Optimal. Leaf size=33 \[ \frac {(1+a) \log (x)}{1-a}-\frac {2 \log (1-a-b x)}{1-a} \]
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Rubi [A]
time = 0.03, 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 = {6298, 78}
\begin {gather*} \frac {(a+1) \log (x)}{1-a}-\frac {2 \log (-a-b x+1)}{1-a} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6298
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 = 0.01, size = 26, normalized size = 0.79 \begin {gather*} \frac {-((1+a) \log (x))+2 \log (1-a-b x)}{-1+a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 29, normalized size = 0.88
method | result | size |
norman | \(\frac {2 \ln \left (b x +a -1\right )}{-1+a}-\frac {\left (1+a \right ) \ln \left (x \right )}{-1+a}\) | \(28\) |
default | \(\frac {\left (-1-a \right ) \ln \left (x \right )}{-1+a}+\frac {2 \ln \left (b x +a -1\right )}{-1+a}\) | \(29\) |
risch | \(-\frac {\ln \left (x \right )}{-1+a}-\frac {\ln \left (x \right ) a}{-1+a}+\frac {2 \ln \left (-b x -a +1\right )}{-1+a}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 27, normalized size = 0.82 \begin {gather*} -\frac {{\left (a + 1\right )} \log \left (x\right )}{a - 1} + \frac {2 \, \log \left (b x + a - 1\right )}{a - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 23, normalized size = 0.70 \begin {gather*} -\frac {{\left (a + 1\right )} \log \left (x\right ) - 2 \, \log \left (b x + a - 1\right )}{a - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs.
\(2 (22) = 44\).
time = 0.28, size = 88, normalized size = 2.67 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 34, normalized size = 1.03 \begin {gather*} \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 {gather*}
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 \begin {gather*} \frac {2\,\ln \left (a+b\,x-1\right )}{a-1}-\frac {2\,\ln \left (x\right )}{a-1}-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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