*exp(I*x*y), x = - 1..1) = ((I*y)^(n)* exp(I*y))/(factorial(n))*(2)^(n + alpha + beta + 1)* Beta(n + alpha + 1, n + beta + 1)*hypergeom([n + alpha + 1], [2*n + alpha + beta + 2], - 2*I*y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - x)^\[Alpha]*(1 + x)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], x]*Exp[I*x*y], {x, - 1, 1}, GenerateConditions->None] == Divide[(I*y)^(n)* Exp[I*y],(n)!]*(2)^(n + \[Alpha]+ \[Beta]+ 1)* Beta[n + \[Alpha]+ 1, n + \[Beta]+ 1]*HypergeometricPFQ[{n + \[Alpha]+ 1}, {2*n + \[Alpha]+ \[Beta]+ 2}, - 2*I*y]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E17 18.17.E17] || <math qid="Q5758">\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n}@{x}\cos@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda}\BesselJ{\lambda+2n}@{y}}{(2n)!\EulerGamma@{\lambda}(2y)^{\lambda}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n}@{x}\cos@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda}\BesselJ{\lambda+2n}@{y}}{(2n)!\EulerGamma@{\lambda}(2y)^{\lambda}}</syntaxhighlight> || <math>\realpart@@{((\lambda+2n)+k+1)} > 0, \realpart@@{(2n+2\lambda)} > 0, \realpart@@{(\lambda)} > 0</math> || <syntaxhighlight lang=mathematica>int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2*n, lambda, x)*cos(x*y), x = 0..1) = ((- 1)^(n)* Pi*GAMMA(2*n + 2*lambda)*BesselJ(lambda + 2*n, y))/(factorial(2*n)*GAMMA(lambda)*(2*y)^(lambda))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2*n, \[Lambda], x]*Cos[x*y], {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pi*Gamma[2*n + 2*\[Lambda]]*BesselJ[\[Lambda]+ 2*n, y],(2*n)!*Gamma[\[Lambda]]*(2*y)^\[Lambda]]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E18 18.17.E18] || <math qid="Q5759">\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n+1}@{x}\sin@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda+1}\BesselJ{2n+\lambda+1}@{y}}{(2n+1)!\EulerGamma@{\lambda}(2y)^{\lambda}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n+1}@{x}\sin@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda+1}\BesselJ{2n+\lambda+1}@{y}}{(2n+1)!\EulerGamma@{\lambda}(2y)^{\lambda}}</syntaxhighlight> || <math>\realpart@@{((2n+\lambda+1)+k+1)} > 0, \realpart@@{(2n+2\lambda+1)} > 0, \realpart@@{(\lambda)} > 0</math> || <syntaxhighlight lang=mathematica>int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2*n + 1, lambda, x)*sin(x*y), x = 0..1) = ((- 1)^(n)* Pi*GAMMA(2*n + 2*lambda + 1)*BesselJ(2*n + lambda + 1, y))/(factorial(2*n + 1)*GAMMA(lambda)*(2*y)^(lambda))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2*n + 1, \[Lambda], x]*Sin[x*y], {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pi*Gamma[2*n + 2*\[Lambda]+ 1]*BesselJ[2*n + \[Lambda]+ 1, y],(2*n + 1)!*Gamma[\[Lambda]]*(2*y)^\[Lambda]]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E19 18.17.E19] || <math qid="Q5760">\int_{-1}^{1}\LegendrepolyP{n}@{x}e^{ixy}\diff{x} = i^{n}\sqrt{\frac{2\pi}{y}}\BesselJ{n+\frac{1}{2}}@{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\LegendrepolyP{n}@{x}e^{ixy}\diff{x} = i^{n}\sqrt{\frac{2\pi}{y}}\BesselJ{n+\frac{1}{2}}@{y}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(LegendreP(n, x)*exp(I*x*y), x = - 1..1) = (I)^(n)*sqrt((2*Pi)/(y))*BesselJ(n +(1)/(2), y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[n, x]*Exp[I*x*y], {x, - 1, 1}, GenerateConditions->None] == (I)^(n)*Sqrt[Divide[2*Pi,y]]*BesselJ[n +Divide[1,2], y]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1455515881e-15-1.584691883*I
