Optimal. Leaf size=66 \[ -\frac {2 (1-a)^3 \log (-a-b x+1)}{b^4}-\frac {2 (1-a)^2 x}{b^3}-\frac {(1-a) x^2}{b^2}-\frac {2 x^3}{3 b}-\frac {x^4}{4} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 77
Rule 6163
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*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 1.00 \[ -\frac {2 (1-a)^3 \log (-a-b x+1)}{b^4}-\frac {2 (1-a)^2 x}{b^3}-\frac {(1-a) x^2}{b^2}-\frac {2 x^3}{3 b}-\frac {x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 67, normalized size = 1.02 \[ -\frac {3 \, b^{4} x^{4} + 8 \, b^{3} x^{3} - 12 \, {\left (a - 1\right )} b^{2} x^{2} + 24 \, {\left (a^{2} - 2 \, a + 1\right )} b x - 24 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 82, normalized size = 1.24 \[ \frac {2 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left ({\left | b x + a - 1 \right |}\right )}{b^{4}} - \frac {3 \, b^{4} x^{4} + 8 \, b^{3} x^{3} - 12 \, a b^{2} x^{2} + 24 \, a^{2} b x + 12 \, b^{2} x^{2} - 48 \, a b x + 24 \, b x}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 108, normalized size = 1.64 \[ -\frac {x^{4}}{4}-\frac {2 x^{3}}{3 b}+\frac {x^{2} a}{b^{2}}-\frac {x^{2}}{b^{2}}-\frac {2 a^{2} x}{b^{3}}+\frac {4 a x}{b^{3}}-\frac {2 x}{b^{3}}+\frac {2 \ln \left (b x +a -1\right ) a^{3}}{b^{4}}-\frac {6 \ln \left (b x +a -1\right ) a^{2}}{b^{4}}+\frac {6 \ln \left (b x +a -1\right ) a}{b^{4}}-\frac {2 \ln \left (b x +a -1\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 68, normalized size = 1.03 \[ -\frac {3 \, b^{3} x^{4} + 8 \, b^{2} x^{3} - 12 \, {\left (a - 1\right )} b x^{2} + 24 \, {\left (a^{2} - 2 \, a + 1\right )} x}{12 \, b^{3}} + \frac {2 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 106, normalized size = 1.61 \[ x^3\,\left (\frac {a-1}{3\,b}-\frac {a+1}{3\,b}\right )-\frac {x^4}{4}+\frac {\ln \left (a+b\,x-1\right )\,\left (2\,a^3-6\,a^2+6\,a-2\right )}{b^4}-\frac {x^2\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )\,\left (a-1\right )}{2\,b}+\frac {x\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )\,{\left (a-1\right )}^2}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 66, normalized size = 1.00 \[ - \frac {x^{4}}{4} - x^{2} \left (- \frac {a}{b^{2}} + \frac {1}{b^{2}}\right ) - x \left (\frac {2 a^{2}}{b^{3}} - \frac {4 a}{b^{3}} + \frac {2}{b^{3}}\right ) - \frac {2 x^{3}}{3 b} + \frac {2 \left (a - 1\right )^{3} \log {\left (a + b x - 1 \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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