\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (e x \right )^{3 n}}{3 a^{2} e n}-\frac {b^{2} \left (e x \right )^{3 n} x^{-n}}{a^{2} \left (a^{2}+b^{2}\right ) d e n}+\frac {2 b^{2} \left (e x \right )^{3 n} \ln \left (1+\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-2 n}}{a^{2} \left (a^{2}+b^{2}\right ) d^{2} e n}+\frac {b^{3} \left (e x \right )^{3 n} \ln \left (1+\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-n}}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d e n}+\frac {2 b^{2} \left (e x \right )^{3 n} \ln \left (1+\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-2 n}}{a^{2} \left (a^{2}+b^{2}\right ) d^{2} e n}-\frac {b^{3} \left (e x \right )^{3 n} \ln \left (1+\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-n}}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d e n}+\frac {2 b^{2} \left (e x \right )^{3 n} \polylog \left (2, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-3 n}}{a^{2} \left (a^{2}+b^{2}\right ) d^{3} e n}+\frac {2 b^{3} \left (e x \right )^{3 n} \polylog \left (2, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-2 n}}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} e n}+\frac {2 b^{2} \left (e x \right )^{3 n} \polylog \left (2, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-3 n}}{a^{2} \left (a^{2}+b^{2}\right ) d^{3} e n}-\frac {2 b^{3} \left (e x \right )^{3 n} \polylog \left (2, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-2 n}}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{2} e n}-\frac {2 b^{3} \left (e x \right )^{3 n} \polylog \left (3, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-3 n}}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{3} e n}+\frac {2 b^{3} \left (e x \right )^{3 n} \polylog \left (3, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-3 n}}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d^{3} e n}-\frac {b^{2} \left (e x \right )^{3 n} \cosh \left (c +d \,x^{n}\right ) x^{-n}}{a \left (a^{2}+b^{2}\right ) d e n \left (b +a \sinh \left (c +d \,x^{n}\right )\right )}-\frac {2 b \left (e x \right )^{3 n} \ln \left (1+\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-n}}{a^{2} d e n \sqrt {a^{2}+b^{2}}}+\frac {2 b \left (e x \right )^{3 n} \ln \left (1+\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-n}}{a^{2} d e n \sqrt {a^{2}+b^{2}}}-\frac {4 b \left (e x \right )^{3 n} \polylog \left (2, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-2 n}}{a^{2} d^{2} e n \sqrt {a^{2}+b^{2}}}+\frac {4 b \left (e x \right )^{3 n} \polylog \left (2, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-2 n}}{a^{2} d^{2} e n \sqrt {a^{2}+b^{2}}}+\frac {4 b \left (e x \right )^{3 n} \polylog \left (3, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b -\sqrt {a^{2}+b^{2}}}\right ) x^{-3 n}}{a^{2} d^{3} e n \sqrt {a^{2}+b^{2}}}-\frac {4 b \left (e x \right )^{3 n} \polylog \left (3, -\frac {a \,{\mathrm e}^{c +d \,x^{n}}}{b +\sqrt {a^{2}+b^{2}}}\right ) x^{-3 n}}{a^{2} d^{3} e n \sqrt {a^{2}+b^{2}}} \]
command
integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]