Optimal. Leaf size=102 \[ \frac {1}{2} x \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}-\frac {e^{-a} x \left (c x^n\right )^{-2/n} \text {csch}^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sqrt {\cosh \left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}}{2 \sqrt {1+e^{-2 a} \left (c x^n\right )^{-4/n}}} \]
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Rubi [A]
time = 0.06, 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 = {5637, 5645,
352, 248, 283, 221} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 248
Rule 283
Rule 352
Rule 5637
Rule 5645
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.22, size = 74, normalized size = 0.73 \begin {gather*} \frac {1}{2} x \left (1-\frac {\tanh ^{-1}\left (\sqrt {1+e^{2 a} \left (c x^n\right )^{4/n}}\right )}{\sqrt {1+e^{2 a} \left (c x^n\right )^{4/n}}}\right ) \sqrt {\cosh \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 = 2.48, 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.
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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 = 0.41, size = 141, normalized size = 1.38 \begin {gather*} \frac {1}{8} \, {\left (4 \, \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 )} + \sqrt {2} e^{\left (\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )} \log \left (\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 2 \, \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}}} + 2}{x^{4}}\right )\right )} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{n}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\cosh {\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.
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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 \sqrt {\mathrm {cosh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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