Optimal. Leaf size=291 \[ -\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a d^2 e n \sqrt{a^2+b^2}}+\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a d^2 e n \sqrt{a^2+b^2}}-\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{(e x)^{2 n}}{2 a e n} \]
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Rubi [A] time = 0.545208, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5441, 5437, 4191, 3322, 2264, 2190, 2279, 2391} \[ -\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a d^2 e n \sqrt{a^2+b^2}}+\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a d^2 e n \sqrt{a^2+b^2}}-\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a d e n \sqrt{a^2+b^2}}+\frac{(e x)^{2 n}}{2 a e n} \]
Antiderivative was successfully verified.
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Rule 5441
Rule 5437
Rule 4191
Rule 3322
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(e x)^{-1+2 n}}{a+b \text{csch}\left (c+d x^n\right )} \, dx &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac{x^{-1+2 n}}{a+b \text{csch}\left (c+d x^n\right )} \, dx}{e}\\ &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x}{a+b \text{csch}(c+d x)} \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \left (\frac{x}{a}-\frac{b x}{a (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{\left (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname{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 e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{\left (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname{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 )}{\sqrt{a^2+b^2} e n}+\frac{\left (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname{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 )}{\sqrt{a^2+b^2} e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{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 \sqrt{a^2+b^2} d e n}+\frac{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 \sqrt{a^2+b^2} d e n}+\frac{\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname{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 \sqrt{a^2+b^2} d e n}-\frac{\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname{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 \sqrt{a^2+b^2} d e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{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 \sqrt{a^2+b^2} d e n}+\frac{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 \sqrt{a^2+b^2} d e n}+\frac{\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname{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 \sqrt{a^2+b^2} d^2 e n}-\frac{\left (b x^{-2 n} (e x)^{2 n}\right ) \operatorname{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 \sqrt{a^2+b^2} d^2 e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{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 \sqrt{a^2+b^2} d e n}+\frac{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 \sqrt{a^2+b^2} d e n}-\frac{b x^{-2 n} (e x)^{2 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{b x^{-2 n} (e x)^{2 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}\\ \end{align*}
Mathematica [C] time = 4.03708, size = 1180, normalized size = 4.05 \[ \frac{(e x)^{2 n} \text{csch}\left (d x^n+c\right ) \left (\frac{2 b \left (\frac{i \pi \tanh ^{-1}\left (\frac{b \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )-a}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}}+\frac{2 \left (c+i \cos ^{-1}\left (-\frac{i b}{a}\right )\right ) \tan ^{-1}\left (\frac{(a-i b) \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )+\left (-2 i d x^n-2 i c+\pi \right ) \tanh ^{-1}\left (\frac{(b-i a) \tan \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )-\left (\cos ^{-1}\left (-\frac{i b}{a}\right )-2 \tan ^{-1}\left (\frac{(a-i b) \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )\right ) \log \left (\frac{(a+i b) \left (a-i b+\sqrt{-a^2-b^2}\right ) \left (i \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )+1\right )}{a \left (a+i b+i \sqrt{-a^2-b^2} \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{i b}{a}\right )+2 \tan ^{-1}\left (\frac{(a-i b) \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )\right ) \log \left (\frac{i (a+i b) \left (-a+i b+\sqrt{-a^2-b^2}\right ) \left (\cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )+i\right )}{a \left (a+i b+i \sqrt{-a^2-b^2} \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )\right )}\right )+\left (\cos ^{-1}\left (-\frac{i b}{a}\right )+2 \tan ^{-1}\left (\frac{(a-i b) \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )-2 i \tanh ^{-1}\left (\frac{(b-i a) \tan \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )\right ) \log \left (-\frac{(-1)^{3/4} \sqrt{-a^2-b^2} e^{-\frac{d x^n}{2}-\frac{c}{2}}}{\sqrt{2} \sqrt{-i a} \sqrt{b+a \sinh \left (d x^n+c\right )}}\right )+\left (\cos ^{-1}\left (-\frac{i b}{a}\right )-2 \tan ^{-1}\left (\frac{(a-i b) \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )+2 i \tanh ^{-1}\left (\frac{(b-i a) \tan \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )}{\sqrt{-a^2-b^2}}\right )\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{-a^2-b^2} e^{\frac{1}{2} \left (d x^n+c\right )}}{\sqrt{2} \sqrt{-i a} \sqrt{b+a \sinh \left (d x^n+c\right )}}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (i b+\sqrt{-a^2-b^2}\right ) \left (a+i b-i \sqrt{-a^2-b^2} \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )\right )}{a \left (a+i b+i \sqrt{-a^2-b^2} \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (b+i \sqrt{-a^2-b^2}\right ) \left (i a-b+\sqrt{-a^2-b^2} \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )\right )}{a \left (a+i b+i \sqrt{-a^2-b^2} \cot \left (\frac{1}{4} \left (2 i d x^n+2 i c+\pi \right )\right )\right )}\right )\right )}{\sqrt{-a^2-b^2}}\right ) x^{-2 n}}{d^2}+1\right ) \left (b+a \sinh \left (d x^n+c\right )\right )}{2 a e n \left (a+b \text{csch}\left (d x^n+c\right )\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.128, size = 577, normalized size = 2. \begin{align*} \text{result too large to display} \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^{2 \, n} \int \frac{e^{\left (d x^{n} + 2 \, 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^{2 \, n - 1} x^{2 \, n}}{2 \, a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.48899, size = 3330, normalized size = 11.44 \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 )^{2 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 )^{2 \, 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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