Optimal. Leaf size=34 \[ -\frac {2 (1-a) \log (-a-b x+1)}{b^2}-\frac {2 x}{b}-\frac {x^2}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 34, 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} \[ -\frac {2 (1-a) \log (-a-b x+1)}{b^2}-\frac {2 x}{b}-\frac {x^2}{2} \]
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 = 34, normalized size = 1.00 \[ -\frac {2 (1-a) \log (-a-b x+1)}{b^2}-\frac {2 x}{b}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 29, normalized size = 0.85 \[ -\frac {b^{2} x^{2} + 4 \, b x - 4 \, {\left (a - 1\right )} \log \left (b x + a - 1\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 1.00 \[ \frac {2 \, {\left (a - 1\right )} \log \left ({\left | b x + a - 1 \right |}\right )}{b^{2}} - \frac {b^{2} x^{2} + 4 \, b x}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 38, normalized size = 1.12 \[ -\frac {x^{2}}{2}-\frac {2 x}{b}+\frac {2 \ln \left (b x +a -1\right ) a}{b^{2}}-\frac {2 \ln \left (b x +a -1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 30, normalized size = 0.88 \[ -\frac {b x^{2} + 4 \, x}{2 \, b} + \frac {2 \, {\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 40, normalized size = 1.18 \[ x\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )-\frac {x^2}{2}+\frac {\ln \left (a+b\,x-1\right )\,\left (2\,a-2\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 26, normalized size = 0.76 \[ - \frac {x^{2}}{2} - \frac {2 x}{b} + \frac {2 \left (a - 1\right ) \log {\left (a + b x - 1 \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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