Optimal. Leaf size=71 \[ -\frac{2 (a+1) x^3}{3 b^2}+\frac{(a+1)^2 x^2}{b^3}-\frac{2 (a+1)^3 x}{b^4}+\frac{2 (a+1)^4 \log (a+b x+1)}{b^5}+\frac{x^4}{2 b}-\frac{x^5}{5} \]
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Rubi [A] time = 0.0785129, antiderivative size = 71, 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} \[ -\frac{2 (a+1) x^3}{3 b^2}+\frac{(a+1)^2 x^2}{b^3}-\frac{2 (a+1)^3 x}{b^4}+\frac{2 (a+1)^4 \log (a+b x+1)}{b^5}+\frac{x^4}{2 b}-\frac{x^5}{5} \]
Antiderivative was successfully verified.
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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 = 0.0570517, size = 71, normalized size = 1. \[ -\frac{2 (a+1) x^3}{3 b^2}+\frac{(a+1)^2 x^2}{b^3}-\frac{2 (a+1)^3 x}{b^4}+\frac{2 (a+1)^4 \log (a+b x+1)}{b^5}+\frac{x^4}{2 b}-\frac{x^5}{5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 159, normalized size = 2.2 \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 = 0.955928, size = 127, normalized size = 1.79 \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 = 1.41104, size = 234, normalized size = 3.3 \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 = 0.447723, size = 78, normalized size = 1.1 \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 [B] time = 1.20884, size = 273, normalized size = 3.85 \begin{align*} \frac{{\left (b x + a + 1\right )}^{5}{\left (\frac{15 \,{\left (2 \, a b + 3 \, b\right )}}{{\left (b x + a + 1\right )} b} - \frac{20 \,{\left (3 \, a^{2} b^{2} + 10 \, a b^{2} + 7 \, b^{2}\right )}}{{\left (b x + a + 1\right )}^{2} b^{2}} + \frac{60 \,{\left (a^{3} b^{3} + 6 \, a^{2} b^{3} + 9 \, a b^{3} + 4 \, b^{3}\right )}}{{\left (b x + a + 1\right )}^{3} b^{3}} - \frac{30 \,{\left (a^{4} b^{4} + 12 \, a^{3} b^{4} + 30 \, a^{2} b^{4} + 28 \, a b^{4} + 9 \, b^{4}\right )}}{{\left (b x + a + 1\right )}^{4} b^{4}} - 6\right )}}{30 \, b^{5}} - \frac{2 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (\frac{{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2}{\left | b \right |}}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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