Integrand size = 19, antiderivative size = 107 \[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3853, 3856, 2719} \[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]
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Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {csch}^{\frac {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\text {Subst}\left (\int \sqrt {i \sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \\ & = -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75 \[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \left (\cosh \left (a+b \log \left (c x^n\right )\right )-E\left (\left .\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
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Time = 0.90 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.98
method | result | size |
derivativedivides | \(\frac {2 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(212\) |
default | \(\frac {2 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(212\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.49 \[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \, {\left (\sqrt {2} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1}} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} + \sqrt {2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )\right )}}{b n} \]
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\[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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\[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}}{x} \,d x \]
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