6.1 Problem number 90

\[ \int \frac {\text {sech}^3(x)}{i+\text {csch}(x)} \, dx \]

Optimal antiderivative \[ -\frac {\mathrm {I} \arctan \! \left (\sinh \! \left (x \right )\right )}{8}-\frac {\mathrm {sech}\! \left (x \right )^{4}}{4}-\frac {\mathrm {I} \,\mathrm {sech}\! \left (x \right ) \tanh \! \left (x \right )}{8}+\frac {\mathrm {I} \mathrm {sech}\! \left (x \right )^{3} \tanh \! \left (x \right )}{4} \]

command

integrate(sech(x)^3/(I+csch(x)),x, algorithm="maxima")

Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output

\[ \text {Exception raised: RuntimeError} \]

Maxima 5.44 via sagemath 9.3 output

\[ \frac {8 \, {\left (i \, e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} - 10 i \, e^{\left (-3 \, x\right )} - 2 \, e^{\left (-4 \, x\right )} + i \, e^{\left (-5 \, x\right )}\right )}}{64 i \, e^{\left (-x\right )} - 32 \, e^{\left (-2 \, x\right )} + 128 i \, e^{\left (-3 \, x\right )} + 32 \, e^{\left (-4 \, x\right )} + 64 i \, e^{\left (-5 \, x\right )} + 32 \, e^{\left (-6 \, x\right )} - 32} - \frac {1}{8} \, \log \left (e^{\left (-x\right )} + i\right ) + \frac {1}{8} \, \log \left (e^{\left (-x\right )} - i\right ) \]