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 = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 72
Rule 6163
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.46, size = 23, normalized size = 0.70 \[ -\frac {{\left (a + 1\right )} \log \relax (x) - 2 \, \log \left (b x + a - 1\right )}{a - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 34, normalized size = 1.03 \[ \frac {2 \, b \log \left ({\left | b x + a - 1 \right |}\right )}{a b - b} - \frac {{\left (a + 1\right )} \log \left ({\left | x \right |}\right )}{a - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 1.06 \[ -\frac {\ln \relax (x )}{a -1}-\frac {\ln \relax (x ) a}{a -1}+\frac {2 \ln \left (b x +a -1\right )}{a -1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 0.82 \[ -\frac {{\left (a + 1\right )} \log \relax (x)}{a - 1} + \frac {2 \, \log \left (b x + a - 1\right )}{a - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 28, normalized size = 0.85 \[ \frac {2\,\ln \left (a+b\,x-1\right )}{a-1}-\frac {2\,\ln \relax (x)}{a-1}-\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 88, normalized size = 2.67 \[ - \frac {\left (a + 1\right ) \log {\left (x + \frac {a^{2} - \frac {a^{2} \left (a + 1\right )}{a - 1} + \frac {2 a \left (a + 1\right )}{a - 1} - 1 - \frac {a + 1}{a - 1}}{a b + 3 b} \right )}}{a - 1} + \frac {2 \log {\left (x + \frac {a^{2} + \frac {2 a^{2}}{a - 1} - \frac {4 a}{a - 1} - 1 + \frac {2}{a - 1}}{a b + 3 b} \right )}}{a - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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