Test Values: {y = -3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5093971348+.7797894631e-16*I
Test Values: {y = -3/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, -1.584691882848889]
Test Values: {Rule[n, 1], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5093971347536326, -3.3306690738754696*^-16]
Test Values: {Rule[n, 2], Rule[y, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/18.17.E20 18.17.E20] || <math qid="Q5761">\int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\cos@{xy}\diff{x} = (-1)^{n}\tfrac{1}{2}\pi\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\BesselJ{-n-\frac{1}{2}}@{\tfrac{1}{2}y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\cos@{xy}\diff{x} = (-1)^{n}\tfrac{1}{2}\pi\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\BesselJ{-n-\frac{1}{2}}@{\tfrac{1}{2}y}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(LegendreP(n, 1 - 2*(x)^(2))*cos(x*y), x = 0..1) = (- 1)^(n)*(1)/(2)*Pi*BesselJ(n +(1)/(2), (1)/(2)*y)*BesselJ(- n -(1)/(2), (1)/(2)*y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[n, 1 - 2*(x)^(2)]*Cos[x*y], {x, 0, 1}, GenerateConditions->None] == (- 1)^(n)*Divide[1,2]*Pi*BesselJ[n +Divide[1,2], Divide[1,2]*y]*BesselJ[- n -Divide[1,2], Divide[1,2]*y]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
|- 
| [https://dlmf.nist.gov/18.17.E21 18.17.E21] || <math qid="Q5762">\int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\sin@{xy}\diff{x} = \tfrac{1}{2}\pi\left(\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\right)^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\sin@{xy}\diff{x} = \tfrac{1}{2}\pi\left(\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\right)^{2}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(LegendreP(n, 1 - 2*(x)^(2))*sin(x*y), x = 0..1) = (1)/(2)*Pi*(BesselJ(n +(1)/(2), (1)/(2)*y))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[n, 1 - 2*(x)^(2)]*Sin[x*y], {x, 0, 1}, GenerateConditions->None] == Divide[1,2]*Pi*(BesselJ[n +Divide[1,2], Divide[1,2]*y])^(2)</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
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| [https://dlmf.nist.gov/18.17.E30 18.17.E30] || <math qid="Q5771">\int_{0}^{\infty}x^{2n}e^{-\frac{1}{2}x^{2}}\LaguerrepolyL[n-\frac{1}{2}]{n}@{\tfrac{1}{2}x^{2}}\cos@{xy}\diff{x} = \sqrt{\tfrac{1}{2}\pi}y^{2n}e^{-\frac{1}{2}y^{2}}\LaguerrepolyL[n-\frac{1}{2}]{n}@{\tfrac{1}{2}y^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}x^{2n}e^{-\frac{1}{2}x^{2}}\LaguerrepolyL[n-\frac{1}{2}]{n}@{\tfrac{1}{2}x^{2}}\cos@{xy}\diff{x} = \sqrt{\tfrac{1}{2}\pi}y^{2n}e^{-\frac{1}{2}y^{2}}\LaguerrepolyL[n-\frac{1}{2}]{n}@{\tfrac{1}{2}y^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((x)^(2*n)* exp(-(1)/(2)*(x)^(2))*LaguerreL(n, n -(1)/(2), (1)/(2)*(x)^(2))*cos(x*y), x = 0..infinity) = sqrt((1)/(2)*Pi)*(y)^(2*n)* exp(-(1)/(2)*(y)^(2))*LaguerreL(n, n -(1)/(2), (1)/(2)*(y)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(2*n)* Exp[-Divide[1,2]*(x)^(2)]*LaguerreL[n, n -Divide[1,2], Divide[1,2]*(x)^(2)]*Cos[x*y], {x, 0, Infinity}, GenerateConditions->None] == Sqrt[Divide[1,2]*Pi]*(y)^(2*n)* Exp[-Divide[1,2]*(y)^(2)]*LaguerreL[n, n -Divide[1,2], Divide[1,2]*(y)^(2)]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E31 18.17.E31] || <math qid="Q5772">\int_{0}^{\infty}e^{-ax}x^{\nu-2n}\LaguerrepolyL[\nu-2n]{2n-1}@{ax}\cos@{xy}\diff{x} = i\frac{(-1)^{n}\EulerGamma@{\nu}}{2(2n-1)!