Bessel Functions - 10.43 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
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Maple 		Numeric
Mathematica
10.43.E4 ∫ 0 x I 0 ⁡ ( t ) - 1 t ⁢ d t = 1 2 ⁢ ∑ k = 1 ∞ ( - 1 ) k - 1 ⁢ ψ ⁡ ( k + 1 ) - ψ ⁡ ( 1 ) k ! ⁢ ( 1 2 ⁢ x ) k ⁢ I k ⁡ ( x ) superscript subscript 0 𝑥 modified-Bessel-first-kind 0 𝑡 1 𝑡 𝑡 1 2 superscript subscript 𝑘 1 superscript 1 𝑘 1 digamma 𝑘 1 digamma 1 𝑘 superscript 1 2 𝑥 𝑘 modified-Bessel-first-kind 𝑘 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\mathrm% {d}t=\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi% \left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)}}
\int_{0}^{x}\frac{\modBesselI{0}@{t}-1}{t}\diff{t} = \frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\modBesselI{k}@{x}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}} 
int((BesselI(0, t)- 1)/(t), t = 0..x) = (1)/(2)*sum((- 1)^(k - 1)*(Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselI(k, x), k = 1..infinity)

Integrate[Divide[BesselI[0, t]- 1,t], {t, 0, x}, GenerateConditions->None] == Divide[1,2]*Sum[(- 1)^(k - 1)*Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselI[k, x], {k, 1, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 3]
Failed [3 / 3]
Result: Plus[DirectedInfinity[-1], Times[-0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.75, k], BesselI[k, 1.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5]}


Result: Plus[DirectedInfinity[-1], Times[-0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.25, k], BesselI[k, 0.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5]}

... skip entries to safe data
10.43.E4 1 2 ⁢ ∑ k = 1 ∞ ( - 1 ) k - 1 ⁢ ψ ⁡ ( k + 1 ) - ψ ⁡ ( 1 ) k ! ⁢ ( 1 2 ⁢ x ) k ⁢ I k ⁡ ( x ) = 2 x ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ ( 2 ⁢ k + 3 ) ⁢ ( ψ ⁡ ( k + 2 ) - ψ ⁡ ( 1 ) ) ⁢ I 2 ⁢ k + 3 ⁡ ( x ) 1 2 superscript subscript 𝑘 1 superscript 1 𝑘 1 digamma 𝑘 1 digamma 1 𝑘 superscript 1 2 𝑥 𝑘 modified-Bessel-first-kind 𝑘 𝑥 2 𝑥 superscript subscript 𝑘 0 superscript 1 𝑘 2 𝑘 3 digamma 𝑘 2 digamma 1 modified-Bessel-first-kind 2 𝑘 3 𝑥 {\displaystyle{\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi% \left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)% =\frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(\psi\left(k+2\right)-\psi\left(1% \right))I_{2k+3}\left(x\right)}}
\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\digamma@{k+1}-\digamma@{1}}{k!}(\tfrac{1}{2}x)^{k}\modBesselI{k}@{x} = \frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(\digamma@{k+2}-\digamma@{1})\modBesselI{2k+3}@{x}		 ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 , ℜ ⁡ ( ( 2 ⁢ k + 3 ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 𝑘 𝑘 1 0 2 𝑘 3 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0,\Re((2k+3)+k+1)>0}} 
(1)/(2)*sum((- 1)^(k - 1)*(Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselI(k, x), k = 1..infinity) = (2)/(x)*sum((- 1)^(k)*(2*k + 3)*(Psi(k + 2)- Psi(1))*BesselI(2*k + 3, x), k = 0..infinity)

Divide[1,2]*Sum[(- 1)^(k - 1)*Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselI[k, x], {k, 1, Infinity}, GenerateConditions->None] == Divide[2,x]*Sum[(- 1)^(k)*(2*k + 3)*(PolyGamma[k + 2]- PolyGamma[1])*BesselI[2*k + 3, x], {k, 0, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 3]
Failed [3 / 3]
Result: Plus[Times[0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.75, k], BesselI[k, 1.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-1.3333333333333333, NSum[Times[Power[-1, k], Plus[3, Times[2, k]], BesselI[Plus[3, Times[2, k]], 1.5], Plus[EulerGamma, PolyGamma[0, Plus[2, k]]]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5]}


Result: Plus[Times[0.5, NSum[Times[Power[-1, Plus[-1, k]], Power[0.25, k], BesselI[k, 0.5], Power[Factorial[k], -1], Plus[EulerGamma, PolyGamma[0, Plus[1, k]]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[-4.0, NSum[Times[Power[-1, k], Plus[3, Times[2, k]], BesselI[Plus[3, Times[2, k]], 0.5], Plus[EulerGamma, PolyGamma[0, Plus[2, k]]]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 0.5]}

... skip entries to safe data
10.43.E5 ∫ x ∞ K 0 ⁡ ( t ) t ⁢ d t = 1 2 ⁢ ( ln ⁡ ( 1 2 ⁢ x ) + γ ) 2 + π 2 24 - ∑ k = 1 ∞ ( ψ ⁡ ( k + 1 ) + 1 2 ⁢ k - ln ⁡ ( 1 2 ⁢ x ) ) ⁢ ( 1 2 ⁢ x ) 2 ⁢ k 2 ⁢ k ⁢ ( k ! ) 2 superscript subscript 𝑥 modified-Bessel-second-kind 0 𝑡 𝑡 𝑡 1 2 superscript 1 2 𝑥 2 superscript 𝜋 2 24 superscript subscript 𝑘 1 digamma 𝑘 1 1 2 𝑘 1 2 𝑥 superscript 1 2 𝑥 2 𝑘 2 𝑘 superscript 𝑘 2 {\displaystyle{\displaystyle\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}% \mathrm{d}t=\frac{1}{2}\left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+% \frac{\pi^{2}}{24}-\sum_{k=1}^{\infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-% \ln\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}}}}
\int_{x}^{\infty}\frac{\modBesselK{0}@{t}}{t}\diff{t} = \frac{1}{2}\left(\ln@{\tfrac{1}{2}x}+\EulerConstant\right)^{2}+\frac{\pi^{2}}{24}-\sum_{k=1}^{\infty}\left(\digamma@{k+1}+\frac{1}{2k}-\ln@{\tfrac{1}{2}x}\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}}		 

