Optimal. Leaf size=33 \[ \frac {2-x}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )+\frac {3}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1814, 635, 203, 260} \begin {gather*} \frac {2-x}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )+\frac {3}{2} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1814
Rubi steps
\begin {align*} \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx &=\frac {2-x}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {-3+2 x}{1+x^2} \, dx\\ &=\frac {2-x}{2 \left (1+x^2\right )}+\frac {3}{2} \int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=\frac {2-x}{2 \left (1+x^2\right )}+\frac {3}{2} \tan ^{-1}(x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.91 \begin {gather*} \frac {1}{2} \left (\frac {2-x}{x^2+1}-\log \left (x^2+1\right )+3 \tan ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-3 x+2 x^2-x^3}{\left (1+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.68, size = 36, normalized size = 1.09 \begin {gather*} \frac {3 \, {\left (x^{2} + 1\right )} \arctan \relax (x) - {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - x + 2}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 25, normalized size = 0.76 \begin {gather*} -\frac {x - 2}{2 \, {\left (x^{2} + 1\right )}} + \frac {3}{2} \, \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.85 \begin {gather*} \frac {3 \arctan \relax (x )}{2}-\frac {\ln \left (x^{2}+1\right )}{2}-\frac {\frac {x}{2}-1}{x^{2}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.15, size = 25, normalized size = 0.76 \begin {gather*} -\frac {x - 2}{2 \, {\left (x^{2} + 1\right )}} + \frac {3}{2} \, \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 32, normalized size = 0.97 \begin {gather*} \frac {3\,\mathrm {atan}\relax (x)}{2}-\frac {\ln \left (x^2+1\right )}{2}-\frac {x}{2\,\left (x^2+1\right )}+\frac {1}{x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 24, normalized size = 0.73 \begin {gather*} - \frac {x - 2}{2 x^{2} + 2} - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {3 \operatorname {atan}{\relax (x )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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