Optimal. Leaf size=13 \[ \frac {1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1673, 1149, 203, 1247, 626, 31} \[ \frac {1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 31
Rule 203
Rule 626
Rule 1149
Rule 1247
Rule 1673
Rubi steps
\begin {align*} \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx &=\int \frac {x \left (1+x^2\right )}{2+3 x^2+x^4} \, dx+\int \frac {2+x^2}{2+3 x^2+x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x}{2+3 x+x^2} \, dx,x,x^2\right )+\int \frac {1}{1+x^2} \, dx\\ &=\tan ^{-1}(x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right )\\ &=\tan ^{-1}(x)+\frac {1}{2} \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 11, normalized size = 0.85 \[ \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 11, normalized size = 0.85 \[ \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 0.92 \[ \arctan \relax (x )+\frac {\ln \left (x^{2}+2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.16, size = 11, normalized size = 0.85 \[ \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 11, normalized size = 0.85 \[ \frac {\ln \left (x^2+2\right )}{2}+\mathrm {atan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 10, normalized size = 0.77 \[ \frac {\log {\left (x^{2} + 2 \right )}}{2} + \operatorname {atan}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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