int((BesselK(0, t))/(t), t = x..infinity) = (1)/(2)*(ln((1)/(2)*x)+ gamma)^(2)+((Pi)^(2))/(24)- sum((Psi(k + 1)+(1)/(2*k)- ln((1)/(2)*x))*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity)

Integrate[Divide[BesselK[0, t],t], {t, x, Infinity}, GenerateConditions->None] == Divide[1,2]*(Log[Divide[1,2]*x]+ EulerGamma)^(2)+Divide[(Pi)^(2),24]- Sum[(PolyGamma[k + 1]+Divide[1,2*k]- Log[Divide[1,2]*x])*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}, GenerateConditions->None]
Failure Aborted Successful [Tested: 3] Skipped - Because timed out
10.43.E6 ∫ 0 x e - t ⁢ I n ⁡ ( t ) ⁢ d t = x ⁢ e - x ⁢ ( I 0 ⁡ ( x ) + I 1 ⁡ ( x ) ) + n ⁢ ( e - x ⁢ I 0 ⁡ ( x ) - 1 ) + 2 ⁢ e - x ⁢ ∑ k = 1 n - 1 ( n - k ) ⁢ I k ⁡ ( x ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 modified-Bessel-first-kind 𝑛 𝑡 𝑡 𝑥 superscript 𝑒 𝑥 modified-Bessel-first-kind 0 𝑥 modified-Bessel-first-kind 1 𝑥 𝑛 superscript 𝑒 𝑥 modified-Bessel-first-kind 0 𝑥 1 2 superscript 𝑒 𝑥 superscript subscript 𝑘 1 𝑛 1 𝑛 𝑘 modified-Bessel-first-kind 𝑘 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}I_{n}\left(t\right)\mathrm{d}t=% xe^{-x}(I_{0}\left(x\right)+I_{1}\left(x\right))+n(e^{-x}I_{0}\left(x\right)-1% )+2e^{-x}\sum_{k=1}^{n-1}(n-k)I_{k}\left(x\right)}}
\int_{0}^{x}e^{-t}\modBesselI{n}@{t}\diff{t} = xe^{-x}(\modBesselI{0}@{x}+\modBesselI{1}@{x})+n(e^{-x}\modBesselI{0}@{x}-1)+2e^{-x}\sum_{k=1}^{n-1}(n-k)\modBesselI{k}@{x}		 ℜ ⁡ ( n + k + 1 ) > 0 , ℜ ⁡ ( 0 + k + 1 ) > 0 , ℜ ⁡ ( 1 + k + 1 ) > 0 , ℜ ⁡ ( k + k + 1 ) > 0 formulae-sequence 𝑛 𝑘 1 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 1 𝑘 1 0 𝑘 𝑘 1 0 {\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(0+k+1)>0,\Re(1+k+1)>0,\Re(k+k+1)>% 0}} 
int(exp(- t)*BesselI(n, t), t = 0..x) = x*exp(- x)*(BesselI(0, x)+ BesselI(1, x))+ n*(exp(- x)*BesselI(0, x)- 1)+ 2*exp(- x)*sum((n - k)*BesselI(k, x), k = 1..n - 1)

Integrate[Exp[- t]*BesselI[n, t], {t, 0, x}, GenerateConditions->None] == x*Exp[- x]*(BesselI[0, x]+ BesselI[1, x])+ n*(Exp[- x]*BesselI[0, x]- 1)+ 2*Exp[- x]*Sum[(n - k)*BesselI[k, x], {k, 1, n - 1}, GenerateConditions->None]
Failure Aborted Successful [Tested: 3]
Failed [2 / 3]
Result: Plus[1.0269197346695518, Times[-0.44626032029685964, Plus[-4.940169569318671, Times[3.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[1.5, []], Times[Plus[-2, Times[-2, ], Times[-1, 1.5]], [Plus[1, ]]], Times[Plus[2, Times[2, ], Times[-1, 1.5]], [Plus[2, ]]], Times[1.5, [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], BesselI[0, 1.5]], Equal[[2], Plus[BesselI[0, 1.5], BesselI[1, 1.5]]]}]][3.0]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], 1.5, []], Times[-1, Plus[2, ], Plus[Times[2, ], 1.5], [Plus[1, ]]], Times[, Plus[4, Times[2, ], Times[-1, 1.5]], [Plus[2, ]]], Times[, 1.5, [Plus[3, ]]]], 0], Equal[[1], 0], Equal[[2], BesselI[1, 1.5]], Equal[[3], Plus[Times[2, Power[1.5, -1], Plus[Times[1.5, BesselI[0, 1.5]], Times[-2, BesselI[1, 1.5]]]], BesselI[1, 1.5]]]}]][3.0]]]]], {Rule[n, 3], Rule[x, 1.5]}


