Integrand size = 24, antiderivative size = 681 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )} \]
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Time = 0.82 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5549, 5545, 4276, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \sqrt {a^2+b^2}}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \sqrt {a^2+b^2}}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sinh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2+b^2\right )}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d e n \sqrt {a^2+b^2}}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d e n \sqrt {a^2+b^2}}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^n\right )+b\right )}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}+\frac {(e x)^{2 n}}{2 a^2 e n} \]
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
Rule 4276
Rule 5545
Rule 5549
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac {x^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx}{e} \\ & = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(a+b \text {csch}(c+d x))^2} \, dx,x,x^n\right )}{e n} \\ & = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(b+a \sinh (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{-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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\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)^{2 n}}{2 a^2 e n}-\frac {b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{-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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{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 (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {2 b x^{-n} (e x)^{2 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 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{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 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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 (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 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)^{2 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^3 x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 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 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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 (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 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)^{2 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^3 x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\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 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\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)^{2 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 (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \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^2 e n}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \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^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 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)^{2 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^3 x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\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 {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\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^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\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 {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\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)^{2 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*}
Result contains complex when optimal does not.
Time = 20.68 (sec) , antiderivative size = 3219, normalized size of antiderivative = 4.73 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (e x \right )^{2 n -1}}{{\left (a +b \,\operatorname {csch}\left (c +d \,x^{n}\right )\right )}^{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 8453 vs. \(2 (645) = 1290\).
Time = 0.41 (sec) , antiderivative size = 8453, normalized size of antiderivative = 12.41 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{2 n - 1}}{\left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{2 \, n - 1}}{{\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{2 \, n - 1}}{{\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{2\,n-1}}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x^n\right )}\right )}^2} \,d x \]
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