Optimal. Leaf size=18 \[ \frac{\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.0171739, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2638} \[ \frac{\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2638
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sinh (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 )}{b n}\\ \end{align*}
Mathematica [B] time = 0.014098, size = 37, normalized size = 2.06 \[ \frac{\sinh (a) \sinh \left (b \log \left (c x^n\right )\right )}{b n}+\frac{\cosh (a) \cosh \left (b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 19, normalized size = 1.1 \begin{align*}{\frac{\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02403, size = 24, normalized size = 1.33 \begin{align*} \frac{\cosh \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.041, size = 53, normalized size = 2.94 \begin{align*} \frac{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.72233, size = 37, normalized size = 2.06 \begin{align*} \begin{cases} \log{\left (x \right )} \sinh{\left (a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \sinh{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\cosh{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10892, size = 54, normalized size = 3. \begin{align*} \frac{{\left (c^{2 \, b} x^{b n} e^{\left (2 \, a\right )} + \frac{1}{x^{b n}}\right )} e^{\left (-a\right )}}{2 \, b c^{b} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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