Integrand size = 15, antiderivative size = 67 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {\text {csch}(2 \log (c x))} \sqrt {i \sinh (2 \log (c x))}} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3853, 3856, 2719} \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))}} \]
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Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \text {csch}^{\frac {3}{2}}(2 x) \, dx,x,\log (c x)\right ) \\ & = -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\text {Subst}\left (\int \frac {1}{\sqrt {\text {csch}(2 x)}} \, dx,x,\log (c x)\right ) \\ & = -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {\text {Subst}\left (\int \sqrt {i \sinh (2 x)} \, dx,x,\log (c x)\right )}{\sqrt {\text {csch}(2 \log (c x))} \sqrt {i \sinh (2 \log (c x))}} \\ & = -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {\text {csch}(2 \log (c x))} \sqrt {i \sinh (2 \log (c x))}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\sqrt {\text {csch}(2 \log (c x))} \left (-\cosh (2 \log (c x))+E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right ) \sqrt {i \sinh (2 \log (c x))}\right ) \]
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Time = 0.45 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.43
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (2 \ln \left (c x \right )\right )^{2}}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}}\) | \(163\) |
default | \(\frac {2 \sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (2 \ln \left (c x \right )\right )^{2}}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}}\) | \(163\) |
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Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} x^{2} - i \, \sqrt {2} c^{2} E(\arcsin \left (c x\right )\,|\,-1) + i \, \sqrt {2} c^{2} F(\arcsin \left (c x\right )\,|\,-1) \]
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\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \]
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\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int { \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \]
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