Optimal. Leaf size=46 \[ i \sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))} F\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 2641
Rule 3771
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*}
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Mathematica [A] time = 0.07, size = 43, normalized size = 0.93 \[ (i \sinh (2 \log (c x)))^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x)) F\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 90, normalized size = 1.96 \[ \frac {i \sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \EllipticF \left (\sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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