Optimal. Leaf size=42 \[ \frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rubi [A] time = 0.0329257, antiderivative size = 42, 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 = {3767} \[ \frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \frac{\text{csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end{align*}
Mathematica [A] time = 0.0648677, size = 56, normalized size = 1.33 \[ \frac{2 \coth \left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\coth \left (a+b \log \left (c x^n\right )\right ) \text{csch}^2\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 36, normalized size = 0.9 \begin{align*}{\frac{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}{bn} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \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 = 1.13217, size = 124, normalized size = 2.95 \begin{align*} -\frac{4 \,{\left (3 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1\right )}}{3 \,{\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.93634, size = 869, normalized size = 20.69 \begin{align*} -\frac{8 \,{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 2 \, \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}}{3 \,{\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 5 \, b n \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 )^{4} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} - 3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} +{\left (10 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \, b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (10 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 9 \, b n \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 )^{2} +{\left (5 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 9 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}} \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{\operatorname{csch}^{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 = 1.17783, size = 63, normalized size = 1.5 \begin{align*} -\frac{4 \,{\left (3 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}}{3 \,{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{3} b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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