Result: Plus[0.6643873281588137, Times[-1.2130613194252668, Plus[-3.19045011222397, Times[3.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[0.5, []], Times[Plus[-2, Times[-2, ], Times[-1, 0.5]], [Plus[1, ]]], Times[Plus[2, Times[2, ], Times[-1, 0.5]], [Plus[2, ]]], Times[0.5, [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], BesselI[0, 0.5]], Equal[[2], Plus[BesselI[0, 0.5], BesselI[1, 0.5]]]}]][3.0]], Times[-1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[2, ], 0.5, []], Times[-1, Plus[2, ], Plus[Times[2, ], 0.5], [Plus[1, ]]], Times[, Plus[4, Times[2, ], Times[-1, 0.5]], [Plus[2, ]]], Times[, 0.5, [Plus[3, ]]]], 0], Equal[[1], 0], Equal[[2], BesselI[1, 0.5]], Equal[[3], Plus[Times[2, Power[0.5, -1], Plus[Times[0.5, BesselI[0, 0.5]], Times[-2, BesselI[1, 0.5]]]], BesselI[1, 0.5]]]}]][3.0]]]]], {Rule[n, 3], Rule[x, 0.5]}

10.43.E7 ∫ 0 x e + t ⁢ t ν ⁢ I ν ⁡ ( t ) ⁢ d t = e + x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( I ν ⁡ ( x ) - I ν + 1 ⁡ ( x ) ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{\nu}I_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{+x}x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)-I_{\nu+1}% \left(x\right))}}
\int_{0}^{x}e^{+ t}t^{\nu}\modBesselI{\nu}@{t}\diff{t} = \frac{e^{+ x}x^{\nu+1}}{2\nu+1}(\modBesselI{\nu}@{x}-\modBesselI{\nu+1}@{x})		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((\nu+1)+k+% 1)>0}} 
int(exp(+ t)*(t)^(nu)* BesselI(nu, t), t = 0..x) = (exp(+ x)*(x)^(nu + 1))/(2*nu + 1)*(BesselI(nu, x)- BesselI(nu + 1, x))

Integrate[Exp[+ t]*(t)^\[Nu]* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[+ x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselI[\[Nu], x]- BesselI[\[Nu]+ 1, x])
Failure Successful Successful [Tested: 15] Successful [Tested: 15]
10.43.E7 ∫ 0 x e - t ⁢ t ν ⁢ I ν ⁡ ( t ) ⁢ d t = e - x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( I ν ⁡ ( x ) + I ν + 1 ⁡ ( x ) ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{\nu}I_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{-x}x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)+I_{\nu+1}% \left(x\right))}}
\int_{0}^{x}e^{- t}t^{\nu}\modBesselI{\nu}@{t}\diff{t} = \frac{e^{- x}x^{\nu+1}}{2\nu+1}(\modBesselI{\nu}@{x}+\modBesselI{\nu+1}@{x})		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν + 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((\nu+1)+k+% 1)>0}} 
int(exp(- t)*(t)^(nu)* BesselI(nu, t), t = 0..x) = (exp(- x)*(x)^(nu + 1))/(2*nu + 1)*(BesselI(nu, x)+ BesselI(nu + 1, x))

Integrate[Exp[- t]*(t)^\[Nu]* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[- x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselI[\[Nu], x]+ BesselI[\[Nu]+ 1, x])
Failure Successful Skipped - Because timed out Successful [Tested: 15]
10.43.E8 ∫ 0 x e + t ⁢ t - ν ⁢ I ν ⁡ ( t ) ⁢ d t = - e + x ⁢ x - ν + 1 2 ⁢ ν - 1 ⁢ ( I ν ⁡ ( x ) - I ν - 1 ⁡ ( x ) ) - 2 - ν + 1 ( 2 ⁢ ν - 1 ) ⁢ Γ ⁡ ( ν ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 superscript 2 𝜈 1 2 𝜈 1 Euler-Gamma 𝜈 {\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{-\nu}I_{\nu}\left(t\right)% \mathrm{d}t=-\frac{e^{+x}x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)-I_{\nu-1}% \left(x\right))-\frac{2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)}}}
\int_{0}^{x}e^{+ t}t^{-\nu}\modBesselI{\nu}@{t}\diff{t} = -\frac{e^{+ x}x^{-\nu+1}}{2\nu-1}(\modBesselI{\nu}@{x}-\modBesselI{\nu-1}@{x})-\frac{2^{-\nu+1}}{(2\nu-1)\EulerGamma@{\nu}}		 ν ≠ 1 2 , ℜ ⁡ ( ν ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν - 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu)>0,\Re(\nu+k+1)>0,\Re(% (\nu-1)+k+1)>0}} 
int(exp(+ t)*(t)^(- nu)* BesselI(nu, t), t = 0..x) = -(exp(+ x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselI(nu, x)- BesselI(nu - 1, x))-((2)^(- nu + 1))/((2*nu - 1)*GAMMA(nu))

Integrate[Exp[+ t]*(t)^(- \[Nu])* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == -Divide[Exp[+ x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselI[\[Nu], x]- BesselI[\[Nu]- 1, x])-Divide[(2)^(- \[Nu]+ 1),(2*\[Nu]- 1)*Gamma[\[Nu]]]
Failure Successful Manual Skip!
Failed [3 / 12]
Result: 0.39894228040143315
Test Values: {Rule[x, 1.5], Rule[ν, 1.5]}


