31.10 Problem number 455

\[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx \]

Optimal antiderivative \[ -\frac {\coth \! \left (f x +e \right ) \mathrm {csch}\! \left (f x +e \right )^{2}}{3 a f \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {\coth \! \left (f x +e \right ) \mathrm {csch}\! \left (f x +e \right )^{4}}{5 a f \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}} \]

command

integrate(coth(f*x+e)^6/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

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

Giac 1.7.0 via sagemath 9.3 output

\[ -\frac {8 \, {\left (5 \, \sqrt {a} e^{\left (7 \, f x + 7 \, e\right )} + 2 \, \sqrt {a} e^{\left (5 \, f x + 5 \, e\right )} + 5 \, \sqrt {a} e^{\left (3 \, f x + 3 \, e\right )}\right )}}{15 \, a^{2} f {\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{5}} \]