}y^{2n-1}\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-ax}x^{\nu-2n}\LaguerrepolyL[\nu-2n]{2n-1}@{ax}\cos@{xy}\diff{x} = i\frac{(-1)^{n}\EulerGamma@{\nu}}{2(2n-1)!}y^{2n-1}\left((a+iy)^{-\nu}-(a-iy)^{-\nu}\right)</syntaxhighlight> || <math>\nu > 2n-1, a > 0, \realpart@@{(\nu)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- a*x)*(x)^(nu - 2*n)* LaguerreL(2*n - 1, nu - 2*n, a*x)*cos(x*y), x = 0..infinity) = I*((- 1)^(n)* GAMMA(nu))/(2*factorial(2*n - 1))*(y)^(2*n - 1)*((a + I*y)^(- nu)-(a - I*y)^(- nu))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- a*x]*(x)^(\[Nu]- 2*n)* LaguerreL[2*n - 1, \[Nu]- 2*n, a*x]*Cos[x*y], {x, 0, Infinity}, GenerateConditions->None] == I*Divide[(- 1)^(n)* Gamma[\[Nu]],2*(2*n - 1)!]*(y)^(2*n - 1)*((a + I*y)^(- \[Nu])-(a - I*y)^(- \[Nu]))</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E32 18.17.E32] || <math qid="Q5773">\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}\LaguerrepolyL[\nu-1-2n]{2n}@{ax}\cos@{xy}\diff{x} = \frac{(-1)^{n}\EulerGamma@{\nu}}{2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}\LaguerrepolyL[\nu-1-2n]{2n}@{ax}\cos@{xy}\diff{x} = \frac{(-1)^{n}\EulerGamma@{\nu}}{2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right)</syntaxhighlight> || <math>\nu > 2n, a > 0, \realpart@@{(\nu)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- a*x)*(x)^(nu - 1 - 2*n)* LaguerreL(2*n, nu - 1 - 2*n, a*x)*cos(x*y), x = 0..infinity) = ((- 1)^(n)* GAMMA(nu))/(2*factorial(2*n))*(y)^(2*n)*((a + I*y)^(- nu)+(a - I*y)^(- nu))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- a*x]*(x)^(\[Nu]- 1 - 2*n)* LaguerreL[2*n, \[Nu]- 1 - 2*n, a*x]*Cos[x*y], {x, 0, Infinity}, GenerateConditions->None] == Divide[(- 1)^(n)* Gamma[\[Nu]],2*(2*n)!]*(y)^(2*n)*((a + I*y)^(- \[Nu])+(a - I*y)^(- \[Nu]))</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E33 18.17.E33] || <math qid="Q5774">\int_{-1}^{1}e^{-(x+1)z}\JacobipolyP{\alpha}{\beta}{n}@{x}(1-x)^{\alpha}(1+x)^{\beta}\diff{x} = \frac{(-1)^{n}2^{\alpha+\beta+n+1}\EulerGamma@{\alpha+n+1}\EulerGamma@{\beta+n+1}}{\EulerGamma@{\alpha+\beta+2n+2}n!}z^{n}\genhyperF{1}{1}@@{\beta+n+1}{\alpha+\beta+2n+2}{-2z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}e^{-(x+1)z}\JacobipolyP{\alpha}{\beta}{n}@{x}(1-x)^{\alpha}(1+x)^{\beta}\diff{x} = \frac{(-1)^{n}2^{\alpha+\beta+n+1}\EulerGamma@{\alpha+n+1}\EulerGamma@{\beta+n+1}}{\EulerGamma@{\alpha+\beta+2n+2}n!}z^{n}\genhyperF{1}{1}@@{\beta+n+1}{\alpha+\beta+2n+2}{-2z}</syntaxhighlight> || <math>\realpart@@{(\alpha+n+1)} > 0, \realpart@@{(\beta+n+1)} > 0, \realpart@@{(\alpha+\beta+2n+2)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(x + 1)*(x + y*I))*JacobiP(n, alpha, beta, x)*(1 - x)^(alpha)*(1 + x)^(beta), x = - 1..1) = ((- 1)^(n)* (2)^(alpha + beta + n + 1)* GAMMA(alpha + n + 1)*GAMMA(beta + n + 1))/(GAMMA(alpha + beta + 2*n + 2)*factorial(n))*(x + y*I)^(n)* hypergeom([beta + n + 1], [alpha + beta + 2*n + 2], - 2*(x + y*I))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-(x + 1)*(x + y*I)]*JacobiP[n, \[Alpha], \[Beta], x]*(1 - x)^\[Alpha]*(1 + x)^\[Beta], {x, - 1, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* (2)^(\[Alpha]+ \[Beta]+ n + 1)* Gamma[\[Alpha]+ n + 1]*Gamma[\[Beta]+ n + 1],Gamma[\[Alpha]+ \[Beta]+ 2*n + 2]*(n)!]