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 = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 77
Rule 6163
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*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.00 \[ -\frac {2 (a+1) \log (a+b x+1)}{b^2}+\frac {2 x}{b}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 29, normalized size = 1.00 \[ -\frac {b^{2} x^{2} - 4 \, b x + 4 \, {\left (a + 1\right )} \log \left (b x + a + 1\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 69, normalized size = 2.38 \[ \frac {\frac {{\left (b x + a + 1\right )}^{2} {\left (\frac {2 \, {\left (a b + 3 \, b\right )}}{{\left (b x + a + 1\right )} b} - 1\right )}}{b} + \frac {4 \, {\left (a + 1\right )} \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2} {\left | b \right |}}\right )}{b}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 38, normalized size = 1.31 \[ -\frac {x^{2}}{2}+\frac {2 x}{b}-\frac {2 \ln \left (b x +a +1\right ) a}{b^{2}}-\frac {2 \ln \left (b x +a +1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 30, normalized size = 1.03 \[ -\frac {b x^{2} - 4 \, x}{2 \, b} - \frac {2 \, {\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 42, normalized size = 1.45 \[ -\frac {x^2}{2}-x\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )-\frac {\ln \left (a+b\,x+1\right )\,\left (2\,a+2\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 26, normalized size = 0.90 \[ - \frac {x^{2}}{2} + \frac {2 x}{b} - \frac {2 \left (a + 1\right ) \log {\left (a + b x + 1 \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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