Optimal. Leaf size=39 \[ \frac{i \text{PolyLog}\left (2,-i a x^n\right )}{2 n}-\frac{i \text{PolyLog}\left (2,i a x^n\right )}{2 n} \]
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Rubi [A] time = 0.0358367, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5031, 4848, 2391} \[ \frac{i \text{PolyLog}\left (2,-i a x^n\right )}{2 n}-\frac{i \text{PolyLog}\left (2,i a x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 5031
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (a x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\tan ^{-1}(a x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{\log (1-i a x)}{x} \, dx,x,x^n\right )}{2 n}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1+i a x)}{x} \, dx,x,x^n\right )}{2 n}\\ &=\frac{i \text{Li}_2\left (-i a x^n\right )}{2 n}-\frac{i \text{Li}_2\left (i a x^n\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0131773, size = 32, normalized size = 0.82 \[ \frac{i \left (\text{PolyLog}\left (2,-i a x^n\right )-\text{PolyLog}\left (2,i a x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 94, normalized size = 2.4 \begin{align*}{\frac{\ln \left ( a{x}^{n} \right ) \arctan \left ( a{x}^{n} \right ) }{n}}+{\frac{{\frac{i}{2}}\ln \left ( a{x}^{n} \right ) \ln \left ( 1+ia{x}^{n} \right ) }{n}}-{\frac{{\frac{i}{2}}\ln \left ( a{x}^{n} \right ) \ln \left ( 1-ia{x}^{n} \right ) }{n}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+ia{x}^{n} \right ) }{n}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1-ia{x}^{n} \right ) }{n}} \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 n \int \frac{x^{n} \log \left (x\right )}{a^{2} x x^{2 \, n} + x}\,{d x} + \arctan \left (a x^{n}\right ) \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.56211, size = 181, normalized size = 4.64 \begin{align*} \frac{2 \, n \arctan \left (a x^{n}\right ) \log \left (x\right ) + i \, n \log \left (i \, a x^{n} + 1\right ) \log \left (x\right ) - i \, n \log \left (-i \, a x^{n} + 1\right ) \log \left (x\right ) - i \,{\rm Li}_2\left (i \, a x^{n}\right ) + i \,{\rm Li}_2\left (-i \, a x^{n}\right )}{2 \, n} \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{atan}{\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{\arctan \left (a x^{n}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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