*(x + y*I)^(n)* HypergeometricPFQ[{\[Beta]+ n + 1}, {\[Alpha]+ \[Beta]+ 2*n + 2}, - 2*(x + y*I)]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E34 18.17.E34] || <math qid="Q5775">\int_{0}^{\infty}e^{-xz}\LaguerrepolyL[\alpha]{n}@{x}e^{-x}x^{\alpha}\diff{x} = \frac{\EulerGamma@{\alpha+n+1}z^{n}}{n!(z+1)^{\alpha+n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-xz}\LaguerrepolyL[\alpha]{n}@{x}e^{-x}x^{\alpha}\diff{x} = \frac{\EulerGamma@{\alpha+n+1}z^{n}}{n!(z+1)^{\alpha+n+1}}</syntaxhighlight> || <math>\realpart@@{z} > -1, \realpart@@{(\alpha+n+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- x*(x + y*I))*LaguerreL(n, alpha, x)*exp(- x)*(x)^(alpha), x = 0..infinity) = (GAMMA(alpha + n + 1)*(x + y*I)^(n))/(factorial(n)*((x + y*I)+ 1)^(alpha + n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- x*(x + y*I)]*LaguerreL[n, \[Alpha], x]*Exp[- x]*(x)^\[Alpha], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]+ n + 1]*(x + y*I)^(n),(n)!*((x + y*I)+ 1)^(\[Alpha]+ n + 1)]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [162 / 162]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.07467065623203636, -0.1489394690482153], NIntegrate[Complex[-0.027140152128725715, 0.033616541935162864]
Test Values: {1.5, 0, DirectedInfinity[1]}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.13823623490446432, -0.16092399439966643], NIntegrate[Complex[-0.006785038032181429, 0.008404135483790716]
Test Values: {1.5, 0, DirectedInfinity[1]}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/18.17.E35 18.17.E35] || <math qid="Q5776">\int_{-\infty}^{\infty}e^{-xz}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \pi^{\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}e^{-xz}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \pi^{\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(exp(- x*(x + y*I))*HermiteH(n, x)*exp(- (x)^(2)), x = - infinity..infinity) = (Pi)^((1)/(2))*(-(x + y*I))^(n)* exp((1)/(4)*(x + y*I)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- x*(x + y*I)]*HermiteH[n, x]*Exp[- (x)^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == (Pi)^(Divide[1,2])*(-(x + y*I))^(n)* Exp[Divide[1,4]*(x + y*I)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.252480791-2.835663866*I
Test Values: {x = 3/2, y = -3/2, n = 1, z = 1+I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 5.718319609+3.439082150*I
Test Values: {x = 3/2, y = -3/2, n = 2, z = 1+I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.25248079113256, -3.5452022239920282], NIntegrate[Complex[-0.020935135800726114, 0.025930837352181123]
Test Values: {1.5, DirectedInfinity[-1], DirectedInfinity[1]}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[z, Complex[1, 1]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[7.196524522686883, 3.4390821492892023], NIntegrate[Complex[-0.048848650201694266, 0.060505287155089287]
