View source for 14.17

{{DISPLAYTITLE:Legendre and Related Functions - 14.17 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/14.17.E1 14.17.E1] || <math qid="Q4882">{\int\left(1-x^{2}\right)^{-\mu/2}\FerrersP[\mu]{\nu}@{x}\diff{x}} = {-\left(1-x^{2}\right)^{-(\mu-1)/2}\FerrersP[\mu-1]{\nu}@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\int\left(1-x^{2}\right)^{-\mu/2}\FerrersP[\mu]{\nu}@{x}\diff{x}} = {-\left(1-x^{2}\right)^{-(\mu-1)/2}\FerrersP[\mu-1]{\nu}@{x}}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>int((1 - (x)^(2))^(- mu/2)* LegendreP(nu, mu, x), x) = -(1 - (x)^(2))^(-(mu - 1)/2)* LegendreP(nu, mu - 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - (x)^(2))^(- \[Mu]/2)* LegendreP[\[Nu], \[Mu], x], x, GenerateConditions->None] == -(1 - (x)^(2))^(-(\[Mu]- 1)/2)* LegendreP[\[Nu], \[Mu]- 1, x]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[3.8842606727900413, 5.104372500552582], Integrate[Complex[-4.747850387868644, -1.1425414738949808], 1.5, Rule[GenerateConditions, None]]]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[3.976584990156878, 2.3595388807039552], Integrate[Complex[-2.482845880898655, 4.683216982349827], 1.5, Rule[GenerateConditions, None]]]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.17.E2 14.17.E2] || <math qid="Q4883">\int\left(1-x^{2}\right)^{\mu/2}\FerrersP[\mu]{\nu}@{x}\diff{x} = \frac{\left(1-x^{2}\right)^{(\mu+1)/2}}{(\nu-\mu)(\nu+\mu+1)}\FerrersP[\mu+1]{\nu}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\left(1-x^{2}\right)^{\mu/2}\FerrersP[\mu]{\nu}@{x}\diff{x} = \frac{\left(1-x^{2}\right)^{(\mu+1)/2}}{(\nu-\mu)(\nu+\mu+1)}\FerrersP[\mu+1]{\nu}@{x}</syntaxhighlight> || <math>\mu \neq \nu, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>int((1 - (x)^(2))^(mu/2)* LegendreP(nu, mu, x), x) = ((1 - (x)^(2))^((mu + 1)/2))/((nu - mu)*(nu + mu + 1))*LegendreP(nu, mu + 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - (x)^(2))^(\[Mu]/2)* LegendreP[\[Nu], \[Mu], x], x, GenerateConditions->None] == Divide[(1 - (x)^(2))^((\[Mu]+ 1)/2),(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)]*LegendreP[\[Nu], \[Mu]+ 1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [270 / 270]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.5646480599960819, 1.3746025553854266], Integrate[Complex[0.23690790481776922, -1.3156471186304795], 1.5, Rule[GenerateConditions, None]]]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.228607897264037, 1.5189132046928975], Integrate[Complex[0.8670522613344679, -2.293703747689092], 1.5, Rule[GenerateConditions, None]]]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.17.E3 14.17.E3] || <math qid="Q4884">\int x\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}\diff{x} = \frac{1}{2\nu(\nu+1)}\left((\mu^{2}-(\nu+1)(\nu+x^{2}))\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}+(\nu+1)(\nu-\mu+1)x(\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x}+\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x})-(\nu-\mu+1)^{2}\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu+1}@{x}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int x\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}\diff{x} = \frac{1}{2\nu(\nu+1)}\left((\mu^{2}-(\nu+1)(\nu+x^{2}))\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}+(\nu+1)(\nu-\mu+1)x(\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x}+\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x})-(\nu-\mu+1)^{2}\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu+1}@{x}\right)</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1, \realpart@@{(\nu+\mu+1)} > 