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 = 0.04, 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 = {5636, 5644,
270} \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 5636
Rule 5644
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 ) \text {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 ) \text {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 = 0.09, size = 61, normalized size = 1.42 \begin {gather*} \frac {-\cosh \left (a-2 \log (x)+\frac {2 \log \left (c x^n\right )}{n}\right )+\sinh \left (a-2 \log (x)+\frac {2 \log \left (c x^n\right )}{n}\right )}{x \sqrt {\sinh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.25, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 68, normalized size = 1.58 \begin {gather*} -\frac {2 \, \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )}}{x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sinh ^{\frac {3}{2}}{\left (a + \frac {2 \log {\left (c x^{n} \right )}}{n} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 41, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {2}}{\sqrt {c^{\frac {4}{n}} e^{\left (3 \, a\right )} - \frac {e^{a}}{x^{4}}} c^{\left (\frac {1}{n}\right )} x^{2} \mathrm {sgn}\left (x\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\mathrm {sinh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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