Optimal. Leaf size=37 \[ \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )+\frac {1}{4} \log \left (1+x^2+x^4\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1607, 1261,
648, 632, 210, 642} \begin {gather*} \frac {1}{2} \sqrt {3} \text {ArcTan}\left (\frac {2 x^2+1}{\sqrt {3}}\right )+\frac {1}{4} \log \left (x^4+x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1261
Rule 1607
Rubi steps
\begin {align*} \int \frac {2 x+x^3}{1+x^2+x^4} \, dx &=\int \frac {x \left (2+x^2\right )}{1+x^2+x^4} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {2+x}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^2\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} \log \left (1+x^2+x^4\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )+\frac {1}{4} \log \left (1+x^2+x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 37, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )+\frac {1}{4} \log \left (1+x^2+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 31, normalized size = 0.84
method | result | size |
default | \(\frac {\ln \left (x^{4}+x^{2}+1\right )}{4}+\frac {\arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2}\) | \(31\) |
risch | \(\frac {\ln \left (4 x^{4}+4 x^{2}+4\right )}{4}+\frac {\arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 53, normalized size = 1.43 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{4} \, \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 30, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{4} + x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 37, normalized size = 1.00 \begin {gather*} \frac {\log {\left (x^{4} + x^{2} + 1 \right )}}{4} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} + \frac {\sqrt {3}}{3} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.73, size = 30, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{4} + 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 = 0.86 \begin {gather*} \frac {\ln \left (x^4+x^2+1\right )}{4}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^2}{3}+\frac {\sqrt {3}}{3}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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