Optimal. Leaf size=46 \[ \frac {\tan ^{-1}\left (\frac {4 x+\sqrt {5}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1161, 618, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {4 x+\sqrt {5}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {5}-4 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+2 x^2}{1-x^2+4 x^4} \, dx &=\frac {1}{4} \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5} x}{2}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5} x}{2}+x^2} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,-\frac {\sqrt {5}}{2}+2 x\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,\frac {\sqrt {5}}{2}+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {5}+4 x}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 101, normalized size = 2.20 \begin {gather*} \frac {\left (\sqrt {15}-5 i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (-1-i \sqrt {15}\right )}}\right )}{\sqrt {30 \left (-1-i \sqrt {15}\right )}}+\frac {\left (\sqrt {15}+5 i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (-1+i \sqrt {15}\right )}}\right )}{\sqrt {30 \left (-1+i \sqrt {15}\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+2 x^2}{1-x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.63, size = 31, normalized size = 0.67 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (4 \, x^{3} + x\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 52, normalized size = 1.13 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {6} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x + \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {6} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x - \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 40, normalized size = 0.87 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x -\sqrt {5}\right ) \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x +\sqrt {5}\right ) \sqrt {3}}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} + 1}{4 \, x^{4} - x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 29, normalized size = 0.63 \begin {gather*} \frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {4\,\sqrt {3}\,x^3}{3}+\frac {\sqrt {3}\,x}{3}\right )+\mathrm {atan}\left (\frac {2\,\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 = 0.14, size = 42, normalized size = 0.91 \begin {gather*} \frac {\sqrt {3} \left (2 \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} \right )} + 2 \operatorname {atan}{\left (\frac {4 \sqrt {3} x^{3}}{3} + \frac {\sqrt {3} x}{3} \right )}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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