Optimal. Leaf size=46 \[ i \sqrt{i \sinh (2 \log (c x))} \sqrt{\text{csch}(2 \log (c x))} \text{EllipticF}\left (\frac{\pi }{4}-i \log (c x),2\right ) \]
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Rubi [A] time = 0.0321486, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3771, 2641} \[ i \sqrt{i \sinh (2 \log (c x))} \sqrt{\text{csch}(2 \log (c x))} F\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right ) \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sqrt{\text{csch}(2 \log (c x))}}{x} \, dx &=\operatorname{Subst}\left (\int \sqrt{\text{csch}(2 x)} \, dx,x,\log (c x)\right )\\ &=\left (\sqrt{\text{csch}(2 \log (c x))} \sqrt{i \sinh (2 \log (c x))}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{i \sinh (2 x)}} \, dx,x,\log (c x)\right )\\ &=i \sqrt{\text{csch}(2 \log (c x))} F\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right ) \sqrt{i \sinh (2 \log (c x))}\\ \end{align*}
Mathematica [A] time = 0.0680655, size = 43, normalized size = 0.93 \[ (i \sinh (2 \log (c x)))^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x)) \text{EllipticF}\left (\frac{\pi }{4}-i \log (c x),2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.168, size = 90, normalized size = 2. \begin{align*}{\frac{{\frac{i}{2}}\sqrt{2}}{\cosh \left ( 2\,\ln \left ( cx \right ) \right ) }\sqrt{-i \left ( i+\sinh \left ( 2\,\ln \left ( cx \right ) \right ) \right ) }\sqrt{-i \left ( -\sinh \left ( 2\,\ln \left ( cx \right ) \right ) +i \right ) }\sqrt{i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }{\it EllipticF} \left ( \sqrt{-i \left ( i+\sinh \left ( 2\,\ln \left ( cx \right ) \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }}}} \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{csch}\left (2 \, \log \left (c x\right )\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{csch}\left (2 \, \log \left (c x\right )\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{csch}{\left (2 \log{\left (c x \right )} \right )}}}{x}\, 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{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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