Optimal. Leaf size=49 \[ -\frac{2 (1-a) x}{b^2}-\frac{2 (1-a)^2 \log (-a-b x+1)}{b^3}-\frac{x^2}{b}-\frac{x^3}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0506131, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6163, 77} \[ -\frac{2 (1-a) x}{b^2}-\frac{2 (1-a)^2 \log (-a-b x+1)}{b^3}-\frac{x^2}{b}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6163
Rule 77
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a+b x)} x^2 \, dx &=\int \frac{x^2 (1+a+b x)}{1-a-b x} \, dx\\ &=\int \left (\frac{2 (-1+a)}{b^2}-\frac{2 x}{b}-x^2-\frac{2 (-1+a)^2}{b^2 (-1+a+b x)}\right ) \, dx\\ &=-\frac{2 (1-a) x}{b^2}-\frac{x^2}{b}-\frac{x^3}{3}-\frac{2 (1-a)^2 \log (1-a-b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0310277, size = 44, normalized size = 0.9 \[ -\frac{b x \left (-6 a+b^2 x^2+3 b x+6\right )+6 (a-1)^2 \log (-a-b x+1)}{3 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.028, size = 68, normalized size = 1.4 \begin{align*} -{\frac{{x}^{3}}{3}}-{\frac{{x}^{2}}{b}}+2\,{\frac{ax}{{b}^{2}}}-2\,{\frac{x}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a-1 \right ){a}^{2}}{{b}^{3}}}+4\,{\frac{\ln \left ( bx+a-1 \right ) a}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a-1 \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.940435, size = 62, normalized size = 1.27 \begin{align*} -\frac{b^{2} x^{3} + 3 \, b x^{2} - 6 \,{\left (a - 1\right )} x}{3 \, b^{2}} - \frac{2 \,{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.73783, size = 115, normalized size = 2.35 \begin{align*} -\frac{b^{3} x^{3} + 3 \, b^{2} x^{2} - 6 \,{\left (a - 1\right )} b x + 6 \,{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.361109, size = 37, normalized size = 0.76 \begin{align*} - \frac{x^{3}}{3} - \frac{x^{2}}{b} + \frac{x \left (2 a - 2\right )}{b^{2}} - \frac{2 \left (a - 1\right )^{2} \log{\left (a + b x - 1 \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15478, size = 70, normalized size = 1.43 \begin{align*} -\frac{2 \,{\left (a^{2} - 2 \, a + 1\right )} \log \left ({\left | b x + a - 1 \right |}\right )}{b^{3}} - \frac{b^{3} x^{3} + 3 \, b^{2} x^{2} - 6 \, a b x + 6 \, b x}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]