Optimal. Leaf size=29 \[ -\frac{2 (a+1) \log (a+b x+1)}{b^2}+\frac{2 x}{b}-\frac{x^2}{2} \]
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Rubi [A] time = 0.0317985, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6163, 77} \[ -\frac{2 (a+1) \log (a+b x+1)}{b^2}+\frac{2 x}{b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 77
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a+b x)} x \, dx &=\int \frac{x (1-a-b x)}{1+a+b x} \, dx\\ &=\int \left (\frac{2}{b}-x-\frac{2 (1+a)}{b (1+a+b x)}\right ) \, dx\\ &=\frac{2 x}{b}-\frac{x^2}{2}-\frac{2 (1+a) \log (1+a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0172854, size = 29, normalized size = 1. \[ -\frac{2 (a+1) \log (a+b x+1)}{b^2}+\frac{2 x}{b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 38, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2}}{2}}+2\,{\frac{x}{b}}-2\,{\frac{\ln \left ( bx+a+1 \right ) a}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a+1 \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958947, size = 41, normalized size = 1.41 \begin{align*} -\frac{b x^{2} - 4 \, x}{2 \, b} - \frac{2 \,{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48488, size = 77, normalized size = 2.66 \begin{align*} -\frac{b^{2} x^{2} - 4 \, b x + 4 \,{\left (a + 1\right )} \log \left (b x + a + 1\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.315509, size = 26, normalized size = 0.9 \begin{align*} - \frac{x^{2}}{2} + \frac{2 x}{b} - \frac{2 \left (a + 1\right ) \log{\left (a + b x + 1 \right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13369, size = 93, normalized size = 3.21 \begin{align*} \frac{\frac{{\left (b x + a + 1\right )}^{2}{\left (\frac{2 \,{\left (a b + 3 \, b\right )}}{{\left (b x + a + 1\right )} b} - 1\right )}}{b} + \frac{4 \,{\left (a + 1\right )} \log \left (\frac{{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2}{\left | b \right |}}\right )}{b}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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