33.12 Problem number 47

\[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx \]

Optimal antiderivative \[ \frac {\left (a +6 b \right ) \arctanh \! \left (\cosh \! \left (d x +c \right )\right )}{2 a^{4} d}-\frac {\coth \! \left (d x +c \right ) \mathrm {csch}\! \left (d x +c \right )}{2 a d \left (a +b -b \mathrm {sech}\! \left (d x +c \right )^{2}\right )^{2}}-\frac {3 b \,\mathrm {sech}\! \left (d x +c \right )}{4 a^{2} d \left (a +b -b \mathrm {sech}\! \left (d x +c \right )^{2}\right )^{2}}-\frac {b \left (11 a +12 b \right ) \mathrm {sech}\! \left (d x +c \right )}{8 a^{3} \left (a +b \right ) d \left (a +b -b \mathrm {sech}\! \left (d x +c \right )^{2}\right )}-\frac {\left (15 a^{2}+40 a b +24 b^{2}\right ) \arctanh \! \left (\frac {\mathrm {sech}\left (d x +c \right ) \sqrt {b}}{\sqrt {a +b}}\right ) \sqrt {b}}{8 a^{4} \left (a +b \right )^{\frac {3}{2}} d} \]

command

integrate(csch(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,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

\[ \text {output too large to display} \]