Optimal. Leaf size=28 \[ \log (x)-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
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Rubi [A] time = 0.0291035, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3473, 8} \[ \log (x)-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \coth ^2(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\log (x)\\ \end{align*}
Mathematica [C] time = 0.103764, size = 49, normalized size = 1.75 \[ -\frac{\coth \left (a d+b d \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2\left (a d+b \log \left (c x^n\right ) d\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 80, normalized size = 2.9 \begin{align*} -{\frac{{\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )}{dbn}}-{\frac{\ln \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )-1 \right ) }{2\,dbn}}+{\frac{\ln \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )+1 \right ) }{2\,dbn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30048, size = 50, normalized size = 1.79 \begin{align*} -\frac{2}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.51242, size = 197, normalized size = 7.04 \begin{align*} \frac{{\left (b d n \log \left (x\right ) + 1\right )} \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \cosh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b d n \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\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{\coth ^{2}{\left (a d + b d \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 = 1.33036, size = 50, normalized size = 1.79 \begin{align*} -\frac{2}{{\left (c^{2 \, b d} x^{2 \, b d n} e^{\left (2 \, a d\right )} - 1\right )} b d n} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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