Optimal. Leaf size=101 \[ -\frac {x \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{6 \cosh ^{\frac {7}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {e^{-2 a} x \left (c x^n\right )^{-4/n} \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{15 \cosh ^{\frac {7}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5637, 5645,
277, 270} \begin {gather*} -\frac {e^{-2 a} x \left (c x^n\right )^{-4/n} \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{15 \cosh ^{\frac {7}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {x \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{6 \cosh ^{\frac {7}{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 277
Rule 5637
Rule 5645
Rubi steps
\begin {align*} \int \frac {1}{\cosh ^{\frac {7}{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}}}{\cosh ^{\frac {7}{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} \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {6}{n}}}{\left (1+e^{-2 a} x^{-4/n}\right )^{7/2}} \, dx,x,c x^n\right )}{n \cosh ^{\frac {7}{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 )}{6 \cosh ^{\frac {7}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}+\frac {\left (2 e^{-2 a} x \left (c x^n\right )^{6/n} \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {10}{n}}}{\left (1+e^{-2 a} x^{-4/n}\right )^{7/2}} \, dx,x,c x^n\right )}{3 n \cosh ^{\frac {7}{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 )}{6 \cosh ^{\frac {7}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {e^{-2 a} x \left (c x^n\right )^{-4/n} \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{15 \cosh ^{\frac {7}{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 = 121, normalized size = 1.20 \begin {gather*} \frac {\left (\left (2+5 x^4\right ) \cosh \left (a-2 \log (x)+\frac {2 \log \left (c x^n\right )}{n}\right )+\left (-2+5 x^4\right ) \sinh \left (a-2 \log (x)+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \left (-\cosh \left (2 a-4 \log (x)+\frac {4 \log \left (c x^n\right )}{n}\right )+\sinh \left (2 a-4 \log (x)+\frac {4 \log \left (c x^n\right )}{n}\right )\right )}{15 x^5 \cosh ^{\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 = 2.42, size = 0, normalized size = 0.00 \[\int \frac {1}{\cosh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )^{\frac {7}{2}}}\, dx\]
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 = 0.40, size = 128, normalized size = 1.27 \begin {gather*} -\frac {8 \, \sqrt {\frac {1}{2}} {\left (5 \, x^{5} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 2 \, x\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 )}}{15 \, {\left (x^{12} e^{\left (\frac {6 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 3 \, x^{8} e^{\left (\frac {4 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 3 \, x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} + 1\right )}} \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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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