31.6 Problem number 447

\[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx \]

Optimal antiderivative \[ -\frac {a^{2}}{7 f \left (a \left (\cosh ^{2}\left (f x +e \right )\right )\right )^{\frac {7}{2}}}+\frac {2 a}{5 f \left (a \left (\cosh ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}-\frac {1}{3 f \left (a \left (\cosh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]

command

integrate(tanh(f*x+e)^5/(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

\[ -\frac {8 \, {\left (35 \, \sqrt {a} e^{\left (11 \, f x + 11 \, e\right )} - 28 \, \sqrt {a} e^{\left (9 \, f x + 9 \, e\right )} + 114 \, \sqrt {a} e^{\left (7 \, f x + 7 \, e\right )} - 28 \, \sqrt {a} e^{\left (5 \, f x + 5 \, e\right )} + 35 \, \sqrt {a} e^{\left (3 \, f x + 3 \, e\right )}\right )}}{105 \, a^{2} f {\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}^{7}} \]