Optimal. Leaf size=13 \[ \tan ^{-1}(x)+\frac {1}{2} \log \left (2+x^2\right ) \]
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Rubi [A]
time = 0.02, 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 = {1687, 1163,
209, 1261, 640, 31} \begin {gather*} \text {ArcTan}(x)+\frac {1}{2} \log \left (x^2+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 209
Rule 640
Rule 1163
Rule 1261
Rule 1687
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} \text {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} \text {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 \begin {gather*} \tan ^{-1}(x)+\frac {1}{2} \log \left (2+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 12, normalized size = 0.92
method | result | size |
default | \(\arctan \left (x \right )+\frac {\ln \left (x^{2}+2\right )}{2}\) | \(12\) |
risch | \(\arctan \left (x \right )+\frac {\ln \left (x^{2}+2\right )}{2}\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 11, normalized size = 0.85 \begin {gather*} \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 11, normalized size = 0.85 \begin {gather*} \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 10, normalized size = 0.77 \begin {gather*} \frac {\log {\left (x^{2} + 2 \right )}}{2} + \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.12, size = 11, normalized size = 0.85 \begin {gather*} \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \end {gather*}
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 \begin {gather*} \frac {\ln \left (x^2+2\right )}{2}+\mathrm {atan}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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