Optimal. Leaf size=111 \[ \frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{5 b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]
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Rubi [A] time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3769, 3771, 2639} \[ \frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{5 b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\text {csch}^{\frac {5}{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 )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3 \operatorname {Subst}\left (\int \sqrt {i \sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 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 )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 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 )}{5 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*}
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Mathematica [A] time = 0.19, size = 95, normalized size = 0.86 \[ \frac {2 \left (\cosh \left (a+b \log \left (c x^n\right )\right )-3 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) E\left (\left .\frac {1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right )\right |2\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 227, normalized size = 2.05 \[ \frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 \left (\cosh ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{5}-\frac {2 \left (\cosh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]
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 \[ \int \frac {1}{x \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\left (\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{x \operatorname {csch}^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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