Optimal. Leaf size=206 \[ \frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^2}-\frac {1}{4} x \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \text {csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 206, 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 = {5526, 5534, 353, 349, 345, 242, 277, 215} \[ \frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^2}-\frac {1}{4} x \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \text {csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 242
Rule 277
Rule 345
Rule 349
Rule 353
Rule 5526
Rule 5534
Rubi steps
\begin {align*} \int \cosh ^{\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 ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \cosh ^{\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} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \operatorname {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 \cosh ^{\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} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \operatorname {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 \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \cosh ^{\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} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \operatorname {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 \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \cosh ^{\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} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \operatorname {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 \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \cosh ^{\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} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \operatorname {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 \cosh ^{\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} \cosh ^{\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 \cosh ^{\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} \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \operatorname {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 \cosh ^{\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} \cosh ^{\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 \cosh ^{\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} \sinh ^{-1}\left (e^{-a} \left (c x^n\right )^{-2/n}\right ) \cosh ^{\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] time = 0.49, size = 85, normalized size = 0.41 \[ \frac {1}{14} e^{2 a} x \left (c x^n\right )^{4/n} \left (e^{2 a} \left (c x^n\right )^{4/n}+1\right ) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};e^{2 a} \left (c x^n\right )^{4/n}+1\right ) \cosh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 187, normalized size = 0.91 \[ \frac {{\left (15 \, \sqrt {2} x^{3} e^{\left (\frac {3 \, {\left (a n + 2 \, \log \relax (c)\right )}}{2 \, n}\right )} \log \left (\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} - 2 \, \sqrt {2} \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} + 1}{x^{2}}} + 2}{x^{4}}\right ) + 4 \, \sqrt {\frac {1}{2}} {\left (2 \, x^{8} e^{\left (\frac {4 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} + 14 \, x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} - 3\right )} \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \relax (c)}{2 \, n}\right )}\right )} e^{\left (-\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )}}{192 \, x^{3}} \]
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 = 0.66, size = 0, normalized size = 0.00 \[ \int \cosh ^{\frac {5}{2}}\left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )\, dx \]
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 \[ \int \cosh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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