Optimal. Leaf size=32 \[ \frac{3}{2} \log \left (x^2-2 x+4\right )+\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.035154, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1872, 634, 618, 204, 628} \[ \frac{3}{2} \log \left (x^2-2 x+4\right )+\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1872
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{-8+2 x+3 x^2}{8+x^3} \, dx &=\frac{1}{2} \int \frac{-8+6 x}{4-2 x+x^2} \, dx\\ &=\frac{3}{2} \int \frac{-2+2 x}{4-2 x+x^2} \, dx-\int \frac{1}{4-2 x+x^2} \, dx\\ &=\frac{3}{2} \log \left (4-2 x+x^2\right )+2 \operatorname{Subst}\left (\int \frac{1}{-12-x^2} \, dx,x,-2+2 x\right )\\ &=\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{3}{2} \log \left (4-2 x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0050206, size = 31, normalized size = 0.97 \[ \frac{3}{2} \log \left (x^2-2 x+4\right )-\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 29, normalized size = 0.9 \begin{align*}{\frac{3\,\ln \left ({x}^{2}-2\,x+4 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{3}}{6}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46215, size = 35, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.976078, size = 90, normalized size = 2.81 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.10787, size = 36, normalized size = 1.12 \begin{align*} \frac{3 \log{\left (x^{2} - 2 x + 4 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13532, size = 35, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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