35.5 Problem number 154

\[ \int \frac {\coth (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx \]

Optimal antiderivative \[ \frac {b^{2}}{2 a^{2} \left (a +b \right ) d \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}+\frac {b \left (2 a +b \right ) \ln \! \left (b +a \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{2 a^{2} \left (a +b \right )^{2} d}+\frac {\ln \! \left (\sinh \! \left (d x +c \right )\right )}{\left (a +b \right )^{2} d} \]

command

integrate(coth(d*x+c)/(a+b*sech(d*x+c)^2)^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

\[ \frac {\frac {{\left (2 \, a b + b^{2}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a^{4} + 2 \, a^{3} b + a^{2} b^{2}} + \frac {2 \, e^{\left (2 \, c\right )} \log \left ({\left | -e^{\left (2 \, d x + 2 \, c\right )} + 1 \right |}\right )}{a^{2} e^{\left (2 \, c\right )} + 2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}} - \frac {2 \, d x}{a^{2}} - \frac {2 \, a b e^{\left (4 \, d x + 4 \, c\right )} + b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b + b^{2}}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}}}{2 \, d} \]