0, \realpart@@{((\nu+1)+\mu+1)} > 0, \realpart@@{(\nu-\mu+1)} > 0, \realpart@@{((\nu+1)-\mu+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(x*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x), x) = (1)/(2*nu*(nu + 1))*(((mu)^(2)-(nu + 1)*(nu + (x)^(2)))*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x)+(nu + 1)*(nu - mu + 1)*x*(LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x)+ LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x))-(nu - mu + 1)^(2)* LegendreP(nu + 1, mu, x)*LegendreQ(nu + 1, mu, x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[x*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x], x, GenerateConditions->None] == Divide[1,2*\[Nu]*(\[Nu]+ 1)]*((\[Mu]^(2)-(\[Nu]+ 1)*(\[Nu]+ (x)^(2)))*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x]+(\[Nu]+ 1)*(\[Nu]- \[Mu]+ 1)*x*(LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x]+ LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu], \[Mu], x])-(\[Nu]- \[Mu]+ 1)^(2)* LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x])</syntaxhighlight> || Error || Aborted || - || Skip - No test values generated
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| [https://dlmf.nist.gov/14.17.E4 14.17.E4] || <math qid="Q4885">\int\frac{x}{\left(1-x^{2}\right)^{3/2}}\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}\diff{x} = \frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{1/2}}\left((1-2\mu^{2}+2\nu(\nu+1))\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}+(2\nu+1)(\mu-\nu-1)x(\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x}+\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x})+2(\mu-\nu-1)^{2}\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu+1}@{x}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\frac{x}{\left(1-x^{2}\right)^{3/2}}\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}\diff{x} = \frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{1/2}}\left((1-2\mu^{2}+2\nu(\nu+1))\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu}@{x}+(2\nu+1)(\mu-\nu-1)x(\FerrersP[\mu]{\nu}@{x}\FerrersQ[\mu]{\nu+1}@{x}+\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu}@{x})+2(\mu-\nu-1)^{2}\FerrersP[\mu]{\nu+1}@{x}\FerrersQ[\mu]{\nu+1}@{x}\right)</syntaxhighlight> || <math>\mu \neq +\frac{1}{2}, \mu \neq -\frac{1}{2}, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1, \realpart@@{(\nu+\mu+1)} > 0, \realpart@@{((\nu+1)+\mu+1)} > 0, \realpart@@{(\nu-\mu+1)} > 0, \realpart@@{((\nu+1)-\mu+1)} > 0</math> || <syntaxhighlight lang=mathematica>int((x)/((1 - (x)^(2))^(3/2))*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x), x) = (1)/((1 - 4*(mu)^(2))*(1 - (x)^(2))^(1/2))*((1 - 2*(mu)^(2)+ 2*nu*(nu + 1))*LegendreP(nu, mu, x)*LegendreQ(nu, mu, x)+(2*nu + 1)*(mu - nu - 1)*x*(LegendreP(nu, mu, x)*LegendreQ(nu + 1, mu, x)+ LegendreP(nu + 1, mu, x)*LegendreQ(nu, mu, x))+ 2*(mu - nu - 1)^(2)* LegendreP(nu + 1, mu, x)*LegendreQ(nu + 1, mu, x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[x,(1 - (x)^(2))^(3/2)]*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x], x, GenerateConditions->None] == Divide[1,(1 - 4*\[Mu]^(2))*(1 - (x)^(2))^(1/2)]*((1 - 2*\[Mu]^(2)+ 2*\[Nu]*(\[Nu]+ 