31.4 Problem number 442

\[ \int \frac {\tanh ^4(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx \]

Optimal antiderivative \[ \frac {3 \arctan \! \left (\sinh \! \left (f x +e \right )\right ) \cosh \! \left (f x +e \right )}{8 f \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {3 \tanh \! \left (f x +e \right )}{8 f \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {\tanh ^{3}\left (f x +e \right )}{4 f \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}} \]

command

integrate(tanh(f*x+e)^4/(a+a*sinh(f*x+e)^2)^(1/2),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

\[ \frac {\frac {3 \, \arctan \left (e^{\left (f x + e\right )}\right )}{\sqrt {a}} - \frac {5 \, \sqrt {a} e^{\left (7 \, f x + 7 \, e\right )} - 3 \, \sqrt {a} e^{\left (5 \, f x + 5 \, e\right )} + 3 \, \sqrt {a} e^{\left (3 \, f x + 3 \, e\right )} - 5 \, \sqrt {a} e^{\left (f x + e\right )}}{a {\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}^{4}}}{4 \, f} \]