Optimal. Leaf size=73 \[ \frac{\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{3 \log (x)}{8} \]
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Rubi [A] time = 0.0471974, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac{\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{3 \log (x)}{8} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cosh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cosh ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int \cosh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac{3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac{3 \log (x)}{8}+\frac{3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end{align*}
Mathematica [A] time = 0.0458092, size = 51, normalized size = 0.7 \[ \frac{12 \left (a+b \log \left (c x^n\right )\right )+8 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 84, normalized size = 1.2 \begin{align*}{\frac{ \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{4\,bn}}+{\frac{3\,\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{8\,bn}}+{\frac{3\,\ln \left ( c{x}^{n} \right ) }{8\,n}}+{\frac{3\,a}{8\,bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04559, size = 126, normalized size = 1.73 \begin{align*} \frac{e^{\left (4 \, b \log \left (c x^{n}\right ) + 4 \, a\right )}}{64 \, b n} + \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{e^{\left (-4 \, b \log \left (c x^{n}\right ) - 4 \, a\right )}}{64 \, b n} + \frac{3}{8} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87579, size = 270, normalized size = 3.7 \begin{align*} \frac{\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 )^{3} + 3 \, b n \log \left (x\right ) +{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, 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{\cosh ^{4}{\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 [A] time = 1.2682, size = 154, normalized size = 2.11 \begin{align*} \frac{{\left (24 \, b c^{4 \, b} n e^{\left (4 \, a\right )} \log \left (x\right ) + c^{8 \, b} x^{4 \, b n} e^{\left (8 \, a\right )} + 8 \, c^{6 \, b} x^{2 \, b n} e^{\left (6 \, a\right )} - \frac{18 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 8 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{4 \, b n}}\right )} e^{\left (-4 \, a\right )}}{64 \, b c^{4 \, b} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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