1))*LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu], \[Mu], x]+(2*\[Nu]+ 1)*(\[Mu]- \[Nu]- 1)*x*(LegendreP[\[Nu], \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x]+ LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu], \[Mu], x])+ 2*(\[Mu]- \[Nu]- 1)^(2)* LegendreP[\[Nu]+ 1, \[Mu], x]*LegendreQ[\[Nu]+ 1, \[Mu], x])</syntaxhighlight> || Failure || Aborted || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [99 / 99]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-15.417707085194902, 19.940158970813897], Integrate[Complex[-9.988309927179525, -1.2041271824131927], 1.5, Rule[GenerateConditions, None]]]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[17.198725078389664, -1.5826141510664629], Integrate[Complex[20.92420958974465, 36.064324396521705], 1.5, Rule[GenerateConditions, None]]]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.17.E5 14.17.E5] || <math qid="Q4886">\int_{0}^{1}x^{\sigma}\left(1-x^{2}\right)^{\mu/2}\FerrersP[-\mu]{\nu}@{x}\diff{x} = \frac{\EulerGamma@{\frac{1}{2}\sigma+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\sigma+1}}{2^{\mu+1}\EulerGamma@{\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}x^{\sigma}\left(1-x^{2}\right)^{\mu/2}\FerrersP[-\mu]{\nu}@{x}\diff{x} = \frac{\EulerGamma@{\frac{1}{2}\sigma+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\sigma+1}}{2^{\mu+1}\EulerGamma@{\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1}\EulerGamma@{\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}}}</syntaxhighlight> || <math>\realpart@@{\sigma} > -1, \realpart@@{\mu} > -1, \realpart@@{(\frac{1}{2}\sigma+\frac{1}{2})} > 0, \realpart@@{(\frac{1}{2}\sigma+1)} > 0, \realpart@@{(\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1)} > 0, \realpart@@{(\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2})} > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>int((x)^(sigma)*(1 - (x)^(2))^(mu/2)* LegendreP(nu, - mu, x), x = 0..1) = (GAMMA((1)/(2)*sigma +(1)/(2))*GAMMA((1)/(2)*sigma + 1))/((2)^(mu + 1)* GAMMA((1)/(2)*sigma -(1)/(2)*nu +(1)/(2)*mu + 1)*GAMMA((1)/(2)*sigma +(1)/(2)*nu +(1)/(2)*mu +(3)/(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^\[Sigma]*(1 - (x)^(2))^(\[Mu]/2)* LegendreP[\[Nu], - \[Mu], x], {x, 0, 1}, GenerateConditions->None] == Divide[Gamma[Divide[1,2]*\[Sigma]+Divide[1,2]]*Gamma[Divide[1,2]*\[Sigma]+ 1],(2)^(\[Mu]+ 1)* Gamma[Divide[1,2]*\[Sigma]-Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]*Gamma[Divide[1,2]*\[Sigma]+Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[3,2]]]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E6 14.17.E6] || <math qid="Q4887">\int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersP[m]{n}@{x}\diff{x} = \frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}\Kroneckerdelta{l}{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersP[m]{n}@{x}\diff{x} = \frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}\Kroneckerdelta{l}{n}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>int(LegendreP(l, m, x)*LegendreP(n, m, x), x = - 1..1) = (factorial(n + m))/(factorial(n - m)*(n +(1)/(2)))*KroneckerDelta[l, n]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[l, m, x]*LegendreP[n, m, x], {x, - 1, 1}, GenerateConditions->None] == Divide[(n + m)!,(n - m)!*(n +Divide[1,2])]*KroneckerDelta[l, n]</syntaxhighlight> || Aborted || Failure || Successful [Tested: 27] || Successful [Tested: 27]
