Optimal. Leaf size=22 \[ \frac{1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \]
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Rubi [A] time = 0.0075908, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {634, 617, 204, 628} \[ \frac{1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \]
Antiderivative was successfully verified.
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Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{2-2 x+x^2} \, dx &=\frac{1}{2} \int \frac{-2+2 x}{2-2 x+x^2} \, dx+\int \frac{1}{2-2 x+x^2} \, dx\\ &=\frac{1}{2} \log \left (2-2 x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-x\right )\\ &=-\tan ^{-1}(1-x)+\frac{1}{2} \log \left (2-2 x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0032798, size = 22, normalized size = 1. \[ \frac{1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 17, normalized size = 0.8 \begin{align*} \arctan \left ( -1+x \right ) +{\frac{\ln \left ({x}^{2}-2\,x+2 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41361, size = 22, normalized size = 1. \begin{align*} \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76874, size = 54, normalized size = 2.45 \begin{align*} \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.091334, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x^{2} - 2 x + 2 \right )}}{2} + \operatorname{atan}{\left (x - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07189, size = 22, normalized size = 1. \begin{align*} \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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