Optimal. Leaf size=43 \[ -\frac {x \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \sinh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5525, 5533, 264} \[ -\frac {x \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \sinh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]
Antiderivative was successfully verified.
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Rule 264
Rule 5525
Rule 5533
Rubi steps
\begin {align*} \int \frac {1}{\sinh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+\frac {1}{n}}}{\sinh ^{\frac {3}{2}}\left (a+\frac {2 \log (x)}{n}\right )} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{2/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^{-1-\frac {2}{n}}}{\left (1-e^{-2 a} x^{-4/n}\right )^{3/2}} \, dx,x,c x^n\right )}{n \sinh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}\\ &=-\frac {x \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \sinh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 61, normalized size = 1.42 \[ \frac {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}-2 \log (x)\right )-\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}-2 \log (x)\right )}{x \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 68, normalized size = 1.58 \[ -\frac {2 \, \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} - 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \relax (c)}{2 \, n}\right )}}{x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} - 1} \]
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 [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sinh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{\sinh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {sinh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\sinh ^{\frac {3}{2}}{\left (a + \frac {2 \log {\left (c x^{n} \right )}}{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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