Optimal. Leaf size=61 \[ -\frac{\text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}\left (a x^n\right )}\right )}{2 n}+\frac{\text{sech}^{-1}\left (a x^n\right )^2}{2 n}-\frac{\text{sech}^{-1}\left (a x^n\right ) \log \left (e^{2 \text{sech}^{-1}\left (a x^n\right )}+1\right )}{n} \]
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Rubi [A] time = 0.105367, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6281, 5660, 3718, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}\left (a x^n\right )}\right )}{2 n}+\frac{\text{sech}^{-1}\left (a x^n\right )^2}{2 n}-\frac{\text{sech}^{-1}\left (a x^n\right ) \log \left (e^{2 \text{sech}^{-1}\left (a x^n\right )}+1\right )}{n} \]
Antiderivative was successfully verified.
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Rule 6281
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}\left (a x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\text{sech}^{-1}(a x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\cosh ^{-1}\left (\frac{x}{a}\right )}{x} \, dx,x,x^{-n}\right )}{n}\\ &=-\frac{\operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}\left (\frac{x^{-n}}{a}\right )\right )}{n}\\ &=\frac{\cosh ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (\frac{x^{-n}}{a}\right )\right )}{n}\\ &=\frac{\cosh ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{\cosh ^{-1}\left (\frac{x^{-n}}{a}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{n}+\frac{\operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (\frac{x^{-n}}{a}\right )\right )}{n}\\ &=\frac{\cosh ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{\cosh ^{-1}\left (\frac{x^{-n}}{a}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{n}+\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{2 n}\\ &=\frac{\cosh ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{\cosh ^{-1}\left (\frac{x^{-n}}{a}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{n}-\frac{\text{Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{2 n}\\ \end{align*}
Mathematica [B] time = 0.956552, size = 219, normalized size = 3.59 \[ \frac{\sqrt{\frac{1-a x^n}{a x^n+1}} \left (\sqrt{1-a^2 x^{2 n}} \left (-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{1-a^2 x^{2 n}}\right )+\log ^2\left (a^2 x^{2 n}\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{1-a^2 x^{2 n}}+1\right )\right )-4 \log \left (\frac{1}{2} \left (\sqrt{1-a^2 x^{2 n}}+1\right )\right ) \log \left (a^2 x^{2 n}\right )\right )+4 \sqrt{a^2 x^{2 n}-1} \left (2 n \log (x)-\log \left (a^2 x^{2 n}\right )\right ) \tan ^{-1}\left (\sqrt{a^2 x^{2 n}-1}\right )\right )}{8 \left (n-a n x^n\right )}+\log (x) \text{sech}^{-1}\left (a x^n\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.333, size = 116, normalized size = 1.9 \begin{align*}{\frac{ \left ({\rm arcsech} \left (a{x}^{n}\right ) \right ) ^{2}}{2\,n}}-{\frac{{\rm arcsech} \left (a{x}^{n}\right )}{n}\ln \left ( 1+ \left ({\frac{1}{a{x}^{n}}}+\sqrt{{\frac{1}{a{x}^{n}}}-1}\sqrt{{\frac{1}{a{x}^{n}}}+1} \right ) ^{2} \right ) }-{\frac{1}{2\,n}{\it polylog} \left ( 2,- \left ({\frac{1}{a{x}^{n}}}+\sqrt{{\frac{1}{a{x}^{n}}}-1}\sqrt{{\frac{1}{a{x}^{n}}}+1} \right ) ^{2} \right ) } \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*} a^{2} n \int \frac{x^{2 \, n} \log \left (x\right )}{a^{2} x x^{2 \, n} +{\left (a^{2} x x^{2 \, n} - x\right )} \sqrt{a x^{n} + 1} \sqrt{-a x^{n} + 1} - x}\,{d x} + n \int \frac{\log \left (x\right )}{2 \,{\left (a x x^{n} + x\right )}}\,{d x} - n \int \frac{\log \left (x\right )}{2 \,{\left (a x x^{n} - x\right )}}\,{d x} + \log \left (\sqrt{a x^{n} + 1} \sqrt{-a x^{n} + 1} + 1\right ) \log \left (x\right ) - \log \left (a\right ) \log \left (x\right ) - \log \left (x\right ) \log \left (x^{n}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\operatorname{asech}{\left (a x^{n} \right )}}{x}\, 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{\operatorname{arsech}\left (a x^{n}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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