Optimal. Leaf size=34 \[ -\frac{2 (1-a) \log (-a-b x+1)}{b^2}-\frac{2 x}{b}-\frac{x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0306465, antiderivative size = 34, 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} \[ -\frac{2 (1-a) \log (-a-b x+1)}{b^2}-\frac{2 x}{b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
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 = 0.0171643, size = 34, normalized size = 1. \[ -\frac{2 (1-a) \log (-a-b x+1)}{b^2}-\frac{2 x}{b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 38, normalized size = 1.1 \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.950921, size = 41, normalized size = 1.21 \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.71797, size = 77, normalized size = 2.26 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.319083, size = 26, normalized size = 0.76 \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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18514, size = 46, normalized size = 1.35 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]