Integrand size = 12, antiderivative size = 159 \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=-8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+4 i \sqrt {2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+8 \sinh (x)-4 \log \left (2+\sinh ^2(x)\right ) \sinh (x)+\log ^2\left (2+\sinh ^2(x)\right ) \sinh (x) \]
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Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4443, 2500, 2526, 2498, 327, 209, 2520, 12, 5040, 4964, 2449, 2352} \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=4 i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2-8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\sinh ^2(x)+2\right )+8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 i \sqrt {2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{i \sinh (x)+\sqrt {2}}\right )+8 \sinh (x)+\sinh (x) \log ^2\left (\sinh ^2(x)+2\right )-4 \sinh (x) \log \left (\sinh ^2(x)+2\right ) \]
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Rule 12
Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 2498
Rule 2500
Rule 2520
Rule 2526
Rule 4443
Rule 4964
Rule 5040
Rubi steps \begin{align*} \text {integral}& = \text {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 \text {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 \text {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 \text {Subst}\left (\int \log \left (2+x^2\right ) \, dx,x,\sinh (x)\right )+8 \text {Subst}\left (\int \frac {\log \left (2+x^2\right )}{2+x^2} \, dx,x,\sinh (x)\right ) \\ & = 4 \sqrt {2} \arctan \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 \text {Subst}\left (\int \frac {x^2}{2+x^2} \, dx,x,\sinh (x)\right )-16 \text {Subst}\left (\int \frac {x \arctan \left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2} \left (2+x^2\right )} \, dx,x,\sinh (x)\right ) \\ & = 4 \sqrt {2} \arctan \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 \text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sinh (x)\right )-\left (8 \sqrt {2}\right ) \text {Subst}\left (\int \frac {x \arctan \left (\frac {x}{\sqrt {2}}\right )}{2+x^2} \, dx,x,\sinh (x)\right ) \\ & = -8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+4 \sqrt {2} \arctan \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 \text {Subst}\left (\int \frac {\arctan \left (\frac {x}{\sqrt {2}}\right )}{i-\frac {x}{\sqrt {2}}} \, dx,x,\sinh (x)\right ) \\ & = -8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \arctan \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 \text {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} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \arctan \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 ) \text {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} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+4 i \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )^2+8 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+i \sinh (x)}\right )+4 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \log \left (2+\sinh ^2(x)\right )+4 i \sqrt {2} \operatorname {PolyLog}\left (2,1-\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*}
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=4 \sqrt {2} \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right ) \left (-2+i \arctan \left (\frac {\sinh (x)}{\sqrt {2}}\right )+2 \log \left (\frac {4 i}{2 i-\sqrt {2} \sinh (x)}\right )+\log \left (2+\sinh ^2(x)\right )\right )+4 i \sqrt {2} \operatorname {PolyLog}\left (2,\frac {2 i+\sqrt {2} \sinh (x)}{-2 i+\sqrt {2} \sinh (x)}\right )+\left (8-4 \log \left (2+\sinh ^2(x)\right )+\log ^2\left (2+\sinh ^2(x)\right )\right ) \sinh (x) \]
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\[\int \cosh \left (x \right ) \ln \left (1+\cosh ^{2}\left (x \right )\right )^{2}d x\]
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\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2} \,d x } \]
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\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int \log {\left (\cosh ^{2}{\left (x \right )} + 1 \right )}^{2} \cosh {\left (x \right )}\, dx \]
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\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2} \,d x } \]
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\[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int { \cosh \left (x\right ) \log \left (\cosh \left (x\right )^{2} + 1\right )^{2} \,d x } \]
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Timed out. \[ \int \cosh (x) \log ^2\left (1+\cosh ^2(x)\right ) \, dx=\int {\ln \left ({\mathrm {cosh}\left (x\right )}^2+1\right )}^2\,\mathrm {cosh}\left (x\right ) \,d x \]
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