Test Values: {1.5, DirectedInfinity[-1], DirectedInfinity[1]}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[z, Complex[1, 1]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/18.17.E36 18.17.E36] || <math qid="Q5777">\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}\diff{x} = \frac{2^{\beta+z}\EulerGamma@{z}\EulerGamma@{1+\beta+n}\Pochhammersym{1+\alpha-z}{n}}{n!\EulerGamma@{1+\beta+z+n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}\diff{x} = \frac{2^{\beta+z}\EulerGamma@{z}\EulerGamma@{1+\beta+n}\Pochhammersym{1+\alpha-z}{n}}{n!\EulerGamma@{1+\beta+z+n}}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{(1+\beta+n)} > 0, \realpart@@{(1+\beta+z+n)} > 0</math> || <syntaxhighlight lang=mathematica>int((1 - x)^((x + y*I)- 1)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x), x = - 1..1) = ((2)^(beta +(x + y*I))* GAMMA(x + y*I)*GAMMA(1 + beta + n)*pochhammer(1 + alpha -(x + y*I), n))/(factorial(n)*GAMMA(1 + beta +(x + y*I)+ n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - x)^((x + y*I)- 1)*(1 + x)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], x], {x, - 1, 1}, GenerateConditions->None] == Divide[(2)^(\[Beta]+(x + y*I))* Gamma[x + y*I]*Gamma[1 + \[Beta]+ n]*Pochhammer[1 + \[Alpha]-(x + y*I), n],(n)!*Gamma[1 + \[Beta]+(x + y*I)+ n]]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E37 18.17.E37] || <math qid="Q5778">\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{n}@{x}x^{z-1}\diff{x} = \frac{\pi\,2^{1-2\lambda-z}\EulerGamma@{n+2\lambda}\EulerGamma@{z}}{n!\EulerGamma@{\lambda}\EulerGamma@{\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z}\EulerGamma@{\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{n}@{x}x^{z-1}\diff{x} = \frac{\pi\,2^{1-2\lambda-z}\EulerGamma@{n+2\lambda}\EulerGamma@{z}}{n!\EulerGamma@{\lambda}\EulerGamma@{\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z}\EulerGamma@{\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n}}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{(n+2\lambda)} > 0, \realpart@@{(\lambda)} > 0, \realpart@@{(\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z)} > 0, \realpart@@{(\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n)} > 0</math> || <syntaxhighlight lang=mathematica>int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x)*(x)^((x + y*I)- 1), x = 0..1) = (Pi*(2)^(1 - 2*lambda -(x + y*I))* GAMMA(n + 2*lambda)*GAMMA(x + y*I))/(factorial(n)*GAMMA(lambda)*GAMMA((1)/(2)+(1)/(2)*n + lambda +(1)/(2)*(x + y*I))*GAMMA((1)/(2)+(1)/(2)*(x + y*I)-(1)/(2)*n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x]*(x)^((x + y*I)- 1), {x, 0, 1}, GenerateConditions->None] == Divide[Pi*(2)^(1 - 2*\[Lambda]-(x + y*I))* Gamma[n + 2*\[Lambda]]*Gamma[x + y*I],(n)!*Gamma[\[Lambda]]*Gamma[Divide[1,2]+Divide[1,2]*n + \[Lambda]+Divide[1,2]*(x + y*I)]*Gamma[Divide[1,2]+Divide[1,2]*(x + y*I)-Divide[1,2]*n]]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [270 / 270]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.2612561594092788, -0.2567131462958256], NIntegrate[Complex[0.3181035727957409, 0.7653241874975689]
Test Values: {1.5, 0, 1}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.264978322932814, -0.1130252321165333], NIntegrate[Complex[0.21035635691874377, 2.1256411810993385]
