\[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sinh \left (x \right )}{\sqrt {\sinh \left (2 x \right )}}\right )}{6}+\arctan \left (\mathrm {sech}\left (x \right ) \sqrt {\cosh \left (x \right ) \sinh \left (x \right )}\right ) \sqrt {2}-\frac {\arctanh \left (\mathrm {sech}\left (x \right ) \sqrt {\cosh \left (x \right ) \sinh \left (x \right )}\right ) \sqrt {2}}{3}+\frac {\cosh \left (x \right )}{\sqrt {\sinh \left (2 x \right )}} \]
command
integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \sqrt {2} \arctan \left (\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {e^{\left (4 \, x\right )} - 1} + e^{\left (2 \, x\right )}\right ) + \frac {\sqrt {2}}{\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )} + 1} + \frac {1}{6} \, \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, \sqrt {e^{\left (4 \, x\right )} - 1} - 3 \, e^{\left (2 \, x\right )} - 1\right )}\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________