Optimal. Leaf size=102 \[ \frac {1}{2} x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {e^{-a} x \left (c x^n\right )^{-2/n} \text {csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5526, 5534, 345, 242, 277, 215} \[ \frac {1}{2} x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {e^{-a} x \left (c x^n\right )^{-2/n} \text {csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {e^{-2 a} \left (c x^n\right )^{-4/n}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 242
Rule 277
Rule 345
Rule 5526
Rule 5534
Rubi steps
\begin {align*} \int \sqrt {\cosh \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}} \sqrt {\cosh \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} \sqrt {\cosh \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 )}{n \sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=\frac {\left (x \left (c x^n\right )^{-2/n} \sqrt {\cosh \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 )}{2 \sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=-\frac {\left (x \left (c x^n\right )^{-2/n} \sqrt {\cosh \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 )}{2 \sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=\frac {1}{2} x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {\left (e^{-2 a} x \left (c x^n\right )^{-2/n} \sqrt {\cosh \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 )}{2 \sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ &=\frac {1}{2} x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {e^{-a} x \left (c x^n\right )^{-2/n} \sinh ^{-1}\left (e^{-a} \left (c x^n\right )^{-2/n}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.33, size = 74, normalized size = 0.73 \[ \frac {1}{2} x \left (1-\frac {\tanh ^{-1}\left (\sqrt {e^{2 a} \left (c x^n\right )^{4/n}+1}\right )}{\sqrt {e^{2 a} \left (c x^n\right )^{4/n}+1}}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 141, normalized size = 1.38 \[ \frac {1}{8} \, {\left (4 \, \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} + 1}{x^{2}}} e^{\left (\frac {a n + 2 \, \log \relax (c)}{2 \, n}\right )} + \sqrt {2} e^{\left (\frac {a n + 2 \, \log \relax (c)}{2 \, n}\right )} \log \left (\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} - 2 \, \sqrt {2} \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \relax (c)\right )}}{n}\right )} + 1}{x^{2}}} + 2}{x^{4}}\right )\right )} e^{\left (-\frac {a n + 2 \, \log \relax (c)}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \sqrt {\cosh }\left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cosh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cosh {\left (a + \frac {2 \log {\left (c x^{n} \right )}}{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________