Optimal. Leaf size=83 \[ -\frac{2 (1-a) x^3}{3 b^2}-\frac{(1-a)^2 x^2}{b^3}-\frac{2 (1-a)^3 x}{b^4}-\frac{2 (1-a)^4 \log (-a-b x+1)}{b^5}-\frac{x^4}{2 b}-\frac{x^5}{5} \]
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Rubi [A] time = 0.0834926, antiderivative size = 83, 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} \[ -\frac{2 (1-a) x^3}{3 b^2}-\frac{(1-a)^2 x^2}{b^3}-\frac{2 (1-a)^3 x}{b^4}-\frac{2 (1-a)^4 \log (-a-b x+1)}{b^5}-\frac{x^4}{2 b}-\frac{x^5}{5} \]
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^4 \, dx &=\int \frac{x^4 (1+a+b x)}{1-a-b x} \, dx\\ &=\int \left (\frac{2 (-1+a)^3}{b^4}-\frac{2 (-1+a)^2 x}{b^3}+\frac{2 (-1+a) x^2}{b^2}-\frac{2 x^3}{b}-x^4-\frac{2 (-1+a)^4}{b^4 (-1+a+b x)}\right ) \, dx\\ &=-\frac{2 (1-a)^3 x}{b^4}-\frac{(1-a)^2 x^2}{b^3}-\frac{2 (1-a) x^3}{3 b^2}-\frac{x^4}{2 b}-\frac{x^5}{5}-\frac{2 (1-a)^4 \log (1-a-b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0636001, size = 75, normalized size = 0.9 \[ \frac{2 (a-1) x^3}{3 b^2}-\frac{(a-1)^2 x^2}{b^3}+\frac{2 (a-1)^3 x}{b^4}-\frac{2 (a-1)^4 \log (-a-b x+1)}{b^5}-\frac{x^4}{2 b}-\frac{x^5}{5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 161, normalized size = 1.9 \begin{align*} -{\frac{{x}^{5}}{5}}-{\frac{{x}^{4}}{2\,b}}+{\frac{2\,{x}^{3}a}{3\,{b}^{2}}}-{\frac{2\,{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{2}{x}^{2}}{{b}^{3}}}+2\,{\frac{a{x}^{2}}{{b}^{3}}}+2\,{\frac{x{a}^{3}}{{b}^{4}}}-{\frac{{x}^{2}}{{b}^{3}}}-6\,{\frac{{a}^{2}x}{{b}^{4}}}+6\,{\frac{ax}{{b}^{4}}}-2\,{\frac{x}{{b}^{4}}}-2\,{\frac{\ln \left ( bx+a-1 \right ){a}^{4}}{{b}^{5}}}+8\,{\frac{\ln \left ( bx+a-1 \right ){a}^{3}}{{b}^{5}}}-12\,{\frac{\ln \left ( bx+a-1 \right ){a}^{2}}{{b}^{5}}}+8\,{\frac{\ln \left ( bx+a-1 \right ) a}{{b}^{5}}}-2\,{\frac{\ln \left ( bx+a-1 \right ) }{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961459, size = 127, normalized size = 1.53 \begin{align*} -\frac{6 \, b^{4} x^{5} + 15 \, b^{3} x^{4} - 20 \,{\left (a - 1\right )} b^{2} x^{3} + 30 \,{\left (a^{2} - 2 \, a + 1\right )} b x^{2} - 60 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x}{30 \, b^{4}} - \frac{2 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88907, size = 234, normalized size = 2.82 \begin{align*} -\frac{6 \, b^{5} x^{5} + 15 \, b^{4} x^{4} - 20 \,{\left (a - 1\right )} b^{3} x^{3} + 30 \,{\left (a^{2} - 2 \, a + 1\right )} b^{2} x^{2} - 60 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} b x + 60 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{30 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.444106, size = 78, normalized size = 0.94 \begin{align*} - \frac{x^{5}}{5} - \frac{x^{4}}{2 b} + \frac{x^{3} \left (2 a - 2\right )}{3 b^{2}} - \frac{x^{2} \left (a^{2} - 2 a + 1\right )}{b^{3}} + \frac{x \left (2 a^{3} - 6 a^{2} + 6 a - 2\right )}{b^{4}} - \frac{2 \left (a - 1\right )^{4} \log{\left (a + b x - 1 \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15296, size = 165, normalized size = 1.99 \begin{align*} -\frac{2 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left ({\left | b x + a - 1 \right |}\right )}{b^{5}} - \frac{6 \, b^{5} x^{5} + 15 \, b^{4} x^{4} - 20 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 20 \, b^{3} x^{3} - 60 \, a^{3} b x - 60 \, a b^{2} x^{2} + 180 \, a^{2} b x + 30 \, b^{2} x^{2} - 180 \, a b x + 60 \, b x}{30 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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