Optimal. Leaf size=89 \[ \frac{e^{2 a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (e^{-2 a} \left (c x^n\right )^{\frac{2}{n (2-p)}}+1\right ) \text{sech}^p\left (a-\frac{\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Rubi [A] time = 0.0894543, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5545, 5549, 261} \[ \frac{e^{2 a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (e^{-2 a} \left (c x^n\right )^{\frac{2}{n (2-p)}}+1\right ) \text{sech}^p\left (a-\frac{\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
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Rule 5545
Rule 5549
Rule 261
Rubi steps
\begin{align*} \int \text{sech}^p\left (a+\frac{\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \text{sech}^p\left (a+\frac{\log (x)}{n (-2+p)}\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{1}{n}+\frac{p}{n (-2+p)}} \left (1+e^{-2 a} \left (c x^n\right )^{-\frac{2}{n (-2+p)}}\right )^p \text{sech}^p\left (a+\frac{\log \left (c x^n\right )}{n (-2+p)}\right )\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}-\frac{p}{n (-2+p)}} \left (1+e^{-2 a} x^{-\frac{2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=\frac{e^{2 a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (1+e^{-2 a} \left (c x^n\right )^{\frac{2}{n (2-p)}}\right ) \text{sech}^p\left (a-\frac{\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\\ \end{align*}
Mathematica [A] time = 0.715443, size = 57, normalized size = 0.64 \[ \frac{(p-2) x \left (e^{2 a} \left (c x^n\right )^{\frac{2}{n (p-2)}}+1\right ) \text{sech}^p\left (a+\frac{\log \left (c x^n\right )}{n (p-2)}\right )}{2 (p-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.165, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (a+{\frac{\ln \left ( c{x}^{n} \right ) }{n \left ( -2+p \right ) }}\right ) \right ) ^{p}\, dx \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*} \int \operatorname{sech}\left (a + \frac{\log \left (c x^{n}\right )}{n{\left (p - 2\right )}}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.20621, size = 1322, normalized size = 14.85 \begin{align*} \frac{{\left (p - 2\right )} x \cosh \left (p \log \left (\frac{2 \,{\left (\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 1}\right )\right ) \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) +{\left (p - 2\right )} x \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (p \log \left (\frac{2 \,{\left (\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 1}\right )\right )}{{\left (p - 1\right )} \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) -{\left (p - 1\right )} \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )} \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 \operatorname{sech}^{p}{\left (a + \frac{\log{\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, 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 \operatorname{sech}\left (a + \frac{\log \left (c x^{n}\right )}{n{\left (p - 2\right )}}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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