Optimal. Leaf size=41 \[ -\frac{2 (a+1) x}{b^2}+\frac{2 (a+1)^2 \log (a+b x+1)}{b^3}+\frac{x^2}{b}-\frac{x^3}{3} \]
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Rubi [A] time = 0.0456562, antiderivative size = 41, 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} \[ -\frac{2 (a+1) x}{b^2}+\frac{2 (a+1)^2 \log (a+b x+1)}{b^3}+\frac{x^2}{b}-\frac{x^3}{3} \]
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^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 = 0.0296369, size = 41, normalized size = 1. \[ -\frac{2 (a+1) x}{b^2}+\frac{2 (a+1)^2 \log (a+b x+1)}{b^3}+\frac{x^2}{b}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 67, normalized size = 1.6 \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.
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Maxima [A] time = 0.966852, size = 62, normalized size = 1.51 \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.
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Fricas [A] time = 1.44515, size = 115, normalized size = 2.8 \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.
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Sympy [A] time = 0.353981, size = 37, normalized size = 0.9 \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.
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Giac [B] time = 1.19284, size = 138, normalized size = 3.37 \begin{align*} \frac{{\left (b x + a + 1\right )}^{3}{\left (\frac{3 \,{\left (a b + 2 \, b\right )}}{{\left (b x + a + 1\right )} b} - \frac{3 \,{\left (a^{2} b^{2} + 6 \, a b^{2} + 5 \, b^{2}\right )}}{{\left (b x + a + 1\right )}^{2} b^{2}} - 1\right )}}{3 \, b^{3}} - \frac{2 \,{\left (a^{2} + 2 \, a + 1\right )} \log \left (\frac{{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2}{\left | b \right |}}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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