Optimal. Leaf size=97 \[ \frac{2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text{sech}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac{6 i \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]
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Rubi [A] time = 0.0702866, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3769, 3771, 2639} \[ \frac{2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text{sech}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac{6 i \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{x \text{sech}^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\text{sech}^{\frac{5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text{sech}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{\text{sech}(a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac{2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text{sech}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\left (3 \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int \sqrt{\cosh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=-\frac{6 i \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )}}{5 b n}+\frac{2 \sinh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text{sech}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.128879, size = 87, normalized size = 0.9 \[ \frac{\sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \left (\sinh \left (a+b \log \left (c x^n\right )\right )+\sinh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-12 i \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{10 b n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.345, size = 256, normalized size = 2.6 \begin{align*}{\frac{2}{5\,bn}\sqrt{ \left ( 2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}} \left ( 8\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{7}-16\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{5}+10\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}-3\,\sqrt{- \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) ,\sqrt{2} \right ) -2\,\cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ){\frac{1}{\sqrt{2\, \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}}}} \left ( \sinh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}}}} \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}{x \operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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