Result: 0.39894228040143254
Test Values: {Rule[x, 0.5], Rule[ν, 1.5]}

... skip entries to safe data
10.43.E8 ∫ 0 x e - t ⁢ t - ν ⁢ I ν ⁡ ( t ) ⁢ d t = - e - x ⁢ x - ν + 1 2 ⁢ ν - 1 ⁢ ( I ν ⁡ ( x ) + I ν - 1 ⁡ ( x ) ) + 2 - ν + 1 ( 2 ⁢ ν - 1 ) ⁢ Γ ⁡ ( ν ) superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 superscript 2 𝜈 1 2 𝜈 1 Euler-Gamma 𝜈 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{-\nu}I_{\nu}\left(t\right)% \mathrm{d}t=-\frac{e^{-x}x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)+I_{\nu-1}% \left(x\right))+\frac{2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)}}}
\int_{0}^{x}e^{- t}t^{-\nu}\modBesselI{\nu}@{t}\diff{t} = -\frac{e^{- x}x^{-\nu+1}}{2\nu-1}(\modBesselI{\nu}@{x}+\modBesselI{\nu-1}@{x})+\frac{2^{-\nu+1}}{(2\nu-1)\EulerGamma@{\nu}}		 ν ≠ 1 2 , ℜ ⁡ ( ν ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( ν - 1 ) + k + 1 ) > 0 formulae-sequence 𝜈 1 2 formulae-sequence 𝜈 0 formulae-sequence 𝜈 𝑘 1 0 𝜈 1 𝑘 1 0 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu)>0,\Re(\nu+k+1)>0,\Re(% (\nu-1)+k+1)>0}} 
int(exp(- t)*(t)^(- nu)* BesselI(nu, t), t = 0..x) = -(exp(- x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselI(nu, x)+ BesselI(nu - 1, x))+((2)^(- nu + 1))/((2*nu - 1)*GAMMA(nu))

Integrate[Exp[- t]*(t)^(- \[Nu])* BesselI[\[Nu], t], {t, 0, x}, GenerateConditions->None] == -Divide[Exp[- x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselI[\[Nu], x]+ BesselI[\[Nu]- 1, x])+Divide[(2)^(- \[Nu]+ 1),(2*\[Nu]- 1)*Gamma[\[Nu]]]
Failure Successful Manual Skip! Successful [Tested: 12]
10.43.E9 ∫ 0 x e + t ⁢ t ν ⁢ K ν ⁡ ( t ) ⁢ d t = e + x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( K ν ⁡ ( x ) + K ν + 1 ⁡ ( x ) ) - 2 ν ⁢ Γ ⁡ ( ν + 1 ) 2 ⁢ ν + 1 superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 superscript 2 𝜈 Euler-Gamma 𝜈 1 2 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{\nu}K_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{+x}x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)+K_{\nu+1}% \left(x\right))-\frac{2^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1}}}
\int_{0}^{x}e^{+ t}t^{\nu}\modBesselK{\nu}@{t}\diff{t} = \frac{e^{+ x}x^{\nu+1}}{2\nu+1}(\modBesselK{\nu}@{x}+\modBesselK{\nu+1}@{x})-\frac{2^{\nu}\EulerGamma@{\nu+1}}{2\nu+1}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 ) > 0 formulae-sequence 𝜈 1 2 𝜈 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+1)>0}} 
int(exp(+ t)*(t)^(nu)* BesselK(nu, t), t = 0..x) = (exp(+ x)*(x)^(nu + 1))/(2*nu + 1)*(BesselK(nu, x)+ BesselK(nu + 1, x))-((2)^(nu)* GAMMA(nu + 1))/(2*nu + 1)

Integrate[Exp[+ t]*(t)^\[Nu]* BesselK[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[+ x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselK[\[Nu], x]+ BesselK[\[Nu]+ 1, x])-Divide[(2)^\[Nu]* Gamma[\[Nu]+ 1],2*\[Nu]+ 1]
Failure Aborted Manual Skip!
Failed [9 / 15]
Result: DirectedInfinity[]
Test Values: {Rule[x, 1.5], Rule[ν, 1.5]}


Result: DirectedInfinity[]
Test Values: {Rule[x, 1.5], Rule[ν, 0.5]}

... skip entries to safe data
10.43.E9 ∫ 0 x e - t ⁢ t ν ⁢ K ν ⁡ ( t ) ⁢ d t = e - x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( K ν ⁡ ( x ) - K ν + 1 ⁡ ( x ) ) + 2 ν ⁢ Γ ⁡ ( ν + 1 ) 2 ⁢ ν + 1 superscript subscript 0 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 superscript 2 𝜈 Euler-Gamma 𝜈 1 2 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{\nu}K_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{-x}x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)-K_{\nu+1}% \left(x\right))+\frac{2^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1}}}
\int_{0}^{x}e^{- t}t^{\nu}\modBesselK{\nu}@{t}\diff{t} = \frac{e^{- x}x^{\nu+1}}{2\nu+1}(\modBesselK{\nu}@{x}-\modBesselK{\nu+1}@{x})+\frac{2^{\nu}\EulerGamma@{\nu+1}}{2\nu+1}		 ℜ ⁡ ν > - 1 2 , ℜ ⁡ ( ν + 1 ) > 0 formulae-sequence 𝜈 1 2 𝜈 1 0 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+1)>0}} 
int(exp(- t)*(t)^(nu)* BesselK(nu, t), t = 0..x) = (exp(- x)*(x)^(nu + 1))/(2*nu + 1)*(BesselK(nu, x)- BesselK(nu + 1, x))+((2)^(nu)* GAMMA(nu + 1))/(2*nu + 1)

Integrate[Exp[- t]*(t)^\[Nu]* BesselK[\[Nu], t], {t, 0, x}, GenerateConditions->None] == Divide[Exp[- x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselK[\[Nu], x]- BesselK[\[Nu]+ 1, x])+Divide[(2)^\[Nu]* Gamma[\[Nu]+ 1],2*\[Nu]+ 1]
Failure Successful Manual Skip!
Failed [3 / 15]
Result: DirectedInfinity[]
Test Values: {Rule[x, 1.5], Rule[ν, 2]}


