Optimal. Leaf size=69 \[ \frac{i \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{2 n}+\frac{i \sec ^{-1}\left (a x^n\right )^2}{2 n}-\frac{\sec ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{n} \]
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Rubi [A] time = 0.0925888, antiderivative size = 69, 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 = {5218, 4626, 3719, 2190, 2279, 2391} \[ \frac{i \text{PolyLog}\left (2,-e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{2 n}+\frac{i \sec ^{-1}\left (a x^n\right )^2}{2 n}-\frac{\sec ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \sec ^{-1}\left (a x^n\right )}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 5218
Rule 4626
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sec ^{-1}\left (a x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sec ^{-1}(a x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x} \, dx,x,x^{-n}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (\frac{x^{-n}}{a}\right )\right )}{n}\\ &=\frac{i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (\frac{x^{-n}}{a}\right )\right )}{n}\\ &=\frac{i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{\cos ^{-1}\left (\frac{x^{-n}}{a}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{n}+\frac{\operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (\frac{x^{-n}}{a}\right )\right )}{n}\\ &=\frac{i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{\cos ^{-1}\left (\frac{x^{-n}}{a}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{n}-\frac{i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{2 n}\\ &=\frac{i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )^2}{2 n}-\frac{\cos ^{-1}\left (\frac{x^{-n}}{a}\right ) \log \left (1+e^{2 i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{n}+\frac{i \text{Li}_2\left (-e^{2 i \cos ^{-1}\left (\frac{x^{-n}}{a}\right )}\right )}{2 n}\\ \end{align*}
Mathematica [C] time = 0.0843848, size = 60, normalized size = 0.87 \[ \frac{x^{-n} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},\frac{x^{-2 n}}{a^2}\right )}{a n}+\log (x) \left (\sin ^{-1}\left (\frac{x^{-n}}{a}\right )+\sec ^{-1}\left (a x^n\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.336, size = 98, normalized size = 1.4 \begin{align*}{\frac{{\frac{i}{2}} \left ({\rm arcsec} \left (a{x}^{n}\right ) \right ) ^{2}}{n}}-{\frac{{\rm arcsec} \left (a{x}^{n}\right )}{n}\ln \left ( 1+ \left ({\frac{1}{a{x}^{n}}}+i\sqrt{1-{\frac{1}{{a}^{2} \left ({x}^{n} \right ) ^{2}}}} \right ) ^{2} \right ) }+{\frac{{\frac{i}{2}}}{n}{\it polylog} \left ( 2,- \left ({\frac{1}{a{x}^{n}}}+i\sqrt{1-{\frac{1}{{a}^{2} \left ({x}^{n} \right ) ^{2}}}} \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{\sqrt{a x^{n} + 1} \sqrt{a x^{n} - 1} \log \left (x\right )}{a^{4} x x^{2 \, n} - a^{2} x}\,{d x} + \arctan \left (\sqrt{a x^{n} + 1} \sqrt{a x^{n} - 1}\right ) \log \left (x\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{asec}{\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{arcsec}\left (a x^{n}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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