Optimal. Leaf size=20 \[ \frac {1}{2} \log \left (x^2+2 x+2\right )-2 \tan ^{-1}(x+1) \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {634, 617, 204, 628} \[ \frac {1}{2} \log \left (x^2+2 x+2\right )-2 \tan ^{-1}(x+1) \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {-1+x}{2+2 x+x^2} \, dx &=\frac {1}{2} \int \frac {2+2 x}{2+2 x+x^2} \, dx-2 \int \frac {1}{2+2 x+x^2} \, dx\\ &=\frac {1}{2} \log \left (2+2 x+x^2\right )+2 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+x\right )\\ &=-2 \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 = 20, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^2+2 x+2\right )-2 \tan ^{-1}(x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 18, normalized size = 0.90 \[ -2 \, \arctan \left (x + 1\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 18, normalized size = 0.90 \[ -2 \, \arctan \left (x + 1\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.95 \[ -2 \arctan \left (x +1\right )+\frac {\ln \left (x^{2}+2 x +2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 18, normalized size = 0.90 \[ -2 \, \arctan \left (x + 1\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 18, normalized size = 0.90 \[ \frac {\ln \left (x^2+2\,x+2\right )}{2}-2\,\mathrm {atan}\left (x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 17, normalized size = 0.85 \[ \frac {\log {\left (x^{2} + 2 x + 2 \right )}}{2} - 2 \operatorname {atan}{\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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