Result: DirectedInfinity[]
Test Values: {Rule[x, 0.5], Rule[ν, 2]}

... skip entries to safe data
10.43.E10 ∫ x ∞ e t ⁢ t - ν ⁢ K ν ⁡ ( t ) ⁢ d t = e x ⁢ x - ν + 1 2 ⁢ ν - 1 ⁢ ( K ν ⁡ ( x ) + K ν - 1 ⁡ ( x ) ) superscript subscript 𝑥 superscript 𝑒 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 𝑒 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{x}^{\infty}e^{t}t^{-\nu}K_{\nu}\left(t\right% )\mathrm{d}t=\frac{e^{x}x^{-\nu+1}}{2\nu-1}(K_{\nu}\left(x\right)+K_{\nu-1}% \left(x\right))}}
\int_{x}^{\infty}e^{t}t^{-\nu}\modBesselK{\nu}@{t}\diff{t} = \frac{e^{x}x^{-\nu+1}}{2\nu-1}(\modBesselK{\nu}@{x}+\modBesselK{\nu-1}@{x})		 ℜ ⁡ ν > 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>\tfrac{1}{2}}} 
int(exp(t)*(t)^(- nu)* BesselK(nu, t), t = x..infinity) = (exp(x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselK(nu, x)+ BesselK(nu - 1, x))

Integrate[Exp[t]*(t)^(- \[Nu])* BesselK[\[Nu], t], {t, x, Infinity}, GenerateConditions->None] == Divide[Exp[x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselK[\[Nu], x]+ BesselK[\[Nu]- 1, x])
Failure Successful Manual Skip!
Failed [3 / 9]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[ν, 2]}


Result: DirectedInfinity[]
Test Values: {Rule[x, 0.5], Rule[ν, 2]}

... skip entries to safe data
10.43.E18 ∫ 0 ∞ K ν ⁡ ( t ) ⁢ d t = 1 2 ⁢ π ⁢ sec ⁡ ( 1 2 ⁢ π ⁢ ν ) superscript subscript 0 modified-Bessel-second-kind 𝜈 𝑡 𝑡 1 2 𝜋 1 2 𝜋 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}K_{\nu}\left(t\right)\mathrm{d}t=% \tfrac{1}{2}\pi\sec\left(\tfrac{1}{2}\pi\nu\right)}}
\int_{0}^{\infty}\modBesselK{\nu}@{t}\diff{t} = \tfrac{1}{2}\pi\sec@{\tfrac{1}{2}\pi\nu}		 | ℜ ⁡ ν | < 1 𝜈 1 {\displaystyle{\displaystyle|\Re\nu|<1}} 
int(BesselK(nu, t), t = 0..infinity) = (1)/(2)*Pi*sec((1)/(2)*Pi*nu)

Integrate[BesselK[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Pi*Sec[Divide[1,2]*Pi*\[Nu]]
Successful Successful - Successful [Tested: 6]
10.43.E19 ∫ 0 ∞ t μ - 1 ⁢ K ν ⁡ ( t ) ⁢ d t = 2 μ - 2 ⁢ Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν ) superscript subscript 0 superscript 𝑡 𝜇 1 modified-Bessel-second-kind 𝜈 𝑡 𝑡 superscript 2 𝜇 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 Euler-Gamma 1 2 𝜇 1 2 𝜈 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}K_{\nu}\left(t\right)% \mathrm{d}t=2^{\mu-2}\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\Gamma% \left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu\right)}}
\int_{0}^{\infty}t^{\mu-1}\modBesselK{\nu}@{t}\diff{t} = 2^{\mu-2}\EulerGamma@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu}\EulerGamma@{\tfrac{1}{2}\mu+\tfrac{1}{2}\nu}		 | ℜ ⁡ ν | < ℜ ⁡ μ , ℜ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) > 0 , ℜ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν ) > 0 formulae-sequence 𝜈 𝜇 formulae-sequence 1 2 𝜇 1 2 𝜈 0 1 2 𝜇 1 2 𝜈 0 {\displaystyle{\displaystyle|\Re\nu|<\Re\mu,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}% \nu)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu)>0}} 
int((t)^(mu - 1)* BesselK(nu, t), t = 0..infinity) = (2)^(mu - 2)* GAMMA((1)/(2)*mu -(1)/(2)*nu)*GAMMA((1)/(2)*mu +(1)/(2)*nu)

Integrate[(t)^(\[Mu]- 1)* BesselK[\[Nu], t], {t, 0, Infinity}, GenerateConditions->None] == (2)^(\[Mu]- 2)* Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]]
Successful Successful - Successful [Tested: 18]
10.43.E20 ∫ 0 ∞ cos ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( t ) ⁢ d t = π 2 ⁢ ( 1 + a 2 ) 1 2 superscript subscript 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑡 𝑡 𝜋 2 superscript 1 superscript 𝑎 2 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(at\right)K_{0}\left(t% \right)\mathrm{d}t=\frac{\pi}{2(1+a^{2})^{\frac{1}{2}}}}}
\int_{0}^{\infty}\cos@{at}\modBesselK{0}@{t}\diff{t} = \frac{\pi}{2(1+a^{2})^{\frac{1}{2}}}		 | ℑ ⁡ a | < 1 𝑎 1 {\displaystyle{\displaystyle|\Im a|<1}} 
int(cos(a*t)*BesselK(0, t), t = 0..infinity) = (Pi)/(2*(1 + (a)^(2))^((1)/(2)))

