Optimal. Leaf size=39 \[ \frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log (x)}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cosh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\operatorname {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac {\log (x)}{2}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.92 \[ \frac {2 \left (a+b \log \left (c x^n\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 39, normalized size = 1.00 \[ \frac {b n \log \relax (x) + \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{2 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 80, normalized size = 2.05 \[ \frac {{\left (4 \, b c^{2 \, b} n e^{\left (2 \, a\right )} \log \relax (x) + c^{4 \, b} x^{2 \, b n} e^{\left (4 \, a\right )} - \frac {2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{2 \, b n}}\right )} e^{\left (-2 \, a\right )}}{8 \, b c^{2 \, b} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 52, normalized size = 1.33 \[ \frac {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}+\frac {\ln \left (c \,x^{n}\right )}{2 n}+\frac {a}{2 b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 49, normalized size = 1.26 \[ \frac {e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} + \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 32, normalized size = 0.82 \[ \frac {\ln \left (x^n\right )}{2\,n}+\frac {\mathrm {sinh}\left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4\,b\,n} \]
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 \[ \int \frac {\cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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