Optimal. Leaf size=55 \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Rubi [A] time = 0.0396532, 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} \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{sech}^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{sech}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\text{sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac{\operatorname{Subst}\left (\int \text{sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\text{sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end{align*}
Mathematica [A] time = 0.0552964, size = 55, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 51, normalized size = 0.9 \begin{align*}{\frac{{\rm sech} \left (a+b\ln \left ( c{x}^{n} \right ) \right )\tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}+{\frac{\arctan \left ({{\rm e}^{a+b\ln \left ( c{x}^{n} \right ) }} \right ) }{bn}} \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*} 8 \, c^{b} \int \frac{e^{\left (b \log \left (x^{n}\right ) + a\right )}}{8 \,{\left (c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + x\right )}}\,{d x} + \frac{c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.1923, size = 1486, normalized size = 27.02 \begin{align*} \text{result too large to display} \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}^{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 [B] time = 1.17208, size = 155, normalized size = 2.82 \begin{align*} c^{3 \, b}{\left (\frac{\arctan \left (\frac{c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-3 \, a\right )}}{b c^{2 \, b} c^{b} n} + \frac{{\left (c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - x^{b n}\right )} e^{\left (-2 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{2} b c^{2 \, b} n}\right )} e^{\left (3 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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