Optimal. Leaf size=42 \[ -\frac{x \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{2 \cosh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \]
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Rubi [A] time = 0.0498428, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5526, 5534, 264} \[ -\frac{x \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{2 \cosh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \]
Antiderivative was successfully verified.
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Rule 5526
Rule 5534
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\cosh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+\frac{1}{n}}}{\cosh ^{\frac{3}{2}}\left (a+\frac{2 \log (x)}{n}\right )} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{2/n} \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-\frac{2}{n}}}{\left (1+e^{-2 a} x^{-4/n}\right )^{3/2}} \, dx,x,c x^n\right )}{n \cosh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\\ &=-\frac{x \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \cosh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\\ \end{align*}
Mathematica [A] time = 0.149444, size = 61, normalized size = 1.45 \[ \frac{\sinh \left (a+\frac{2 \log \left (c x^n\right )}{n}-2 \log (x)\right )-\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}-2 \log (x)\right )}{x \sqrt{\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.149, size = 0, normalized size = 0. \begin{align*} \int \left ( \cosh \left ( a+2\,{\frac{\ln \left ( c{x}^{n} \right ) }{n}} \right ) \right ) ^{-{\frac{3}{2}}}\, 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 \frac{1}{\cosh \left (a + \frac{2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80027, size = 167, normalized size = 3.98 \begin{align*} -\frac{2 \, \sqrt{\frac{1}{2}} x \sqrt{\frac{x^{4} e^{\left (\frac{2 \,{\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (-\frac{a n + 2 \, \log \left (c\right )}{2 \, n}\right )}}{x^{4} e^{\left (\frac{2 \,{\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1} \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{1}{\cosh ^{\frac{3}{2}}{\left (a + \frac{2 \log{\left (c x^{n} \right )}}{n} \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 \frac{1}{\cosh \left (a + \frac{2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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