Optimal. Leaf size=159 \[ 4 i \sqrt {2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \sinh (x) \log \left (\sinh ^2(x)+2\right )+4 i \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )^2-8 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 \sqrt {2} \log \left (\sinh ^2(x)+2\right ) \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )+8 \sqrt {2} \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right ) \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4358, 2450, 2476, 2448, 321, 203, 2470, 12, 4920, 4854, 2402, 2315} \[ 4 i \sqrt {2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \sinh (x) \log \left (\sinh ^2(x)+2\right )+4 i \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )^2-8 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 \sqrt {2} \log \left (\sinh ^2(x)+2\right ) \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )+8 \sqrt {2} \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right ) \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 321
Rule 2315
Rule 2402
Rule 2448
Rule 2450
Rule 2470
Rule 2476
Rule 4358
Rule 4854
Rule 4920
Rubi steps
\begin {align*} \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx &=\operatorname {Subst}\left (\int \log ^2\left (2+x^2\right ) \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-4 \operatorname {Subst}\left (\int \frac {x^2 \log \left (2+x^2\right )}{2+x^2} \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-4 \operatorname {Subst}\left (\int \left (\log \left (2+x^2\right )-\frac {2 \log \left (2+x^2\right )}{2+x^2}\right ) \, dx,x,\sinh (x)\right )\\ &=\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-4 \operatorname {Subst}\left (\int \log \left (2+x^2\right ) \, dx,x,\sinh (x)\right )+8 \operatorname {Subst}\left (\int \frac {\log \left (2+x^2\right )}{2+x^2} \, dx,x,\sinh (x)\right )\\ &=4 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)+8 \operatorname {Subst}\left (\int \frac {x^2}{2+x^2} \, dx,x,\sinh (x)\right )-16 \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2} \left (2+x^2\right )} \, dx,x,\sinh (x)\right )\\ &=4 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-16 \operatorname {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sinh (x)\right )-\left (8 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2+x^2} \, dx,x,\sinh (x)\right )\\ &=-8 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 i \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+4 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)+8 \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{i-\frac {x}{\sqrt {2}}} \, dx,x,\sinh (x)\right )\\ &=-8 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {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 ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)-8 \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i x}{\sqrt {2}}}\right )}{1+\frac {x^2}{2}} \, dx,x,\sinh (x)\right )\\ &=-8 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {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 ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)+\left (8 i \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sinh (x)}{\sqrt {2}}}\right )\\ &=-8 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {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 ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+4 i \sqrt {2} \text {Li}_2\left (1-\frac {4}{2+i \sqrt {2} \sinh (x)}\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 122, normalized size = 0.77 \[ 4 i \sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \sinh (x)+2 i}{\sqrt {2} \sinh (x)-2 i}\right )+\sinh (x) \left (\log ^2\left (\sinh ^2(x)+2\right )-4 \log \left (\sinh ^2(x)+2\right )+8\right )+4 \sqrt {2} \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right ) \left (\log \left (\sinh ^2(x)+2\right )+2 \log \left (\frac {4 i}{-\sqrt {2} \sinh (x)+2 i}\right )+i \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {2}}\right )-2\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \relax (x) \log \left (\cosh \relax (x)^{2} + 1\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \relax (x) \log \left (\cosh \relax (x)^{2} + 1\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.11, size = 0, normalized size = 0.00 \[ \int \cosh \relax (x ) \ln \left (\cosh ^{2}\relax (x )+1\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\ln \left ({\mathrm {cosh}\relax (x)}^2+1\right )}^2\,\mathrm {cosh}\relax (x) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (\cosh ^{2}{\relax (x )} + 1 \right )}^{2} \cosh {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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