Optimal. Leaf size=20 \[ -\frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.0175591, antiderivative size = 20, 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 = {3770} \[ -\frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end{align*}
Mathematica [B] time = 0.0638428, size = 54, normalized size = 2.7 \[ \frac{\log \left (\sinh \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\log \left (\cosh \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 23, normalized size = 1.2 \begin{align*}{\frac{1}{bn}\ln \left ( \tanh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09665, size = 30, normalized size = 1.5 \begin{align*} \frac{\log \left (\tanh \left (\frac{1}{2} \, b \log \left (c x^{n}\right ) + \frac{1}{2} \, a\right )\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84175, size = 219, normalized size = 10.95 \begin{align*} -\frac{\log \left (\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 ) + 1\right ) - \log \left (\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 ) - 1\right )}{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{\operatorname{csch}{\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 = 1.21073, size = 190, normalized size = 9.5 \begin{align*} -\frac{1}{2} \, c^{b}{\left (\frac{c^{b} e^{\left (-a\right )} \log \left (2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \frac{1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n}{\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1\right )}{b c^{2 \, b} n} - \frac{c^{b} e^{\left (-a\right )} \log \left (-2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \frac{1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n}{\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1\right )}{b c^{2 \, b} n}\right )} e^{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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