10.43

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.43.E4	${\displaystyle{\displaystyle\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\mathrm{d}t=\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)}}$ \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}	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}}$	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	${\displaystyle{\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}I_{k}\left(x\right)=\frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(\psi\left(k+2\right)-\psi\left(1\right))I_{2k+3}\left(x\right)}}$ \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}	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0,\Re((2k+3)+k+1)>0}}$	(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	${\displaystyle{\displaystyle\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}\mathrm{d}t=\frac{1}{2}\left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+\frac{\pi^{2}}{24}-\sum_{k=1}^{\infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}}}}$ \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	${\displaystyle{\displaystyle\int_{0}^{x}e^{-t}I_{n}\left(t\right)\mathrm{d}t=xe^{-x}(I_{0}\left(x\right)+I_{1}\left(x\right))+n(e^{-x}I_{0}\left(x\right)-1)+2e^{-x}\sum_{k=1}^{n-1}(n-k)I_{k}\left(x\right)}}$ \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}	${\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(0+k+1)>0,\Re(1+k+1)>0,\Re(k+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{\nu}I_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{+x}x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)-I_{\nu+1}\left(x\right))}}$ \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})	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{\nu}I_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{-x}x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)+I_{\nu+1}\left(x\right))}}$ \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})	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+k+1)>0,\Re((\nu+1)+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{-\nu}I_{\nu}\left(t\right)\mathrm{d}t=-\frac{e^{+x}x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)-I_{\nu-1}\left(x\right))-\frac{2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)}}}$ \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}}	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu)>0,\Re(\nu+k+1)>0,\Re((\nu-1)+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{-\nu}I_{\nu}\left(t\right)\mathrm{d}t=-\frac{e^{-x}x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)+I_{\nu-1}\left(x\right))+\frac{2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)}}}$ \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}}	${\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\Re(\nu)>0,\Re(\nu+k+1)>0,\Re((\nu-1)+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{x}e^{+t}t^{\nu}K_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{+x}x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)+K_{\nu+1}\left(x\right))-\frac{2^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1}}}$ \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}	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{x}e^{-t}t^{\nu}K_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{-x}x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)-K_{\nu+1}\left(x\right))+\frac{2^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1}}}$ \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}	${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2},\Re(\nu+1)>0}}$	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	${\displaystyle{\displaystyle\int_{x}^{\infty}e^{t}t^{-\nu}K_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{x}x^{-\nu+1}}{2\nu-1}(K_{\nu}\left(x\right)+K_{\nu-1}\left(x\right))}}$ \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})	${\displaystyle{\displaystyle\Re\nu>\tfrac{1}{2}}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}K_{\nu}\left(t\right)\mathrm{d}t=\tfrac{1}{2}\pi\sec\left(\tfrac{1}{2}\pi\nu\right)}}$ \int_{0}^{\infty}\modBesselK{\nu}@{t}\diff{t} = \tfrac{1}{2}\pi\sec@{\tfrac{1}{2}\pi\nu}	${\displaystyle{\displaystyle|\Re\nu|<1}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}K_{\nu}\left(t\right)\mathrm{d}t=2^{\mu-2}\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu\right)}}$ \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}	${\displaystyle{\displaystyle|\Re\nu|<\Re\mu,\Re(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu)>0,\Re(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(at\right)K_{0}\left(t\right)\mathrm{d}t=\frac{\pi}{2(1+a^{2})^{\frac{1}{2}}}}}$ \int_{0}^{\infty}\cos@{at}\modBesselK{0}@{t}\diff{t} = \frac{\pi}{2(1+a^{2})^{\frac{1}{2}}}	${\displaystyle{\displaystyle|\Im a|<1}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(at\right)K_{0}\left(t\right)\mathrm{d}t=\frac{\operatorname{arcsinh}a}{(1+a^{2})^{\frac{1}{2}}}}}$ \int_{0}^{\infty}\sin@{at}\modBesselK{0}@{t}\diff{t} = \frac{\asinh@@{a}}{(1+a^{2})^{\frac{1}{2}}}	${\displaystyle{\displaystyle|\Im a|<1}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\nu+1}I_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)\mathrm{d}t=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp\left(\frac{b^{2}}{4p^{2}}\right)}}$ \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}}}	${\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}I_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2p}\exp\left(\frac{b^{2}}{8p^{2}}\right)I_{\frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}\right)}}$ \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}}}	${\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0,\Re((\frac{1}{2}\nu)+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}K_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{4p}\sec\left(\tfrac{1}{2}\pi\nu\right)\exp\left(\frac{b^{2}}{8p^{2}}\right)K_{\frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}\right)}}$ \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}}}	${\displaystyle{\displaystyle|\Re\nu|<1,\Re\left(p^{2}\right)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{\nu}\left(bt\right)}{t^{\lambda}}\mathrm{d}t=\frac{b^{\nu}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{\nu-\lambda+1}}\*\mathbf{F}\left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda-\mu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right)}}$ \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}}}	${\displaystyle{\displaystyle\Re\left(\nu+1-\lambda\right)>|\Re\mu|,\Re a>|\Im b|,\Re(\nu+k+1)>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2})>0,\Re(\frac{1}{2}\nu-\frac{1}{2}\lambda-\frac{1}{2}\mu+\frac{1}{2})>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu+\nu+1}K_{\mu}\left(at\right)J_{\nu}\left(bt\right)\mathrm{d}t=\frac{(2a)^{\mu}(2b)^{\nu}\Gamma\left(\mu+\nu+1\right)}{(a^{2}+b^{2})^{\mu+\nu+1}}}}$ \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}}	${\displaystyle{\displaystyle\Re\left(\nu+1\right)>|\Re\mu|,\Re a>|\Im b|,\Re(\nu+k+1)>0,\Re(\mu+\nu+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{\nu}\left(at\right)I_{\nu}\left(bt\right)\mathrm{d}t=\frac{1}{2p^{2}}\exp\left(\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{\nu}\left(\frac{ab}{2p^{2}}\right)}}$ \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}}}	${\displaystyle{\displaystyle\Re\nu>-1,\Re\left(p^{2}\right)>0,\Re(\nu+k+1)>0}}$	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	${\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{0}\left(at\right)K_{0}\left(at\right)\mathrm{d}t=\frac{1}{4p^{2}}\exp\left(\frac{a^{2}}{2p^{2}}\right)K_{0}\left(\frac{a^{2}}{2p^{2}}\right)}}$ \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}}}	${\displaystyle{\displaystyle\Re\left(p^{2}\right)>0,\Re(0+k+1)>0}}$	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]