3.28 \(\int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx\)

Optimal. Leaf size=159 \[ 4 i \sqrt{2} \text{PolyLog}\left (2,1-\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \sinh (x) \log \left (\sinh ^2(x)+2\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 \sqrt{2} \log \left (\sinh ^2(x)+2\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+8 \sqrt{2} \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \]

[Out]

-8*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]] + (4*I)*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]^2 + 8
*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[(2*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 4*Sqrt
[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[2 + Sinh[x]^2] + (4*I)*Sqrt[2]*PolyLog[2, 1 - (2
*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 8*Sinh[x] - 4*Log[2 + Sinh[x]^2]*Sinh[x] + Lo
g[2 + Sinh[x]^2]^2*Sinh[x]

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Rubi [A]  time = 0.340039, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1. \[ 4 i \sqrt{2} \text{PolyLog}\left (2,1-\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \sinh (x) \log \left (\sinh ^2(x)+2\right )+4 i \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )^2-8 \sqrt{2} \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+4 \sqrt{2} \log \left (\sinh ^2(x)+2\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right )+8 \sqrt{2} \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right ) \tan ^{-1}\left (\frac{\sinh (x)}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Cosh[x]*Log[1 + Cosh[x]^2]^2,x]

[Out]

-8*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]] + (4*I)*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]^2 + 8
*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[(2*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 4*Sqrt
[2]*ArcTan[Sinh[x]/Sqrt[2]]*Log[2 + Sinh[x]^2] + (4*I)*Sqrt[2]*PolyLog[2, 1 - (2
*Sqrt[2])/(Sqrt[2] + I*Sinh[x])] + 8*Sinh[x] - 4*Log[2 + Sinh[x]^2]*Sinh[x] + Lo
g[2 + Sinh[x]^2]^2*Sinh[x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cosh(x)*ln(1+cosh(x)**2)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 7.89292, size = 0, normalized size = 0. \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Cosh[x]*Log[1 + Cosh[x]^2]^2,x]

[Out]

Integrate[Cosh[x]*Log[1 + Cosh[x]^2]^2, x]

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Maple [F]  time = 3.404, size = 0, normalized size = 0. \[ \int \cosh \left ( x \right ) \left ( \ln \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cosh(x)*ln(1+cosh(x)^2)^2,x)

[Out]

int(cosh(x)*ln(1+cosh(x)^2)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)*log(cosh(x)^2 + 1)^2,x, algorithm="maxima")

[Out]

1/2*(e^(2*x) - 1)*e^(-x)*log(e^(4*x) + 6*e^(2*x) + 1)^2 - 2*(e^(-x) + integrate(
(e^(2*x) + 6)*e^x/(e^(4*x) + 6*e^(2*x) + 1), x))*log(2)^2 + 2*(e^x - integrate((
6*e^(2*x) + 1)*e^x/(e^(4*x) + 6*e^(2*x) + 1), x))*log(2)^2 + 14*integrate(e^(3*x
)/(e^(4*x) + 6*e^(2*x) + 1), x)*log(2)^2 + 14*integrate(e^x/(e^(4*x) + 6*e^(2*x)
+ 1), x)*log(2)^2 + 4*integrate(x*e^(6*x)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2
) + 28*integrate(x*e^(4*x)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) + 28*integrate
(x*e^(2*x)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 2*integrate(e^(6*x)*log(e^(4
*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 14*integrate(e^(4*
x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 14*inte
grate(e^(2*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2
) + 4*integrate(x/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) - 2*integrate(log(e^(4*
x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)*log(2) + 2*integrate(x^2*e^(
6*x)/(e^(5*x) + 6*e^(3*x) + e^x), x) + 14*integrate(x^2*e^(4*x)/(e^(5*x) + 6*e^(
3*x) + e^x), x) + 14*integrate(x^2*e^(2*x)/(e^(5*x) + 6*e^(3*x) + e^x), x) - 2*i
ntegrate(x*e^(6*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)
- 14*integrate(x*e^(4*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x
), x) - 14*integrate(x*e^(2*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x)
+ e^x), x) + 2*integrate(x^2/(e^(5*x) + 6*e^(3*x) + e^x), x) - 2*integrate(x*lo
g(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) - 4*integrate(e^(6*x)
*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) - 8*integrate(e^(4
*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x) + 12*integrate(
e^(2*x)*log(e^(4*x) + 6*e^(2*x) + 1)/(e^(5*x) + 6*e^(3*x) + e^x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)*log(cosh(x)^2 + 1)^2,x, algorithm="fricas")

[Out]

integral(cosh(x)*log(cosh(x)^2 + 1)^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)*ln(1+cosh(x)**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cosh(x)*log(cosh(x)^2 + 1)^2,x, algorithm="giac")

[Out]

integrate(cosh(x)*log(cosh(x)^2 + 1)^2, x)