Optimal. Leaf size=209 \[ -\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \csc ^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5636, 5644,
360, 356, 352, 248, 283, 222} \begin {gather*} -\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \csc ^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 248
Rule 283
Rule 352
Rule 356
Rule 360
Rule 5636
Rule 5644
Rubi steps
\begin {align*} \int \sinh ^{\frac {5}{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 x^{-1+\frac {1}{n}} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log (x)}{n}\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {6}{n}} \left (1-e^{-2 a} x^{-4/n}\right )^{5/2} \, dx,x,c x^n\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\\ &=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {\left (5 x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {6}{n}} \left (1-e^{-2 a} x^{-4/n}\right )^{3/2} \, dx,x,c x^n\right )}{2 n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\\ &=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {\left (5 e^{-2 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sqrt {1-e^{-2 a} x^{-4/n}} \, dx,x,c x^n\right )}{2 n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\\ &=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {\left (5 e^{-2 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int \sqrt {1-\frac {e^{-2 a}}{x^2}} \, dx,x,\left (c x^n\right )^{2/n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\\ &=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}+\frac {\left (5 e^{-2 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {1-e^{-2 a} x^2}}{x^2} \, dx,x,\left (c x^n\right )^{-2/n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\\ &=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {\left (5 e^{-4 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-e^{-2 a} x^2}} \, dx,x,\left (c x^n\right )^{-2/n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\\ &=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \sin ^{-1}\left (e^{-a} \left (c x^n\right )^{-2/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.21, size = 86, normalized size = 0.41 \begin {gather*} \frac {1}{14} e^{2 a} x \left (c x^n\right )^{4/n} \left (-1+e^{2 a} \left (c x^n\right )^{4/n}\right ) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};1-e^{2 a} \left (c x^n\right )^{4/n}\right ) \sinh ^{\frac {5}{2}}\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 = 2.21, size = 0, normalized size = 0.00 \[\int \sinh ^{\frac {5}{2}}\left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )\, dx\]
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 = 0.50, size = 162, normalized size = 0.78 \begin {gather*} \frac {{\left (15 \, \sqrt {2} x^{3} \arctan \left (\sqrt {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}}}\right ) e^{\left (\frac {3 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{2 \, n}\right )} + 2 \, \sqrt {\frac {1}{2}} {\left (2 \, x^{8} e^{\left (\frac {4 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 14 \, x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 3\right )} \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 )}\right )} e^{\left (-\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )}}{96 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int {\mathrm {sinh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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