Optimal. Leaf size=32 \[ \frac{1}{2} \log \left (x^2-x+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.029412, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1872, 634, 618, 204, 628} \[ \frac{1}{2} \log \left (x^2-x+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1872
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{-1+x^2}{1+x^3} \, dx &=\int \frac{-1+x}{1-x+x^2} \, dx\\ &=-\left (\frac{1}{2} \int \frac{1}{1-x+x^2} \, dx\right )+\frac{1}{2} \int \frac{-1+2 x}{1-x+x^2} \, dx\\ &=\frac{1}{2} \log \left (1-x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{-1+2 x}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{1}{2} \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.005401, size = 33, normalized size = 1.03 \[ \frac{1}{2} \log \left (x^2-x+1\right )-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{2}-x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45092, size = 38, normalized size = 1.19 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40583, size = 90, normalized size = 2.81 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101599, size = 34, normalized size = 1.06 \begin{align*} \frac{\log{\left (x^{2} - x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \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 = 1.1297, size = 38, normalized size = 1.19 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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