\[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx \]
Optimal antiderivative \[ \frac {\coth \left (x \right )}{6}-\frac {\mathrm {arccot}\left (\cosh \left (x \right )\right ) \mathrm {csch}\left (x \right )^{3}}{3}+\frac {\arctanh \left (\frac {\sqrt {2}\, \tanh \left (x \right )}{2}\right ) \sqrt {2}}{12} \]
command
integrate(acot(cosh(x))*cosh(x)/sinh(x)**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} + \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} - \frac {\tanh ^{3}{\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24} + \frac {\tanh {\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8} + \frac {\tanh {\left (\frac {x}{2} \right )}}{12} - \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8 \tanh {\left (\frac {x}{2} \right )}} + \frac {1}{12 \tanh {\left (\frac {x}{2} \right )}} + \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24 \tanh ^{3}{\left (\frac {x}{2} \right )}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________