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| [https://dlmf.nist.gov/14.17.E7 14.17.E7] || <math qid="Q4888">\int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersP[-m]{n}@{x}\diff{x} = \frac{(-1)^{m}}{l+\frac{1}{2}}\Kroneckerdelta{l}{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersP[-m]{n}@{x}\diff{x} = \frac{(-1)^{m}}{l+\frac{1}{2}}\Kroneckerdelta{l}{n}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>int(LegendreP(l, m, x)*LegendreP(n, - m, x), x = - 1..1) = ((- 1)^(m))/(l +(1)/(2))*KroneckerDelta[l, n]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[l, m, x]*LegendreP[n, - m, x], {x, - 1, 1}, GenerateConditions->None] == Divide[(- 1)^(m),l +Divide[1,2]]*KroneckerDelta[l, n]</syntaxhighlight> || Aborted || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6666666667
Test Values: {l = 1, m = 2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .6666666667
Test Values: {l = 1, m = 3, n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -0.6666666666666666
Test Values: {Rule[l, 1], Rule[m, 2], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0.6666666666666666
Test Values: {Rule[l, 1], Rule[m, 3], Rule[n, 1]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.17.E8 14.17.E8] || <math qid="Q4889">\int_{-1}^{1}\frac{\FerrersP[l]{n}@{x}\FerrersP[m]{n}@{x}}{1-x^{2}}\diff{x} = \frac{(n+m)!}{(n-m)!m}\Kroneckerdelta{l}{m}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\frac{\FerrersP[l]{n}@{x}\FerrersP[m]{n}@{x}}{1-x^{2}}\diff{x} = \frac{(n+m)!}{(n-m)!m}\Kroneckerdelta{l}{m}</syntaxhighlight> || <math>m > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>int((LegendreP(n, l, x)*LegendreP(n, m, x))/(1 - (x)^(2)), x = - 1..1) = (factorial(n + m))/(factorial(n - m)*m)*KroneckerDelta[l, m]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[LegendreP[n, l, x]*LegendreP[n, m, x],1 - (x)^(2)], {x, - 1, 1}, GenerateConditions->None] == Divide[(n + m)!,(n - m)!*m]*KroneckerDelta[l, m]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Successful [Tested: 27]
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| [https://dlmf.nist.gov/14.17.E9 14.17.E9] || <math qid="Q4890">\int_{-1}^{1}\frac{\FerrersP[l]{n}@{x}\FerrersP[-m]{n}@{x}}{1-x^{2}}\diff{x} = \frac{(-1)^{l}}{l}\Kroneckerdelta{l}{m}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\frac{\FerrersP[l]{n}@{x}\FerrersP[-m]{n}@{x}}{1-x^{2}}\diff{x} = \frac{(-1)^{l}}{l}\Kroneckerdelta{l}{m}</syntaxhighlight> || <math>l > 0, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>int((LegendreP(n, l, x)*LegendreP(n, - m, x))/(1 - (x)^(2)), x = - 1..1) = ((- 1)^(l))/(l)*KroneckerDelta[l, m]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[LegendreP[n, l, x]*LegendreP[n, - m, x],1 - (x)^(2)], {x, - 1, 1}, GenerateConditions->None] == Divide[(- 1)^(l),l]*KroneckerDelta[l, m]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E10 14.17.E10] || <math qid="Q4891">\int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersP[]{\lambda}@{x}\diff{x} = \frac{2\left(2\sin@{\nu\pi}\sin@{\lambda\pi}\left(\digamma@{\nu+1}-\digamma@{\lambda+1}\right)+\pi\sin@{(\lambda-\nu)\pi}\right)}{\pi^{2}(\lambda-\nu)(\lambda+\nu+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersP[]{\lambda}@{x}\diff{x} = \frac{2\left(2\sin@{\nu\pi}\sin@{\lambda\pi}\left(\digamma@{\nu+1}-\digamma@{\lambda+1}\right)+\pi\sin@{(\lambda-\nu)\pi}\right)}{\pi^{2}(\lambda-\nu)(\lambda+\nu+1)}</syntaxhighlight> || <math>\lambda \neq \nu</math> || <syntaxhighlight lang=mathematica>int(LegendreP(nu, x)*LegendreP(lambda, x), x = - 1..1) = (2*(2*sin(nu*Pi)*sin(lambda*Pi)*(Psi(nu + 1)- Psi(lambda + 1))+ Pi*sin((lambda - nu)*Pi)))/((Pi)^(2)*(lambda - nu)*(lambda + nu + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[\[Nu], x]*LegendreP[\[Lambda], x], {x, - 1, 1}, GenerateConditions->None] == Divide[2*(2*Sin[\[Nu]*Pi]*Sin[\[Lambda]*Pi]*(PolyGamma[\[Nu]+ 1]- PolyGamma[\[Lambda]+ 1])+ Pi*Sin[(\[Lambda]- \[Nu])*Pi]),(Pi)^(2)*(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E11 14.17.E11] || <math qid="Q4892">\int_{-1}^{1}\left(\FerrersP[]{\nu}@{x}\right)^{2}\diff{x} = \frac{\pi^{2}-2\sin^{2}@{\nu\pi}\digamma'@{\nu+1}}{\pi^{2}\left(\nu+\frac{1}{2}\right)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\left(\FerrersP[]{\nu}@{x}\right)^{2}\diff{x} = \frac{\pi^{2}-2\sin^{2}@{\nu\pi}\digamma'@{\nu+1}}{\pi^{2}\left(\nu+\frac{1}{2}\right)}</syntaxhighlight> || <math>\nu \neq -\frac{1}{2}</math> || <syntaxhighlight lang=mathematica>int((LegendreP(nu, x))^(2), x = - 1..1) = ((Pi)^(2)- 2*(sin(nu*Pi))^(2)* subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/((Pi)^(2)*(nu +(1)/(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(LegendreP[\[Nu], x])^(2), {x, - 1, 1}, GenerateConditions->None] == Divide[(Pi)^(2)- 2*(Sin[\[Nu]*Pi])^(2)* (D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),(Pi)^(2)*(\[Nu]+Divide[1,2])]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {nu = -2}</syntaxhighlight><br></div></div> || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E12 14.17.E12] || <math qid="Q4893">\int_{-1}^{1}\FerrersQ[]{\nu}@{x}\FerrersQ[]{\lambda}@{x}\diff{x} = \frac{\left((\digamma@{\nu+1}-\digamma@{\lambda+1})(1+\cos@{\nu\pi}\cos@{\lambda\pi})+\frac{1}{2}\pi\sin@{(\lambda-\nu)\pi}\right)}{(\lambda-\nu)(\lambda+\nu+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\FerrersQ[]{\nu}@{x}\FerrersQ[]{\lambda}@{x}\diff{x} = \frac{\left((\digamma@{\nu+1}-\digamma@{\lambda+1})(1+\cos@{\nu\pi}\cos@{\lambda\pi})+\frac{1}{2}\pi\sin@{(\lambda-\nu)\pi}\right)}{(\lambda-\nu)(\lambda+\nu+1)}</syntaxhighlight> || <math>\lambda \neq \nu</math> || <syntaxhighlight lang=mathematica>int(LegendreQ(nu, x)*LegendreQ(lambda, x), x = - 1..1) = ((Psi(nu + 1)- Psi(lambda + 1))*(1 + cos(nu*Pi)*cos(lambda*Pi))+(1)/(2)*Pi*sin((lambda - nu)*Pi))/((lambda - nu)*(lambda + nu + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreQ[\[Nu], x]*LegendreQ[\[Lambda], x], {x, - 1, 1}, GenerateConditions->None] == Divide[(PolyGamma[\[Nu]+ 1]- PolyGamma[\[Lambda]+ 1])*(1 + Cos[\[Nu]*Pi]*Cos[\[Lambda]*Pi])+Divide[1,2]*Pi*Sin[(\[Lambda]- \[Nu])*Pi],(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]</syntaxhighlight> || Aborted || Failure || Manual Skip! || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E13 14.17.E13] || <math qid="Q4894">\int_{-1}^{1}\left(\FerrersQ[]{\nu}@{x}\right)^{2}\diff{x} = \frac{\pi^{2}-2\left(1+\cos^{2}@{\nu\pi}\right)\digamma'@{\nu+1}}{2(2\nu+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\left(\FerrersQ[]{\nu}@{x}\right)^{2}\diff{x} = \frac{\pi^{2}-2\left(1+\cos^{2}@{\nu\pi}\right)\digamma'@{\nu+1}}{2(2\nu+1)}</syntaxhighlight> || <math>\nu \neq -\frac{1}{2}</math> || <syntaxhighlight lang=mathematica>int((LegendreQ(nu, x))^(2), x = - 1..1) = ((Pi)^(2)- 2*(1 + (cos(nu*Pi))^(2))*subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/(2*(2*nu + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(LegendreQ[\[Nu], x])^(2), {x, - 1, 1}, GenerateConditions->None] == Divide[(Pi)^(2)- 2*(1 + (Cos[\[Nu]*Pi])^(2))*(D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),2*(2*\[Nu]+ 1)]</syntaxhighlight> || Aborted || Failure || Manual Skip! || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E14 14.17.E14] || <math qid="Q4895">\int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersQ[]{\lambda}@{x}\diff{x} = \frac{2\sin@{\nu\pi}\cos@{\lambda\pi}\left(\digamma@{\nu+1}-\digamma@{\lambda+1}\right)+\pi\cos@{(\lambda-\nu)\pi}-\pi}{\pi(\lambda-\nu)(\lambda+\nu+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersQ[]{\lambda}@{x}\diff{x} = \frac{2\sin@{\nu\pi}\cos@{\lambda\pi}\left(\digamma@{\nu+1}-\digamma@{\lambda+1}\right)+\pi\cos@{(\lambda-\nu)\pi}-\pi}{\pi(\lambda-\nu)(\lambda+\nu+1)}</syntaxhighlight> || <math>\realpart@@{\lambda} > 0, \realpart@@{\nu} > 0, \lambda \neq \nu</math> || <syntaxhighlight lang=mathematica>int(LegendreP(nu, x)*LegendreQ(lambda, x), x = - 1..1) = (2*sin(nu*Pi)*cos(lambda*Pi)*(Psi(nu + 1)- Psi(lambda + 1))+ Pi*cos((lambda - nu)*Pi)- Pi)/(Pi*(lambda - nu)*(lambda + nu + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[\[Nu], x]*LegendreQ[\[Lambda], x], {x, - 1, 1}, GenerateConditions->None] == Divide[2*Sin[\[Nu]*Pi]*Cos[\[Lambda]*Pi]*(PolyGamma[\[Nu]+ 1]- PolyGamma[\[Lambda]+ 1])+ Pi*Cos[(\[Lambda]- \[Nu])*Pi]- Pi,Pi*(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E15 14.17.E15] || <math qid="Q4896">\int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersQ[]{\nu}@{x}\diff{x} = -\frac{\sin@{2\nu\pi}\digamma'@{\nu+1}}{\pi(2\nu+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\FerrersP[]{\nu}@{x}\FerrersQ[]{\nu}@{x}\diff{x} = -\frac{\sin@{2\nu\pi}\digamma'@{\nu+1}}{\pi(2\nu+1)}</syntaxhighlight> || <math>\realpart@@{\nu} > 0</math> || <syntaxhighlight lang=mathematica>int(LegendreP(nu, x)*LegendreQ(nu, x), x = - 1..1) = -(sin(2*nu*Pi)*subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/(Pi*(2*nu + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[\[Nu], x]*LegendreQ[\[Nu], x], {x, - 1, 1}, GenerateConditions->None] == -Divide[Sin[2*\[Nu]*Pi]*(D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1),Pi*(2*\[Nu]+ 1)]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E16 14.17.E16] || <math qid="Q4897">\int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersQ[m]{n}@{x}\diff{x} = \frac{\left(1-(-1)^{l+n}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-1}^{1}\FerrersP[m]{l}@{x}\FerrersQ[m]{n}@{x}\diff{x} = \frac{\left(1-(-1)^{l+n}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!}</syntaxhighlight> || <math>l \neq n, |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1, \realpart@@{(n+\mu+1)} > 0, \realpart@@{(\nu+m+1)} > 0, \realpart@@{(n-\mu+1)} > 0, \realpart@@{(\nu-m+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(LegendreP(l, m, x)*LegendreQ(n, m, x), x = - 1..1) = ((1 -(- 1)^(l + n))*factorial(l + m))/((l - n)*(l + n + 1)*factorial(l - m))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[l, m, x]*LegendreQ[n, m, x], {x, - 1, 1}, GenerateConditions->None] == Divide[(1 -(- 1)^(l + n))*(l + m)!,(l - n)*(l + n + 1)*(l - m)!]