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 = 0.03, antiderivative size = 28, 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} \[ \frac {(1-a) \log (x)}{a+1}-\frac {2 \log (a+b x+1)}{a+1} \]
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 = 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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fricas [A] time = 0.59, size = 23, normalized size = 0.82 \[ -\frac {{\left (a - 1\right )} \log \relax (x) + 2 \, \log \left (b x + a + 1\right )}{a + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 66, normalized size = 2.36 \[ -b {\left (\frac {{\left (a - 1\right )} \log \left ({\left | -\frac {a}{b x + a + 1} - \frac {1}{b x + a + 1} + 1 \right |}\right )}{a b + b} - \frac {\log \left (\frac {{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2} {\left | b \right |}}\right )}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 34, normalized size = 1.21 \[ \frac {\ln \relax (x )}{1+a}-\frac {\ln \relax (x ) a}{1+a}-\frac {2 \ln \left (b x +a +1\right )}{1+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 0.96 \[ -\frac {{\left (a - 1\right )} \log \relax (x)}{a + 1} - \frac {2 \, \log \left (b x + a + 1\right )}{a + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 28, normalized size = 1.00 \[ \frac {2\,\ln \relax (x)}{a+1}-\ln \relax (x)-\frac {2\,\ln \left (a+b\,x+1\right )}{a+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 90, normalized size = 3.21 \[ - \frac {\left (a - 1\right ) \log {\left (x + \frac {- \frac {a^{2} \left (a - 1\right )}{a + 1} + a^{2} - \frac {2 a \left (a - 1\right )}{a + 1} - \frac {a - 1}{a + 1} - 1}{a b - 3 b} \right )}}{a + 1} - \frac {2 \log {\left (x + \frac {a^{2} - \frac {2 a^{2}}{a + 1} - \frac {4 a}{a + 1} - 1 - \frac {2}{a + 1}}{a b - 3 b} \right )}}{a + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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