Optimal. Leaf size=30 \[ -\frac {x^2}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1804, 774, 635, 203, 260} \[ -\frac {x^2}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 774
Rule 1804
Rubi steps
\begin {align*} \int \frac {x^2 \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx &=-\frac {x^2}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {(-2-2 x) x}{1+x^2} \, dx\\ &=x-\frac {x^2}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {2-2 x}{1+x^2} \, dx\\ &=x-\frac {x^2}{2 \left (1+x^2\right )}-\int \frac {1}{1+x^2} \, dx+\int \frac {x}{1+x^2} \, dx\\ &=x-\frac {x^2}{2 \left (1+x^2\right )}-\tan ^{-1}(x)+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 0.90 \[ \frac {1}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 40, normalized size = 1.33 \[ \frac {2 \, x^{3} - 2 \, {\left (x^{2} + 1\right )} \arctan \relax (x) + {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2 \, x + 1}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 23, normalized size = 0.77 \[ x + \frac {1}{2 \, {\left (x^{2} + 1\right )}} - \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.80 \[ x -\arctan \relax (x )+\frac {\ln \left (x^{2}+1\right )}{2}+\frac {1}{2 x^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 23, normalized size = 0.77 \[ x + \frac {1}{2 \, {\left (x^{2} + 1\right )}} - \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 23, normalized size = 0.77 \[ x+\frac {\ln \left (x^2+1\right )}{2}-\mathrm {atan}\relax (x)+\frac {1}{2\,\left (x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.67 \[ x + \frac {\log {\left (x^{2} + 1 \right )}}{2} - \operatorname {atan}{\relax (x )} + \frac {1}{2 x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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