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 = 0.0292156, antiderivative size = 39, normalized size of antiderivative = 1., 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 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sinh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sinh ^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*}
Mathematica [A] time = 0.0273888, size = 36, normalized size = 0.92 \[ \frac{\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 52, normalized size = 1.3 \begin{align*}{\frac{\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}-{\frac{\ln \left ( c{x}^{n} \right ) }{2\,n}}-{\frac{a}{2\,bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04754, size = 66, normalized size = 1.69 \begin{align*} \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 \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06571, size = 123, normalized size = 3.15 \begin{align*} -\frac{b n \log \left (x\right ) - \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{2 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21402, size = 109, normalized size = 2.79 \begin{align*} -\frac{{\left (4 \, b c^{2 \, b} n e^{\left (2 \, a\right )} \log \left (x\right ) - 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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