Optimal. Leaf size=28 \[ \frac{(1-a) \log (x)}{a+1}-\frac{2 \log (a+b x+1)}{a+1} \]
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Rubi [A] time = 0.0346215, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6163, 72} \[ \frac{(1-a) \log (x)}{a+1}-\frac{2 \log (a+b x+1)}{a+1} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a+b x)}}{x} \, dx &=\int \frac{1-a-b x}{x (1+a+b x)} \, dx\\ &=\int \left (\frac{1-a}{(1+a) x}-\frac{2 b}{(1+a) (1+a+b x)}\right ) \, dx\\ &=\frac{(1-a) \log (x)}{1+a}-\frac{2 \log (1+a+b x)}{1+a}\\ \end{align*}
Mathematica [A] time = 0.0160087, size = 23, normalized size = 0.82 \[ \frac{-2 \log (a+b x+1)-a \log (x)+\log (x)}{a+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 34, normalized size = 1.2 \begin{align*} -2\,{\frac{\ln \left ( bx+a+1 \right ) }{1+a}}+{\frac{\ln \left ( x \right ) }{1+a}}-{\frac{a\ln \left ( x \right ) }{1+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953797, size = 36, normalized size = 1.29 \begin{align*} -\frac{{\left (a - 1\right )} \log \left (x\right )}{a + 1} - \frac{2 \, \log \left (b x + a + 1\right )}{a + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52606, size = 65, normalized size = 2.32 \begin{align*} -\frac{{\left (a - 1\right )} \log \left (x\right ) + 2 \, \log \left (b x + a + 1\right )}{a + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.523708, size = 90, normalized size = 3.21 \begin{align*} - \frac{\left (a - 1\right ) \log{\left (x + \frac{- \frac{a^{2} \left (a - 1\right )}{a + 1} + a^{2} - \frac{2 a \left (a - 1\right )}{a + 1} - \frac{a - 1}{a + 1} - 1}{a b - 3 b} \right )}}{a + 1} - \frac{2 \log{\left (x + \frac{a^{2} - \frac{2 a^{2}}{a + 1} - \frac{4 a}{a + 1} - 1 - \frac{2}{a + 1}}{a b - 3 b} \right )}}{a + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1946, size = 89, normalized size = 3.18 \begin{align*} -b{\left (\frac{{\left (a - 1\right )} \log \left ({\left | -\frac{a}{b x + a + 1} - \frac{1}{b x + a + 1} + 1 \right |}\right )}{a b + b} - \frac{\log \left (\frac{{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2}{\left | b \right |}}\right )}{b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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