Test Values: {1.5, 0, 1}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/18.17.E38 18.17.E38] || <math qid="Q5779">\int_{0}^{1}\LegendrepolyP{2n}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{\frac{1}{2}-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}z}{n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\LegendrepolyP{2n}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{\frac{1}{2}-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}z}{n+1}}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>int(LegendreP(2*n, x)*(x)^((x + y*I)- 1), x = 0..1) = ((- 1)^(n)* pochhammer((1)/(2)-(1)/(2)*(x + y*I), n))/(2*pochhammer((1)/(2)*(x + y*I), n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[2*n, x]*(x)^((x + y*I)- 1), {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pochhammer[Divide[1,2]-Divide[1,2]*(x + y*I), n],2*Pochhammer[Divide[1,2]*(x + y*I), n + 1]]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.19540229885057472, 0.011494252873563225], NIntegrate[Complex[2.8897275468024644, -2.0119423961065603]
Test Values: {1.5, 0, 1}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.03978779840848807, 0.061007957559681705], NIntegrate[Complex[14.158094475230552, -9.85742429396774]
Test Values: {1.5, 0, 1}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/18.17.E39 18.17.E39] || <math qid="Q5780">\int_{0}^{1}\LegendrepolyP{2n+1}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{1-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}+\frac{1}{2}z}{n+1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\LegendrepolyP{2n+1}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{1-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}+\frac{1}{2}z}{n+1}}</syntaxhighlight> || <math>\realpart@@{z} > -1</math> || <syntaxhighlight lang=mathematica>int(LegendreP(2*n + 1, x)*(x)^((x + y*I)- 1), x = 0..1) = ((- 1)^(n)* pochhammer(1 -(1)/(2)*(x + y*I), n))/(2*pochhammer((1)/(2)+(1)/(2)*(x + y*I), n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[2*n + 1, x]*(x)^((x + y*I)- 1), {x, 0, 1}, GenerateConditions->None] == Divide[(- 1)^(n)* Pochhammer[1 -Divide[1,2]*(x + y*I), n],2*Pochhammer[Divide[1,2]+Divide[1,2]*(x + y*I), n + 1]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1141366199-.1434447856*I
Test Values: {x = 3/2, y = -3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1797435469+.6231194668e-1*I
Test Values: {x = 3/2, y = -3/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.058823529411764705, 0.0980392156862745], NIntegrate[Complex[6.21919624203139, -4.330049939446727]
Test Values: {1.5, 0, 1}]], {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.04824851288830139, -0.012998457810090328], NIntegrate[Complex[33.25149808949738, -23.151005642155518]
Test Values: {1.5, 0, 1}]], {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/18.17.E40 18.17.E40] || <math qid="Q5781">\int_{0}^{\infty}e^{-ax}\LaguerrepolyL[\alpha]{n}@{bx}x^{z-1}\diff{x} = \frac{\EulerGamma@{z+n}}{n!}\*{(a-b)^{n}}a^{-n-z}\*\genhyperF{2}{1}@@{-n,1+\alpha-z}{1-n-z}{\frac{a}{a-b}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-ax}\LaguerrepolyL[\alpha]{n}@{bx}x^{z-1}\diff{x} = \frac{\EulerGamma@{z+n}}{n!}\*{(a-b)^{n}}a^{-n-z}\*\genhyperF{2}{1}@@{-n,1+\alpha-z}{1-n-z}{\frac{a}{a-b}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{z} > 0, \realpart@@{(z+n)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- a*x)*LaguerreL(n, alpha, b*x)*(x)^((x + y*I)- 1), x = 0..infinity) = (GAMMA((x + y*I)+ n))/(factorial(n))*(a - b)^(n)*(a)^(- n -(x + y*I))* hypergeom([- n , 1 + alpha -(x + y*I)], [1 - n -(x + y*I)], (a)/(a - b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- a*x]*LaguerreL[n, \[Alpha], b*x]*(x)^((x + y*I)- 1), {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[(x + y*I)+ n],(n)!]*(a - b)^(n)*(a)^(- n -(x + y*I))* HypergeometricPFQ[{- n , 1 + \[Alpha]-(x + y*I)}, {1 - n -(x + y*I)}, Divide[a,a - b]]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
