Optimal. Leaf size=67 \[ -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3768, 3771, 2639} \[ -\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx &=\operatorname {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))}+\operatorname {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 {\operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 54, normalized size = 0.81 \[ \sqrt {\text {csch}(2 \log (c x))} \left (-\cosh (2 \log (c x))+\sqrt {i \sinh (2 \log (c x))} E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.41, size = 163, normalized size = 2.43 \[ \frac {2 \sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (2 \ln \left (c x \right )\right )\right )}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________