Optimal. Leaf size=49 \[ -\frac {2 (1-a)^2 \log (-a-b x+1)}{b^3}-\frac {2 (1-a) x}{b^2}-\frac {x^2}{b}-\frac {x^3}{3} \]
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Rubi [A] time = 0.05, antiderivative size = 49, 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} \[ -\frac {2 (1-a) x}{b^2}-\frac {2 (1-a)^2 \log (-a-b x+1)}{b^3}-\frac {x^2}{b}-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6163
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*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 0.90 \[ -\frac {b x \left (-6 a+b^2 x^2+3 b x+6\right )+6 (a-1)^2 \log (-a-b x+1)}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 45, normalized size = 0.92 \[ -\frac {b^{3} x^{3} + 3 \, b^{2} x^{2} - 6 \, {\left (a - 1\right )} b x + 6 \, {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 52, normalized size = 1.06 \[ -\frac {2 \, {\left (a^{2} - 2 \, a + 1\right )} \log \left ({\left | b x + a - 1 \right |}\right )}{b^{3}} - \frac {b^{3} x^{3} + 3 \, b^{2} x^{2} - 6 \, a b x + 6 \, b x}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 68, normalized size = 1.39 \[ -\frac {x^{3}}{3}-\frac {x^{2}}{b}+\frac {2 a x}{b^{2}}-\frac {2 x}{b^{2}}-\frac {2 \ln \left (b x +a -1\right ) a^{2}}{b^{3}}+\frac {4 \ln \left (b x +a -1\right ) a}{b^{3}}-\frac {2 \ln \left (b x +a -1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 46, normalized size = 0.94 \[ -\frac {b^{2} x^{3} + 3 \, b x^{2} - 6 \, {\left (a - 1\right )} x}{3 \, b^{2}} - \frac {2 \, {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 74, normalized size = 1.51 \[ x^2\,\left (\frac {a-1}{2\,b}-\frac {a+1}{2\,b}\right )-\frac {x^3}{3}-\frac {\ln \left (a+b\,x-1\right )\,\left (2\,a^2-4\,a+2\right )}{b^3}-\frac {x\,\left (\frac {a-1}{b}-\frac {a+1}{b}\right )\,\left (a-1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 42, normalized size = 0.86 \[ - \frac {x^{3}}{3} - x \left (- \frac {2 a}{b^{2}} + \frac {2}{b^{2}}\right ) - \frac {x^{2}}{b} - \frac {2 \left (a - 1\right )^{2} \log {\left (a + b x - 1 \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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