Optimal. Leaf size=19 \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.0162663, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3770} \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{sech}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0480611, size = 19, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 20, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07044, size = 26, normalized size = 1.37 \begin{align*} \frac{\arctan \left (\sinh \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.13184, size = 112, normalized size = 5.89 \begin{align*} \frac{2 \, \arctan \left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b 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{sech}{\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 [A] time = 1.12854, size = 36, normalized size = 1.89 \begin{align*} \frac{2 \, \arctan \left (\frac{c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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