Optimal. Leaf size=66 \[ -\frac{(1-a) x^2}{b^2}-\frac{2 (1-a)^2 x}{b^3}-\frac{2 (1-a)^3 \log (-a-b x+1)}{b^4}-\frac{2 x^3}{3 b}-\frac{x^4}{4} \]
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Rubi [A] time = 0.0600754, antiderivative size = 66, 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} \[ -\frac{(1-a) x^2}{b^2}-\frac{2 (1-a)^2 x}{b^3}-\frac{2 (1-a)^3 \log (-a-b x+1)}{b^4}-\frac{2 x^3}{3 b}-\frac{x^4}{4} \]
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^3 \, dx &=\int \frac{x^3 (1+a+b x)}{1-a-b x} \, dx\\ &=\int \left (-\frac{2 (-1+a)^2}{b^3}+\frac{2 (-1+a) x}{b^2}-\frac{2 x^2}{b}-x^3+\frac{2 (-1+a)^3}{b^3 (-1+a+b x)}\right ) \, dx\\ &=-\frac{2 (1-a)^2 x}{b^3}-\frac{(1-a) x^2}{b^2}-\frac{2 x^3}{3 b}-\frac{x^4}{4}-\frac{2 (1-a)^3 \log (1-a-b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0426213, size = 66, normalized size = 1. \[ -\frac{(1-a) x^2}{b^2}-\frac{2 (1-a)^2 x}{b^3}-\frac{2 (1-a)^3 \log (-a-b x+1)}{b^4}-\frac{2 x^3}{3 b}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 108, normalized size = 1.6 \begin{align*} -{\frac{{x}^{4}}{4}}-{\frac{2\,{x}^{3}}{3\,b}}+{\frac{a{x}^{2}}{{b}^{2}}}-{\frac{{x}^{2}}{{b}^{2}}}-2\,{\frac{{a}^{2}x}{{b}^{3}}}+4\,{\frac{ax}{{b}^{3}}}-2\,{\frac{x}{{b}^{3}}}+2\,{\frac{\ln \left ( bx+a-1 \right ){a}^{3}}{{b}^{4}}}-6\,{\frac{\ln \left ( bx+a-1 \right ){a}^{2}}{{b}^{4}}}+6\,{\frac{\ln \left ( bx+a-1 \right ) a}{{b}^{4}}}-2\,{\frac{\ln \left ( bx+a-1 \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966067, size = 92, normalized size = 1.39 \begin{align*} -\frac{3 \, b^{3} x^{4} + 8 \, b^{2} x^{3} - 12 \,{\left (a - 1\right )} b x^{2} + 24 \,{\left (a^{2} - 2 \, a + 1\right )} x}{12 \, b^{3}} + \frac{2 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83136, size = 171, normalized size = 2.59 \begin{align*} -\frac{3 \, b^{4} x^{4} + 8 \, b^{3} x^{3} - 12 \,{\left (a - 1\right )} b^{2} x^{2} + 24 \,{\left (a^{2} - 2 \, a + 1\right )} b x - 24 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{12 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.40406, size = 56, normalized size = 0.85 \begin{align*} - \frac{x^{4}}{4} - \frac{2 x^{3}}{3 b} + \frac{x^{2} \left (a - 1\right )}{b^{2}} - \frac{x \left (2 a^{2} - 4 a + 2\right )}{b^{3}} + \frac{2 \left (a - 1\right )^{3} \log{\left (a + b x - 1 \right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21377, size = 111, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left ({\left | b x + a - 1 \right |}\right )}{b^{4}} - \frac{3 \, b^{4} x^{4} + 8 \, b^{3} x^{3} - 12 \, a b^{2} x^{2} + 24 \, a^{2} b x + 12 \, b^{2} x^{2} - 48 \, a b x + 24 \, b x}{12 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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