|- 
| [https://dlmf.nist.gov/18.17.E45 18.17.E45] || <math qid="Q5786">(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n+1}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n+1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(n +(1)/(2))*(1 + x)^((1)/(2))* int((x - t)^(-(1)/(2))* LegendreP(n, t), t = - 1..x) = ChebyshevT(n, x)+ ChebyshevT(n + 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n +Divide[1,2])*(1 + x)^(Divide[1,2])* Integrate[(x - t)^(-Divide[1,2])* LegendreP[n, t], {t, - 1, x}, GenerateConditions->None] == ChebyshevT[n, x]+ ChebyshevT[n + 1, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|- 
| [https://dlmf.nist.gov/18.17.E46 18.17.E46] || <math qid="Q5787">(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-x)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}-\ChebyshevpolyT{n+1}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-x)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}-\ChebyshevpolyT{n+1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(n +(1)/(2))*(1 - x)^((1)/(2))* int((t - x)^(-(1)/(2))* LegendreP(n, t), t = x..1) = ChebyshevT(n, x)- ChebyshevT(n + 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n +Divide[1,2])*(1 - x)^(Divide[1,2])* Integrate[(t - x)^(-Divide[1,2])* LegendreP[n, t], {t, x, 1}, GenerateConditions->None] == ChebyshevT[n, x]- ChebyshevT[n + 1, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| [https://dlmf.nist.gov/18.17.E47 18.17.E47] || <math qid="Q5788">\int_{0}^{x}t^{\alpha}\frac{\LaguerrepolyL[\alpha]{m}@{t}}{\LaguerrepolyL[\alpha]{m}@{0}}(x-t)^{\beta}\frac{\LaguerrepolyL[\beta]{n}@{x-t}}{\LaguerrepolyL[\beta]{n}@{0}}\diff{t} = \frac{\EulerGamma@{\alpha+1}\EulerGamma@{\beta+1}}{\EulerGamma@{\alpha+\beta+2}}x^{\alpha+\beta+1}\frac{\LaguerrepolyL[\alpha+\beta+1]{m+n}@{x}}{\LaguerrepolyL[\alpha+\beta+1]{m+n}@{0}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}t^{\alpha}\frac{\LaguerrepolyL[\alpha]{m}@{t}}{\LaguerrepolyL[\alpha]{m}@{0}}(x-t)^{\beta}\frac{\LaguerrepolyL[\beta]{n}@{x-t}}{\LaguerrepolyL[\beta]{n}@{0}}\diff{t} = \frac{\EulerGamma@{\alpha+1}\EulerGamma@{\beta+1}}{\EulerGamma@{\alpha+\beta+2}}x^{\alpha+\beta+1}\frac{\LaguerrepolyL[\alpha+\beta+1]{m+n}@{x}}{\LaguerrepolyL[\alpha+\beta+1]{m+n}@{0}}</syntaxhighlight> || <math>\realpart@@{(\alpha+1)} > 0, \realpart@@{(\beta+1)} > 0, \realpart@@{(\alpha+\beta+2)} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(alpha)*(LaguerreL(m, alpha, t))/(LaguerreL(m, alpha, 0))*(x - t)^(beta)*(LaguerreL(n, beta, x - t))/(LaguerreL(n, beta, 0)), t = 0..x) = (GAMMA(alpha + 1)*GAMMA(beta + 1))/(GAMMA(alpha + beta + 2))*(x)^(alpha + beta + 1)*(LaguerreL(m + n, alpha + beta + 1, x))/(LaguerreL(m + n, alpha + beta + 1, 0))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^\[Alpha]*Divide[LaguerreL[m, \[Alpha], t],LaguerreL[m, \[Alpha], 0]]*(x - t)^\[Beta]*Divide[LaguerreL[n, \[Beta], x - t],LaguerreL[n, \[Beta], 0]], {t, 0, x}, GenerateConditions->None] == Divide[Gamma[\[Alpha]+ 1]*Gamma[\[Beta]+ 1],Gamma[\[Alpha]+ \[Beta]+ 2]]*(x)^(\[Alpha]+ \[Beta]+ 1)*Divide[LaguerreL[m + n, \[Alpha]+ \[Beta]+ 1, x],LaguerreL[m + n, \[Alpha]+ \[Beta]+ 1, 0]]</syntaxhighlight> || Missing Macro Error || Failure || - || Manual Skip!
