Optimal. Leaf size=27 \[ \frac {4}{a (1-a x)}+\frac {4 \log (1-a x)}{a}+x \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6125, 43} \[ \frac {4}{a (1-a x)}+\frac {4 \log (1-a x)}{a}+x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6125
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \, dx &=\int \frac {(1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (1+\frac {4}{(-1+a x)^2}+\frac {4}{-1+a x}\right ) \, dx\\ &=x+\frac {4}{a (1-a x)}+\frac {4 \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 0.96 \[ -\frac {4}{a (a x-1)}+\frac {4 \log (1-a x)}{a}+x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 38, normalized size = 1.41 \[ \frac {a^{2} x^{2} - a x + 4 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 4}{a^{2} x - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 26, normalized size = 0.96 \[ x + \frac {4 \, \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac {4}{{\left (a x - 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 26, normalized size = 0.96 \[ x +\frac {4 \ln \left (a x -1\right )}{a}-\frac {4}{a \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 26, normalized size = 0.96 \[ x + \frac {4 \, \log \left (a x - 1\right )}{a} - \frac {4}{a^{2} x - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 25, normalized size = 0.93 \[ x-\frac {4}{a\,\left (a\,x-1\right )}+\frac {4\,\ln \left (a\,x-1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 19, normalized size = 0.70 \[ x - \frac {4}{a^{2} x - a} + \frac {4 \log {\left (a x - 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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