Optimal. Leaf size=55 \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rubi [A] time = 0.0384104, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sec ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac{\operatorname{Subst}\left (\int \sec (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end{align*}
Mathematica [A] time = 0.0726142, size = 55, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 64, normalized size = 1.2 \begin{align*}{\frac{\sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}+{\frac{\ln \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +\tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{2\,bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.507579, size = 311, normalized size = 5.65 \begin{align*} \frac{\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (-\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) + 2 \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}} \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{\sec ^{3}{\left (a + b \log{\left (c x^{n} \right )} \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{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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