Integrate[Cos[a*t]*BesselK[0, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2*(1 + (a)^(2))^(Divide[1,2])]
Successful Aborted - Successful [Tested: 6]
10.43.E21 ∫ 0 ∞ sin ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( t ) ⁢ d t = arcsinh ⁡ a ( 1 + a 2 ) 1 2 superscript subscript 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑡 𝑡 hyperbolic-inverse-sine 𝑎 superscript 1 superscript 𝑎 2 1 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(at\right)K_{0}\left(t% \right)\mathrm{d}t=\frac{\operatorname{arcsinh}a}{(1+a^{2})^{\frac{1}{2}}}}}
\int_{0}^{\infty}\sin@{at}\modBesselK{0}@{t}\diff{t} = \frac{\asinh@@{a}}{(1+a^{2})^{\frac{1}{2}}}		 | ℑ ⁡ a | < 1 𝑎 1 {\displaystyle{\displaystyle|\Im a|<1}} 
int(sin(a*t)*BesselK(0, t), t = 0..infinity) = (arcsinh(a))/((1 + (a)^(2))^((1)/(2)))

Integrate[Sin[a*t]*BesselK[0, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[ArcSinh[a],(1 + (a)^(2))^(Divide[1,2])]
Failure Successful Successful [Tested: 0] Successful [Tested: 6]
10.43.E23 ∫ 0 ∞ t ν + 1 ⁢ I ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = b ν ( 2 ⁢ p 2 ) ν + 1 ⁢ exp ⁡ ( b 2 4 ⁢ p 2 ) superscript subscript 0 superscript 𝑡 𝜈 1 modified-Bessel-first-kind 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 superscript 𝑏 𝜈 superscript 2 superscript 𝑝 2 𝜈 1 superscript 𝑏 2 4 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\nu+1}I_{\nu}\left(bt\right)% \exp\left(-p^{2}t^{2}\right)\mathrm{d}t=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp% \left(\frac{b^{2}}{4p^{2}}\right)}}
\int_{0}^{\infty}t^{\nu+1}\modBesselI{\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = \frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp@{\frac{b^{2}}{4p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}} 
int((t)^(nu + 1)* BesselI(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = ((b)^(nu))/((2*(p)^(2))^(nu + 1))*exp(((b)^(2))/(4*(p)^(2)))

Integrate[(t)^(\[Nu]+ 1)* BesselI[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[(b)^\[Nu],(2*(p)^(2))^(\[Nu]+ 1)]*Exp[Divide[(b)^(2),4*(p)^(2)]]
Error Aborted - Skip - No test values generated
10.43.E24 ∫ 0 ∞ I ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = π 2 ⁢ p ⁢ exp ⁡ ( b 2 8 ⁢ p 2 ) ⁢ I 1 2 ⁢ ν ⁡ ( b 2 8 ⁢ p 2 ) superscript subscript 0 modified-Bessel-first-kind 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 2 𝑝 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-first-kind 1 2 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}I_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2p}\exp\left(\frac{b^{2}}{8p^{2% }}\right)I_{\frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}\right)}}
\int_{0}^{\infty}\modBesselI{\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = \frac{\sqrt{\pi}}{2p}\exp@{\frac{b^{2}}{8p^{2}}}\modBesselI{\frac{1}{2}\nu}@{\frac{b^{2}}{8p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( ( 1 2 ⁢ ν ) + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 formulae-sequence 𝜈 𝑘 1 0 1 2 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0,% \Re((\frac{1}{2}\nu)+k+1)>0}} 
int(BesselI(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = (sqrt(Pi))/(2*p)*exp(((b)^(2))/(8*(p)^(2)))*BesselI((1)/(2)*nu, ((b)^(2))/(8*(p)^(2)))

Integrate[BesselI[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],2*p]*Exp[Divide[(b)^(2),8*(p)^(2)]]*BesselI[Divide[1,2]*\[Nu], Divide[(b)^(2),8*(p)^(2)]]
Failure Aborted
Failed [228 / 300]
Result: -.7585567167+3.675115279*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I}


Result: -.9489546609+2.381017603*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [152 / 300]
Result: Complex[-0.19039794459564638, -1.294097675814569]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.992047945390181, -4.249025046528451]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.43.E25 ∫ 0 ∞ K ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = π 4 ⁢ p ⁢ sec ⁡ ( 1 2 ⁢ π ⁢ ν ) ⁢ exp ⁡ ( b 2 8 ⁢ p 2 ) ⁢ K 1 2 ⁢ ν ⁡ ( b 2 8 ⁢ p 2 ) superscript subscript 0 modified-Bessel-second-kind 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 4 𝑝 1 2 𝜋 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-second-kind 1 2 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}K_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{4p}\sec\left(\tfrac{1}{2}\pi\nu% \right)\exp\left(\frac{b^{2}}{8p^{2}}\right)K_{\frac{1}{2}\nu}\left(\frac{b^{2% }}{8p^{2}}\right)}}
\int_{0}^{\infty}\modBesselK{\nu}@{bt}\exp@{-p^{2}t^{2}}\diff{t} = \frac{\sqrt{\pi}}{4p}\sec@{\tfrac{1}{2}\pi\nu}\exp@{\frac{b^{2}}{8p^{2}}}\modBesselK{\frac{1}{2}\nu}@{\frac{b^{2}}{8p^{2}}}		 | ℜ ⁡ ν | < 1 , ℜ ⁡ ( p 2 ) > 0 formulae-sequence 𝜈 1 superscript 𝑝 2 0 {\displaystyle{\displaystyle|\Re\nu|<1,\Re\left(p^{2}\right)>0}} 
int(BesselK(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity) = (sqrt(Pi))/(4*p)*sec((1)/(2)*Pi*nu)*exp(((b)^(2))/(8*(p)^(2)))*BesselK((1)/(2)*nu, ((b)^(2))/(8*(p)^(2)))

