Optimal. Leaf size=307 \[ -\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 e n \sqrt{b^2-a^2}}+\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 e n \sqrt{b^2-a^2}}-\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}+1\right )}{a d e n \sqrt{b^2-a^2}}+\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}+1\right )}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{2 n}}{2 a e n} \]
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Rubi [A] time = 0.564444, antiderivative size = 307, 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 = {5440, 5436, 4191, 3320, 2264, 2190, 2279, 2391} \[ -\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 e n \sqrt{b^2-a^2}}+\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 e n \sqrt{b^2-a^2}}-\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{b^2-a^2}}+1\right )}{a d e n \sqrt{b^2-a^2}}+\frac{b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{b^2-a^2}+b}+1\right )}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{2 n}}{2 a e n} \]
Antiderivative was successfully verified.
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Rule 5440
Rule 5436
Rule 4191
Rule 3320
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(e x)^{-1+2 n}}{a+b \text{sech}\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{sech}\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{sech}(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 \cosh (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 \cosh (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 = 2.03922, size = 859, normalized size = 2.8 \[ \frac{(e x)^{2 n} \left (b+a \cosh \left (d x^n+c\right )\right ) \left (\frac{2 b \left (2 \left (d x^n+c\right ) \tan ^{-1}\left (\frac{(a+b) \coth \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )+2 \left (c-i \cos ^{-1}\left (-\frac{b}{a}\right )\right ) \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )+\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 \left (\tan ^{-1}\left (\frac{(a+b) \coth \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )+\tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right )\right ) \log \left (\frac{\sqrt{a^2-b^2} e^{-\frac{d x^n}{2}-\frac{c}{2}}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cosh \left (d x^n+c\right )}}\right )+\left (\cos ^{-1}\left (-\frac{b}{a}\right )-2 \left (\tan ^{-1}\left (\frac{(a+b) \coth \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )+\tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right )\right ) \log \left (\frac{\sqrt{a^2-b^2} e^{\frac{1}{2} \left (d x^n+c\right )}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cosh \left (d x^n+c\right )}}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{(a+b) \left (-a+b+i \sqrt{a^2-b^2}\right ) \left (\tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )-1\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )-2 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{(a+b) \left (a-b+i \sqrt{a^2-b^2}\right ) \left (\tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )+1\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (b-i \sqrt{a^2-b^2}\right ) \left (a+b-i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (b+i \sqrt{a^2-b^2}\right ) \left (a+b-i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )\right )}{a \left (a+b+i \sqrt{a^2-b^2} \tanh \left (\frac{1}{2} \left (d x^n+c\right )\right )\right )}\right )\right )\right ) x^{-2 n}}{\sqrt{a^2-b^2} d^2}+1\right ) \text{sech}\left (d x^n+c\right )}{2 a e n \left (a+b \text{sech}\left (d x^n+c\right )\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.117, size = 585, normalized size = 1.9 \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 = 2.42191, size = 3371, normalized size = 10.98 \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{sech}{\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{sech}\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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