Optimal. Leaf size=14 \[ \log (x)-\frac{1}{2} \tanh (a+2 \log (x)) \]
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Rubi [A] time = 0.023689, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3473, 8} \[ \log (x)-\frac{1}{2} \tanh (a+2 \log (x)) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\tanh ^2(a+2 \log (x))}{x} \, dx &=\operatorname{Subst}\left (\int \tanh ^2(a+2 x) \, dx,x,\log (x)\right )\\ &=-\frac{1}{2} \tanh (a+2 \log (x))+\operatorname{Subst}(\int 1 \, dx,x,\log (x))\\ &=\log (x)-\frac{1}{2} \tanh (a+2 \log (x))\\ \end{align*}
Mathematica [A] time = 0.0377657, size = 24, normalized size = 1.71 \[ \frac{1}{2} \tanh ^{-1}(\tanh (a+2 \log (x)))-\frac{1}{2} \tanh (a+2 \log (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 35, normalized size = 2.5 \begin{align*} -{\frac{\tanh \left ( a+2\,\ln \left ( x \right ) \right ) }{2}}-{\frac{\ln \left ( \tanh \left ( a+2\,\ln \left ( x \right ) \right ) -1 \right ) }{4}}+{\frac{\ln \left ( \tanh \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 = 1.1484, size = 28, normalized size = 2. \begin{align*} \frac{1}{2} \, a - \frac{1}{e^{\left (-2 \, a - 4 \, \log \left (x\right )\right )} + 1} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97699, size = 68, normalized size = 4.86 \begin{align*} \frac{{\left (x^{4} e^{\left (2 \, a\right )} + 1\right )} \log \left (x\right ) + 1}{x^{4} e^{\left (2 \, a\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.499783, size = 12, normalized size = 0.86 \begin{align*} \log{\left (x \right )} - \frac{\tanh{\left (a + 2 \log{\left (x \right )} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4718, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{x^{4} e^{\left (2 \, a\right )} + 1} + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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