Optimal. Leaf size=19 \[ -\frac{2 \log (-a-b x+1)}{b}-x \]
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Rubi [A] time = 0.0124833, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6161, 43} \[ -\frac{2 \log (-a-b x+1)}{b}-x \]
Antiderivative was successfully verified.
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Rule 6161
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a+b x)} \, dx &=\int \frac{1+a+b x}{1-a-b x} \, dx\\ &=\int \left (-1-\frac{2}{-1+a+b x}\right ) \, dx\\ &=-x-\frac{2 \log (1-a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0099302, size = 19, normalized size = 1. \[ -\frac{2 \log (-a-b x+1)}{b}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 17, normalized size = 0.9 \begin{align*} -x-2\,{\frac{\ln \left ( bx+a-1 \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952147, size = 22, normalized size = 1.16 \begin{align*} -x - \frac{2 \, \log \left (b x + a - 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87019, size = 42, normalized size = 2.21 \begin{align*} -\frac{b x + 2 \, \log \left (b x + a - 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.122734, size = 14, normalized size = 0.74 \begin{align*} - x - \frac{2 \log{\left (a + b x - 1 \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16458, size = 23, normalized size = 1.21 \begin{align*} -x - \frac{2 \, \log \left ({\left | b x + a - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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