Optimal. Leaf size=33 \[ \frac{(a+1) \log (x)}{1-a}-\frac{2 \log (-a-b x+1)}{1-a} \]
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Rubi [A] time = 0.0377747, antiderivative size = 33, 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} \[ \frac{(a+1) \log (x)}{1-a}-\frac{2 \log (-a-b x+1)}{1-a} \]
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 = 0.0173799, size = 26, normalized size = 0.79 \[ \frac{2 \log (-a-b x+1)-(a+1) \log (x)}{a-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 35, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( x \right ) }{a-1}}-{\frac{a\ln \left ( x \right ) }{a-1}}+2\,{\frac{\ln \left ( bx+a-1 \right ) }{a-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953696, size = 36, normalized size = 1.09 \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 = 1.88123, size = 65, normalized size = 1.97 \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 = 0.532849, size = 88, normalized size = 2.67 \begin{align*} - \frac{\left (a + 1\right ) \log{\left (x + \frac{a^{2} - \frac{a^{2} \left (a + 1\right )}{a - 1} + \frac{2 a \left (a + 1\right )}{a - 1} - 1 - \frac{a + 1}{a - 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 [A] time = 1.20083, size = 46, normalized size = 1.39 \begin{align*} \frac{2 \, b \log \left ({\left | b x + a - 1 \right |}\right )}{a b - b} - \frac{{\left (a + 1\right )} \log \left ({\left | x \right |}\right )}{a - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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