Integrate[BesselK[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi],4*p]*Sec[Divide[1,2]*Pi*\[Nu]]*Exp[Divide[(b)^(2),8*(p)^(2)]]*BesselK[Divide[1,2]*\[Nu], Divide[(b)^(2),8*(p)^(2)]]
Failure Aborted
Failed [144 / 288]
Result: -.4056916296-1.844454275*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I}


Result: -.2830456904e-1-1.996429597*I
Test Values: {b = -3/2, nu = 1/2*3^(1/2)+1/2*I, p = 3/2}

... skip entries to safe data
Failed [144 / 288]
Result: Complex[0.40569163152223653, 1.8444542715605226]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4232355421098407, -0.8203643961026106]
Test Values: {Rule[b, -1.5], Rule[p, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.43.E26 ∫ 0 ∞ K μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) t λ ⁢ d t = b ν ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ - 1 2 ⁢ μ + 1 2 ) 2 λ + 1 ⁢ a ν - λ + 1 ⁢ 𝐅 ⁡ ( ν - λ + μ + 1 2 , ν - λ - μ + 1 2 ; ν + 1 ; - b 2 a 2 ) superscript subscript 0 modified-Bessel-second-kind 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑡 𝜆 𝑡 superscript 𝑏 𝜈 Euler-Gamma 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 superscript 2 𝜆 1 superscript 𝑎 𝜈 𝜆 1 scaled-hypergeometric-bold-F 𝜈 𝜆 𝜇 1 2 𝜈 𝜆 𝜇 1 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{% \nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{b^{\nu}\Gamma\left(\frac{1}% {2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1% }{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{% \nu-\lambda+1}}\*\mathbf{F}\left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda% -\mu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right)}}
\int_{0}^{\infty}\frac{\modBesselK{\mu}@{at}\BesselJ{\nu}@{bt}}{t^{\lambda}}\diff{t} = \frac{b^{\nu}\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}}}{2^{\lambda+1}a^{\nu-\lambda+1}}\*\hyperOlverF@{\frac{\nu-\lambda+\mu+1}{2}}{\frac{\nu-\lambda-\mu+1}{2}}{\nu+1}{-\frac{b^{2}}{a^{2}}}		 ℜ ⁡ ( ν + 1 - λ ) > | ℜ ⁡ μ | , ℜ ⁡ a > | ℑ ⁡ b | , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ + 1 2 ⁢ μ + 1 2 ) > 0 , ℜ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ λ - 1 2 ⁢ μ + 1 2 ) > 0 formulae-sequence 𝜈 1 𝜆 𝜇 formulae-sequence 𝑎 𝑏 formulae-sequence 𝜈 𝑘 1 0 formulae-sequence 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 0 1 2 𝜈 1 2 𝜆 1 2 𝜇 1 2 0 {\displaystyle{\displaystyle\Re\left(\nu+1-\lambda\right)>|\Re\mu|,\Re a>|\Im b% |,\Re(\nu+k+1)>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}% {2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2})>0}} 
int((BesselK(mu, a*t)*BesselJ(nu, b*t))/((t)^(lambda)), t = 0..infinity) = ((b)^(nu)* GAMMA((1)/(2)*nu -(1)/(2)*lambda +(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*nu -(1)/(2)*lambda -(1)/(2)*mu +(1)/(2)))/((2)^(lambda + 1)* (a)^(nu - lambda + 1))* hypergeom([(nu - lambda + mu + 1)/(2), (nu - lambda - mu + 1)/(2)], [nu + 1], -((b)^(2))/((a)^(2)))/GAMMA(nu + 1)

Integrate[Divide[BesselK[\[Mu], a*t]*BesselJ[\[Nu], b*t],(t)^\[Lambda]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(b)^\[Nu]* Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]-Divide[1,2]*\[Mu]+Divide[1,2]],(2)^(\[Lambda]+ 1)* (a)^(\[Nu]- \[Lambda]+ 1)]* Hypergeometric2F1Regularized[Divide[\[Nu]- \[Lambda]+ \[Mu]+ 1,2], Divide[\[Nu]- \[Lambda]- \[Mu]+ 1,2], \[Nu]+ 1, -Divide[(b)^(2),(a)^(2)]]
Error Aborted - Skip - No test values generated
10.43.E27 ∫ 0 ∞ t μ + ν + 1 ⁢ K μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ d t = ( 2 ⁢ a ) μ ⁢ ( 2 ⁢ b ) ν ⁢ Γ ⁡ ( μ + ν + 1 ) ( a 2 + b 2 ) μ + ν + 1 superscript subscript 0 superscript 𝑡 𝜇 𝜈 1 modified-Bessel-second-kind 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 superscript 2 𝑎 𝜇 superscript 2 𝑏 𝜈 Euler-Gamma 𝜇 𝜈 1 superscript superscript 𝑎 2 superscript 𝑏 2 𝜇 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu+\nu+1}K_{\mu}\left(at% \right)J_{\nu}\left(bt\right)\mathrm{d}t=\frac{(2a)^{\mu}(2b)^{\nu}\Gamma\left% (\mu+\nu+1\right)}{(a^{2}+b^{2})^{\mu+\nu+1}}}}
\int_{0}^{\infty}t^{\mu+\nu+1}\modBesselK{\mu}@{at}\BesselJ{\nu}@{bt}\diff{t} = \frac{(2a)^{\mu}(2b)^{\nu}\EulerGamma@{\mu+\nu+1}}{(a^{2}+b^{2})^{\mu+\nu+1}}		 ℜ ⁡ ( ν + 1 ) > | ℜ ⁡ μ | , ℜ ⁡ a > | ℑ ⁡ b | , ℜ ⁡ ( ν + k + 1 ) > 0 , ℜ ⁡ ( μ + ν + 1 ) > 0 formulae-sequence 𝜈 1 𝜇 formulae-sequence 𝑎 𝑏 formulae-sequence 𝜈 𝑘 1 0 𝜇 𝜈 1 0 {\displaystyle{\displaystyle\Re\left(\nu+1\right)>|\Re\mu|,\Re a>|\Im b|,\Re(% \nu+k+1)>0,\Re(\mu+\nu+1)>0}} 
int((t)^(mu + nu + 1)* BesselK(mu, a*t)*BesselJ(nu, b*t), t = 0..infinity) = ((2*a)^(mu)*(2*b)^(nu)* GAMMA(mu + nu + 1))/(((a)^(2)+ (b)^(2))^(mu + nu + 1))

