Optimal. Leaf size=42 \[ -\frac {x \left (1+e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 42, 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 = {5637, 5645,
270} \begin {gather*} -\frac {x \left (e^{-2 a} \left (c x^n\right )^{-4/n}+1\right )}{2 \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 5637
Rule 5645
Rubi steps
\begin {align*} \int \frac {1}{\cosh ^{\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}}}{\cosh ^{\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 \cosh ^{\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 \cosh ^{\frac {3}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 61, normalized size = 1.45 \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 {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 2.38, size = 0, normalized size = 0.00 \[\int \frac {1}{\cosh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.46, size = 68, normalized size = 1.62 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\cosh ^{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\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.
[In]
[Out]
________________________________________________________________________________________