28.7 Problem number 740

\[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx \]

Optimal antiderivative \[ \frac {2 \csc \! \left (5 x \right ) \sqrt {1+\sin ^{2}\left (5 x \right )}}{15}-\frac {\left (\csc ^{3}\left (5 x \right )\right ) \sqrt {1+\sin ^{2}\left (5 x \right )}}{15} \]

command

integrate(cot(5*x)*csc(5*x)^3/(1+sin(5*x)^2)^(1/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {could not integrate} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \frac {4 \, {\left (3 \, {\left (\sqrt {\sin \left (5 \, x\right )^{2} + 1} - \sin \left (5 \, x\right )\right )}^{2} - 1\right )}}{15 \, {\left ({\left (\sqrt {\sin \left (5 \, x\right )^{2} + 1} - \sin \left (5 \, x\right )\right )}^{2} - 1\right )}^{3}} \]