Optimal. Leaf size=22 \[ \frac {x}{x^2+1}+\frac {1}{2} \log \left (x^2+1\right )+\tan ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {28, 1814, 635, 203, 260} \[ \frac {x}{x^2+1}+\frac {1}{2} \log \left (x^2+1\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 260
Rule 635
Rule 1814
Rubi steps
\begin {align*} \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx &=\int \frac {2+x+x^3}{\left (1+x^2\right )^2} \, dx\\ &=\frac {x}{1+x^2}-\frac {1}{2} \int \frac {-2-2 x}{1+x^2} \, dx\\ &=\frac {x}{1+x^2}+\int \frac {1}{1+x^2} \, dx+\int \frac {x}{1+x^2} \, dx\\ &=\frac {x}{1+x^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 = 22, normalized size = 1.00 \[ \frac {x}{x^2+1}+\frac {1}{2} \log \left (x^2+1\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 34, normalized size = 1.55 \[ \frac {2 \, {\left (x^{2} + 1\right )} \arctan \relax (x) + {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2 \, x}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 20, normalized size = 0.91 \[ \frac {x}{x^{2} + 1} + \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.95 \[ \frac {x}{x^{2}+1}+\arctan \relax (x )+\frac {\ln \left (x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.23, size = 20, normalized size = 0.91 \[ \frac {x}{x^{2} + 1} + \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 20, normalized size = 0.91 \[ \frac {\ln \left (x^2+1\right )}{2}+\mathrm {atan}\relax (x)+\frac {x}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 17, normalized size = 0.77 \[ \frac {x}{x^{2} + 1} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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