Optimal. Leaf size=1218 \[ \frac {2 b^2 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{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^2 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac {4 b (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {2 b^3 (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 b (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \sqrt {a^2+b^2} d^3 e n}+\frac {2 b^3 (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {4 b (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {2 b^3 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {2 b^3 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {b^2 (e x)^{3 n} x^{-n}}{a^2 \left (a^2+b^2\right ) d e n}-\frac {2 b (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \sqrt {a^2+b^2} d e n}+\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \sqrt {a^2+b^2} d e n}-\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {b^2 (e x)^{3 n} \cosh \left (d x^n+c\right ) x^{-n}}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (d x^n+c\right )\right )}+\frac {(e x)^{3 n}}{3 a^2 e n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.12, antiderivative size = 1218, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5441, 5437, 4191, 3324, 3322, 2264, 2190, 2531, 2282, 6589, 5561, 2279, 2391} \[ \frac {2 b^2 (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{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^2 (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac {4 b (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {2 b^3 (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 b (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \sqrt {a^2+b^2} d^3 e n}+\frac {2 b^3 (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {4 b (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {2 b^3 (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {2 b^3 (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {b^2 (e x)^{3 n} x^{-n}}{a^2 \left (a^2+b^2\right ) d e n}-\frac {2 b (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \sqrt {a^2+b^2} d e n}+\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \sqrt {a^2+b^2} d e n}-\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^{-n}}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {b^2 (e x)^{3 n} \cosh \left (d x^n+c\right ) x^{-n}}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (d x^n+c\right )\right )}+\frac {(e x)^{3 n}}{3 a^2 e n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3322
Rule 3324
Rule 4191
Rule 5437
Rule 5441
Rule 5561
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac {x^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{(a+b \text {csch}(c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{a^2}+\frac {b^2 x^2}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x^2}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{(b+a \sinh (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}+\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) d e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt {a^2+b^2} e n}+\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt {a^2+b^2} e n}+\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) d e n}+\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) d e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right )^{3/2} e n}-\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right )^{3/2} e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {a^2+b^2} d e n}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {a^2+b^2} d e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {4 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {4 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}+\frac {2 b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {2 b^3 x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac {4 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}+\frac {2 b^3 x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3 e n}-\frac {b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 118.96, size = 0, normalized size = 0.00 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{3 n -1}}{\left (a +b \,\mathrm {csch}\left (c +d \,x^{n}\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {6 \, a b^{2} e^{3 \, n} x^{2 \, n} + {\left (a^{3} d e^{3 \, n} + a b^{2} d e^{3 \, n}\right )} x^{3 \, n} - {\left (a^{3} d e^{3 \, n} e^{\left (2 \, c\right )} + a b^{2} d e^{3 \, n} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n} + 3 \, n \log \relax (x)\right )} - 2 \, {\left (3 \, b^{3} e^{3 \, n} e^{\left (2 \, n \log \relax (x) + c\right )} + {\left (a^{2} b d e^{3 \, n} e^{c} + b^{3} d e^{3 \, n} e^{c}\right )} x^{3 \, n}\right )} e^{\left (d x^{n}\right )}}{3 \, {\left (a^{5} d e n + a^{3} b^{2} d e n - {\left (a^{5} d e n e^{\left (2 \, c\right )} + a^{3} b^{2} d e n e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n}\right )} - 2 \, {\left (a^{4} b d e n e^{c} + a^{2} b^{3} d e n e^{c}\right )} e^{\left (d x^{n}\right )}\right )}} - \int -\frac {2 \, {\left (2 \, a b^{2} e^{3 \, n} x^{2 \, n} - {\left (2 \, b^{3} e^{3 \, n} e^{\left (2 \, n \log \relax (x) + c\right )} + {\left (2 \, a^{2} b d e^{3 \, n} e^{c} + b^{3} d e^{3 \, n} e^{c}\right )} x^{3 \, n}\right )} e^{\left (d x^{n}\right )}\right )}}{{\left (a^{5} d e e^{\left (2 \, c\right )} + a^{3} b^{2} d e e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d x^{n}\right )} + 2 \, {\left (a^{4} b d e e^{c} + a^{2} b^{3} d e e^{c}\right )} x e^{\left (d x^{n}\right )} - {\left (a^{5} d e + a^{3} b^{2} d e\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{3\,n-1}}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x^n\right )}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 n - 1}}{\left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________