Optimal. Leaf size=12 \[ \frac{1}{2} \log (\sinh (a+2 \log (x))) \]
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Rubi [A] time = 0.0137687, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3475} \[ \frac{1}{2} \log (\sinh (a+2 \log (x))) \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin{align*} \int \frac{\coth (a+2 \log (x))}{x} \, dx &=\operatorname{Subst}(\int \coth (a+2 x) \, dx,x,\log (x))\\ &=\frac{1}{2} \log (\sinh (a+2 \log (x)))\\ \end{align*}
Mathematica [A] time = 0.0266171, size = 21, normalized size = 1.75 \[ \frac{1}{2} (\log (\tanh (a+2 \log (x)))+\log (\cosh (a+2 \log (x)))) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 26, normalized size = 2.2 \begin{align*} -{\frac{\ln \left ({\rm coth} \left (a+2\,\ln \left ( x \right ) \right )-1 \right ) }{4}}-{\frac{\ln \left ({\rm coth} \left (a+2\,\ln \left ( x \right ) \right )+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05091, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, \log \left (\sinh \left (a + 2 \, \log \left (x\right )\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53463, size = 47, normalized size = 3.92 \begin{align*} \frac{1}{2} \, \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.68852, size = 27, normalized size = 2.25 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (\tanh{\left (a + 2 \log{\left (x \right )} \right )} + 1 \right )}}{2} + \frac{\log{\left (\tanh{\left (a + 2 \log{\left (x \right )} \right )} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09541, size = 28, normalized size = 2.33 \begin{align*} -\frac{1}{4} \, \log \left (x^{4}\right ) + \frac{1}{2} \, \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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