Optimal. Leaf size=16 \[ \frac{2 \log (a+b x+1)}{b}-x \]
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Rubi [A] time = 0.0121516, antiderivative size = 16, 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 = 0.0111211, size = 16, normalized size = 1. \[ \frac{2 \log (a+b x+1)}{b}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 17, normalized size = 1.1 \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 = 0.959308, size = 22, normalized size = 1.38 \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 = 1.44847, size = 42, normalized size = 2.62 \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 = 0.123091, size = 12, normalized size = 0.75 \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 [B] time = 1.17688, size = 51, normalized size = 3.19 \begin{align*} -\frac{b x + a + 1}{b} - \frac{2 \, \log \left (\frac{{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2}{\left | b \right |}}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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