31.3 Problem number 439

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

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

command

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

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