Optimal. Leaf size=58 \[ -\frac{2 i \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
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Rubi [A] time = 0.0692647, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3771, 2641} \[ -\frac{2 i \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{\text{sech}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\left (\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 \frac{1}{\sqrt{\cosh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 i \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} F\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 )}}{b n}\\ \end{align*}
Mathematica [A] time = 0.0661949, size = 58, normalized size = 1. \[ -\frac{2 i \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.268, size = 183, normalized size = 3.2 \begin{align*} 2\,{\frac{\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 ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\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 EllipticF} \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) ,\sqrt{2} \right ) }{n\sqrt{2\, \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}+ \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}b}} \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{\sqrt{\operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{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{\sqrt{\operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right )}}{x}, x\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 \frac{\sqrt{\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 [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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