Optimal. Leaf size=38 \[ \frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1161, 618, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+x^2}{1+x^2+x^4} \, dx &=\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x+x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {\tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 99, normalized size = 2.61 \begin {gather*} \frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{1+x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 31, normalized size = 0.82 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x^{3} + 2 \, x\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 26, normalized size = 0.68 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\relax (x) + 2 \, \arctan \left (\frac {\sqrt {3} {\left (x^{2} - 1\right )}}{3 \, x}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 0.89 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.40, size = 33, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 29, normalized size = 0.76 \begin {gather*} \frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {\sqrt {3}\,x^3}{3}+\frac {2\,\sqrt {3}\,x}{3}\right )+\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}\right )\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 41, normalized size = 1.08 \begin {gather*} \frac {\sqrt {3} \left (2 \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} \right )} + 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{3}}{3} + \frac {2 \sqrt {3} x}{3} \right )}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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