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| [https://dlmf.nist.gov/18.17.E48 18.17.E48] || <math qid="Q5789">\int_{-\infty}^{\infty}\HermitepolyH{m}@{y}e^{-y^{2}}\HermitepolyH{n}@{x-y}e^{-(x-y)^{2}}\diff{y} = \pi^{\frac{1}{2}}2^{-\frac{1}{2}(m+n+1)}\HermitepolyH{m+n}@{2^{-\frac{1}{2}}x}e^{-\frac{1}{2}x^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\HermitepolyH{m}@{y}e^{-y^{2}}\HermitepolyH{n}@{x-y}e^{-(x-y)^{2}}\diff{y} = \pi^{\frac{1}{2}}2^{-\frac{1}{2}(m+n+1)}\HermitepolyH{m+n}@{2^{-\frac{1}{2}}x}e^{-\frac{1}{2}x^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(HermiteH(m, y)*exp(- (y)^(2))*HermiteH(n, x - y)*exp(-(x - y)^(2)), y = - infinity..infinity) = (Pi)^((1)/(2))* (2)^(-(1)/(2)*(m + n + 1))* HermiteH(m + n, (2)^(-(1)/(2))* x)*exp(-(1)/(2)*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[HermiteH[m, y]*Exp[- (y)^(2)]*HermiteH[n, x - y]*Exp[-(x - y)^(2)], {y, - Infinity, Infinity}, GenerateConditions->None] == (Pi)^(Divide[1,2])* (2)^(-Divide[1,2]*(m + n + 1))* HermiteH[m + n, (2)^(-Divide[1,2])* x]*Exp[-Divide[1,2]*(x)^(2)]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 27] || Skipped - Because timed out
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| [https://dlmf.nist.gov/18.17.E49 18.17.E49] || <math qid="Q5790">\int_{-\infty}^{\infty}\HermitepolyH{\ell}@{x}\HermitepolyH{m}@{x}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \frac{2^{\frac{1}{2}(\ell+m+n)}\ell\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\HermitepolyH{\ell}@{x}\HermitepolyH{m}@{x}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \frac{2^{\frac{1}{2}(\ell+m+n)}\ell\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(HermiteH(ell, x)*HermiteH(m, x)*HermiteH(n, x)*exp(- (x)^(2)), x = - infinity..infinity) = ((2)^((1)/(2)*(ell + m + n))* factorial(ell)*factorial(m)*factorial(n)*sqrt(Pi))/(factorial((1)/(2)*ell +(1)/(2)*m -(1)/(2)*n)*factorial((1)/(2)*m +(1)/(2)*n -(1)/(2)*ell)*factorial((1)/(2)*n +(1)/(2)*ell -(1)/(2)*m))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[HermiteH[\[ScriptL], x]*HermiteH[m, x]*HermiteH[n, x]*Exp[- (x)^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == Divide[(2)^(Divide[1,2]*(\[ScriptL]+ m + n))* (\[ScriptL])!*(m)!*(n)!*Sqrt[Pi],(Divide[1,2]*\[ScriptL]+Divide[1,2]*m -Divide[1,2]*n)!*(Divide[1,2]*m +Divide[1,2]*n -Divide[1,2]*\[ScriptL])!*(Divide[1,2]*n +Divide[1,2]*\[ScriptL]-Divide[1,2]*m)!]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
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