Optimal. Leaf size=47 \[ \frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
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Rubi [A] time = 0.0398971, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3476, 329, 212, 206, 203} \[ \frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\coth (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (-1+x^2\right )} \, dx,x,\coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.110676, size = 47, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac{\tanh ^{-1}\left (\sqrt{\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{bn}\arctan \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )} \right ) }+{\frac{1}{bn}{\it Artanh} \left ( \sqrt{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\coth \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.34765, size = 1000, normalized size = 21.28 \begin{align*} -\frac{2 \, \arctan \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \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 ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \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 ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt{\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 )}}\right ) + \log \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \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 ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \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 ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt{\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 )}}\right )}{2 \, 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{1}{x \sqrt{\coth{\left (a + b \log{\left (c x^{n} \right )} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\coth \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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