Optimal. Leaf size=428 \[ \frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a d^3 e n \sqrt{a^2+b^2}}-\frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a d^3 e n \sqrt{a^2+b^2}}-\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a d^2 e n \sqrt{a^2+b^2}}+\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a d^2 e n \sqrt{a^2+b^2}}-\frac{b x^{-n} (e x)^{3 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{b x^{-n} (e x)^{3 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{(e x)^{3 n}}{3 a e n} \]
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Rubi [A] time = 0.843823, antiderivative size = 428, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5441, 5437, 4191, 3322, 2264, 2190, 2531, 2282, 6589} \[ \frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a d^3 e n \sqrt{a^2+b^2}}-\frac{2 b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a d^3 e n \sqrt{a^2+b^2}}-\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a d^2 e n \sqrt{a^2+b^2}}+\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a d^2 e n \sqrt{a^2+b^2}}-\frac{b x^{-n} (e x)^{3 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{b x^{-n} (e x)^{3 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{(e x)^{3 n}}{3 a e n} \]
Antiderivative was successfully verified.
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Rule 5441
Rule 5437
Rule 4191
Rule 3322
Rule 2264
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(e x)^{-1+3 n}}{a+b \text{csch}\left (c+d x^n\right )} \, dx &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac{x^{-1+3 n}}{a+b \text{csch}\left (c+d x^n\right )} \, 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)} \, 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}-\frac{b x^2}{a (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{\left (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 e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{\left (2 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 e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{\left (2 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 )}{\sqrt{a^2+b^2} e n}+\frac{\left (2 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 )}{\sqrt{a^2+b^2} e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{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 \sqrt{a^2+b^2} d e n}+\frac{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 \sqrt{a^2+b^2} d e n}+\frac{\left (2 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 \sqrt{a^2+b^2} d e n}-\frac{\left (2 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 \sqrt{a^2+b^2} d e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{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 \sqrt{a^2+b^2} d e n}+\frac{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 \sqrt{a^2+b^2} d e n}-\frac{2 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 \sqrt{a^2+b^2} d^2 e n}+\frac{2 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 \sqrt{a^2+b^2} d^2 e n}+\frac{\left (2 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 \sqrt{a^2+b^2} d^2 e n}-\frac{\left (2 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 \sqrt{a^2+b^2} d^2 e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{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 \sqrt{a^2+b^2} d e n}+\frac{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 \sqrt{a^2+b^2} d e n}-\frac{2 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 \sqrt{a^2+b^2} d^2 e n}+\frac{2 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 \sqrt{a^2+b^2} d^2 e n}+\frac{\left (2 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 \sqrt{a^2+b^2} d^3 e n}-\frac{\left (2 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 \sqrt{a^2+b^2} d^3 e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{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 \sqrt{a^2+b^2} d e n}+\frac{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 \sqrt{a^2+b^2} d e n}-\frac{2 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 \sqrt{a^2+b^2} d^2 e n}+\frac{2 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 \sqrt{a^2+b^2} d^2 e n}+\frac{2 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 \sqrt{a^2+b^2} d^3 e n}-\frac{2 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 \sqrt{a^2+b^2} d^3 e n}\\ \end{align*}
Mathematica [F] time = 6.71648, size = 0, normalized size = 0. \[ \int \frac{(e x)^{-1+3 n}}{a+b \text{csch}\left (c+d x^n\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.243, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{-1+3\,n}}{a+b{\rm csch} \left (c+d{x}^{n}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -2 \, b e^{3 \, n} \int \frac{e^{\left (d x^{n} + 3 \, n \log \left (x\right ) + c\right )}}{a^{2} e x e^{\left (2 \, d x^{n} + 2 \, c\right )} + 2 \, a b e x e^{\left (d x^{n} + c\right )} - a^{2} e x}\,{d x} + \frac{e^{3 \, n - 1} x^{3 \, n}}{3 \, a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.5993, size = 5175, normalized size = 12.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{3 n - 1}}{a + b \operatorname{csch}{\left (c + d x^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{3 \, n - 1}}{b \operatorname{csch}\left (d x^{n} + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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