</syntaxhighlight> || Aborted || Failure || Error || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E17 14.17.E17] || <math qid="Q4898">\int_{0}^{\pi}\FerrersQ[]{l}@{\cos@@{\theta}}\FerrersP[]{m}@{\cos@@{\theta}}\FerrersP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}\FerrersQ[]{l}@{\cos@@{\theta}}\FerrersP[]{m}@{\cos@@{\theta}}\FerrersP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta} = 0</syntaxhighlight> || <math>|m-n| < l, l < m+n</math> || <syntaxhighlight lang=mathematica>int(LegendreQ(l, cos(theta))*LegendreP(m, cos(theta))*LegendreP(n, cos(theta))*sin(theta), theta = 0..Pi) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreQ[l, Cos[\[Theta]]]*LegendreP[m, Cos[\[Theta]]]*LegendreP[n, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None] == 0</syntaxhighlight> || Aborted || Aborted || Error || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E18 14.17.E18] || <math qid="Q4899">\int_{1}^{\infty}\assLegendreP[]{\nu}@{x}\assLegendreQ[]{\lambda}@{x}\diff{x} = \frac{1}{(\lambda-\nu)(\nu+\lambda+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{1}^{\infty}\assLegendreP[]{\nu}@{x}\assLegendreQ[]{\lambda}@{x}\diff{x} = \frac{1}{(\lambda-\nu)(\nu+\lambda+1)}</syntaxhighlight> || <math>\realpart@@{\lambda} > \realpart@@{\nu}, \realpart@@{\nu} > 0</math> || <syntaxhighlight lang=mathematica>int(LegendreP(nu, x)*LegendreQ(lambda, x), x = 1..infinity) = (1)/((lambda - nu)*(nu + lambda + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreP[\[Nu], 0, 3, x]*LegendreQ[\[Lambda], 0, 3, x], {x, 1, Infinity}, GenerateConditions->None] == Divide[1,(\[Lambda]- \[Nu])*(\[Nu]+ \[Lambda]+ 1)]</syntaxhighlight> || Error || Failure || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E19 14.17.E19] || <math qid="Q4900">\int_{1}^{\infty}\assLegendreQ[]{\nu}@{x}\assLegendreQ[]{\lambda}@{x}\diff{x} = \frac{\digamma@{\lambda+1}-\digamma@{\nu+1}}{(\lambda-\nu)(\lambda+\nu+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{1}^{\infty}\assLegendreQ[]{\nu}@{x}\assLegendreQ[]{\lambda}@{x}\diff{x} = \frac{\digamma@{\lambda+1}-\digamma@{\nu+1}}{(\lambda-\nu)(\lambda+\nu+1)}</syntaxhighlight> || <math>\realpart@{\lambda+\nu} > -1, \lambda \neq \nu</math> || <syntaxhighlight lang=mathematica>int(LegendreQ(nu, x)*LegendreQ(lambda, x), x = 1..infinity) = (Psi(lambda + 1)- Psi(nu + 1))/((lambda - nu)*(lambda + nu + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[LegendreQ[\[Nu], 0, 3, x]*LegendreQ[\[Lambda], 0, 3, x], {x, 1, Infinity}, GenerateConditions->None] == Divide[PolyGamma[\[Lambda]+ 1]- PolyGamma[\[Nu]+ 1],(\[Lambda]- \[Nu])*(\[Lambda]+ \[Nu]+ 1)]</syntaxhighlight> || Aborted || Failure || Manual Skip! || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.17.E20 14.17.E20] || <math qid="Q4901">\int_{1}^{\infty}(\assLegendreQ[]{\nu}@{x})^{2}\diff{x} = \frac{\digamma'@{\nu+1}}{2\nu+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{1}^{\infty}(\assLegendreQ[]{\nu}@{x})^{2}\diff{x} = \frac{\digamma'@{\nu+1}}{2\nu+1}</syntaxhighlight> || <math>\realpart@@{\nu} > -\tfrac{1}{2}</math> || <syntaxhighlight lang=mathematica>int((LegendreQ(nu, x))^(2), x = 1..infinity) = (subs( temp=nu + 1, diff( Psi(temp), temp$(1) ) ))/(2*nu + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(LegendreQ[\[Nu], 0, 3, x])^(2), {x, 1, Infinity}, GenerateConditions->None] == Divide[D[PolyGamma[temp], {temp, 1}]/.temp-> \[Nu]+ 1,2*\[Nu]+ 1]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 5]
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