Optimal. Leaf size=22 \[ -\tan ^{-1}(1-x)+\frac {1}{2} \log \left (2-2 x+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {648, 631, 210,
642} \begin {gather*} \frac {1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 648
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 )+\text {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*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} -\tan ^{-1}(1-x)+\frac {1}{2} \log \left (2-2 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.74, size = 16, normalized size = 0.73 \begin {gather*} \text {ArcTan}\left [-1+x\right ]+\frac {\text {Log}\left [2-2 x+x^2\right ]}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 17, normalized size = 0.77
method | result | size |
default | \(\arctan \left (-1+x \right )+\frac {\ln \left (x^{2}-2 x +2\right )}{2}\) | \(17\) |
risch | \(\arctan \left (-1+x \right )+\frac {\ln \left (x^{2}-2 x +2\right )}{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 16, normalized size = 0.73 \begin {gather*} \arctan \left (x - 1\right ) + \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 16, normalized size = 0.73 \begin {gather*} \arctan \left (x - 1\right ) + \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 15, normalized size = 0.68 \begin {gather*} \frac {\log {\left (x^{2} - 2 x + 2 \right )}}{2} + \operatorname {atan}{\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 18, normalized size = 0.82 \begin {gather*} \frac {\ln \left (x^{2}-2 x+2\right )}{2}+\arctan \left (x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 16, normalized size = 0.73 \begin {gather*} \mathrm {atan}\left (x-1\right )+\frac {\ln \left (x^2-2\,x+2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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