Optimal. Leaf size=43 \[ \frac{\cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.0353946, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2633} \[ \frac{\cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \frac{\sinh ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sinh ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac{\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac{\cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end{align*}
Mathematica [A] time = 0.0138707, size = 45, normalized size = 1.05 \[ \frac{\cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{12 b n}-\frac{3 \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 36, normalized size = 0.8 \begin{align*}{\frac{\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03604, size = 116, normalized size = 2.7 \begin{align*} \frac{e^{\left (3 \, b \log \left (c x^{n}\right ) + 3 \, a\right )}}{24 \, b n} - \frac{3 \, e^{\left (b \log \left (c x^{n}\right ) + a\right )}}{8 \, b n} - \frac{3 \, e^{\left (-b \log \left (c x^{n}\right ) - a\right )}}{8 \, b n} + \frac{e^{\left (-3 \, b \log \left (c x^{n}\right ) - 3 \, a\right )}}{24 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09657, size = 208, normalized size = 4.84 \begin{align*} \frac{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \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} - 9 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{12 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.0679, size = 82, normalized size = 1.91 \begin{align*} \begin{cases} \log{\left (x \right )} \sinh ^{3}{\left (a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \sinh ^{3}{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\sinh ^{2}{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )} \cosh{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} - \frac{2 \cosh ^{3}{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{3 b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18802, size = 109, normalized size = 2.53 \begin{align*} \frac{{\left (c^{6 \, b} x^{3 \, b n} e^{\left (6 \, a\right )} - 9 \, c^{4 \, b} x^{b n} e^{\left (4 \, a\right )} - \frac{9 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1}{x^{3 \, b n}}\right )} e^{\left (-3 \, a\right )}}{24 \, b c^{3 \, b} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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