Integrate[(t)^(\[Mu]+ \[Nu]+ 1)* BesselK[\[Mu], a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[(2*a)^\[Mu]*(2*b)^\[Nu]* Gamma[\[Mu]+ \[Nu]+ 1],((a)^(2)+ (b)^(2))^(\[Mu]+ \[Nu]+ 1)]
Error Aborted - Skip - No test values generated
10.43.E28 ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ I ν ⁡ ( a ⁢ t ) ⁢ I ν ⁡ ( b ⁢ t ) ⁢ d t = 1 2 ⁢ p 2 ⁢ exp ⁡ ( a 2 + b 2 4 ⁢ p 2 ) ⁢ I ν ⁡ ( a ⁢ b 2 ⁢ p 2 ) superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 modified-Bessel-first-kind 𝜈 𝑎 𝑡 modified-Bessel-first-kind 𝜈 𝑏 𝑡 𝑡 1 2 superscript 𝑝 2 superscript 𝑎 2 superscript 𝑏 2 4 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 𝑎 𝑏 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{% \nu}\left(at\right)I_{\nu}\left(bt\right)\mathrm{d}t=\frac{1}{2p^{2}}\exp\left% (\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{\nu}\left(\frac{ab}{2p^{2}}\right)}}
\int_{0}^{\infty}t\exp@{-p^{2}t^{2}}\modBesselI{\nu}@{at}\modBesselI{\nu}@{bt}\diff{t} = \frac{1}{2p^{2}}\exp@{\frac{a^{2}+b^{2}}{4p^{2}}}\modBesselI{\nu}@{\frac{ab}{2p^{2}}}		 ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( ν + k + 1 ) > 0 formulae-sequence 𝜈 1 formulae-sequence superscript 𝑝 2 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}} 
int(t*exp(- (p)^(2)* (t)^(2))*BesselI(nu, a*t)*BesselI(nu, b*t), t = 0..infinity) = (1)/(2*(p)^(2))*exp(((a)^(2)+ (b)^(2))/(4*(p)^(2)))*BesselI(nu, (a*b)/(2*(p)^(2)))

Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselI[\[Nu], a*t]*BesselI[\[Nu], b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2*(p)^(2)]*Exp[Divide[(a)^(2)+ (b)^(2),4*(p)^(2)]]*BesselI[\[Nu], Divide[a*b,2*(p)^(2)]]
Error Aborted - Skipped - Because timed out
10.43.E29 ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ I 0 ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( a ⁢ t ) ⁢ d t = 1 4 ⁢ p 2 ⁢ exp ⁡ ( a 2 2 ⁢ p 2 ) ⁢ K 0 ⁡ ( a 2 2 ⁢ p 2 ) superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 modified-Bessel-first-kind 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑎 𝑡 𝑡 1 4 superscript 𝑝 2 superscript 𝑎 2 2 superscript 𝑝 2 modified-Bessel-second-kind 0 superscript 𝑎 2 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{0% }\left(at\right)K_{0}\left(at\right)\mathrm{d}t=\frac{1}{4p^{2}}\exp\left(% \frac{a^{2}}{2p^{2}}\right)K_{0}\left(\frac{a^{2}}{2p^{2}}\right)}}
\int_{0}^{\infty}t\exp@{-p^{2}t^{2}}\modBesselI{0}@{at}\modBesselK{0}@{at}\diff{t} = \frac{1}{4p^{2}}\exp@{\frac{a^{2}}{2p^{2}}}\modBesselK{0}@{\frac{a^{2}}{2p^{2}}}		 ℜ ⁡ ( p 2 ) > 0 , ℜ ⁡ ( 0 + k + 1 ) > 0 formulae-sequence superscript 𝑝 2 0 0 𝑘 1 0 {\displaystyle{\displaystyle\Re\left(p^{2}\right)>0,\Re(0+k+1)>0}} 
int(t*exp(- (p)^(2)* (t)^(2))*BesselI(0, a*t)*BesselK(0, a*t), t = 0..infinity) = (1)/(4*(p)^(2))*exp(((a)^(2))/(2*(p)^(2)))*BesselK(0, ((a)^(2))/(2*(p)^(2)))

Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselI[0, a*t]*BesselK[0, a*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,4*(p)^(2)]*Exp[Divide[(a)^(2),2*(p)^(2)]]*BesselK[0, Divide[(a)^(2),2*(p)^(2)]]
Failure Aborted Skipped - Because timed out Successful [Tested: 48]
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