Optimal. Leaf size=46 \[ \frac {\tan ^{-1}\left (\frac {4 x+\sqrt {3}}{\sqrt {5}}\right )}{\sqrt {5}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3}-4 x}{\sqrt {5}}\right )}{\sqrt {5}} \]
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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1161, 618, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {4 x+\sqrt {3}}{\sqrt {5}}\right )}{\sqrt {5}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3}-4 x}{\sqrt {5}}\right )}{\sqrt {5}} \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 {3} x}{2}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {\sqrt {3} x}{2}+x^2} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {5}{4}-x^2} \, dx,x,-\frac {\sqrt {3}}{2}+2 x\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {5}{4}-x^2} \, dx,x,\frac {\sqrt {3}}{2}+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3}-4 x}{\sqrt {5}}\right )}{\sqrt {5}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}+4 x}{\sqrt {5}}\right )}{\sqrt {5}}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 97, normalized size = 2.11 \begin {gather*} \frac {\left (\sqrt {15}-3 i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {15}\right )}}\right )}{\sqrt {30-30 i \sqrt {15}}}+\frac {\left (\sqrt {15}+3 i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {15}\right )}}\right )}{\sqrt {30+30 i \sqrt {15}}} \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 = 1.77, size = 33, normalized size = 0.72 \begin {gather*} \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (4 \, x^{3} + 3 \, x\right )}\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {2}{5} \, \sqrt {5} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 52, normalized size = 1.13 \begin {gather*} \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {2}{5} \, \sqrt {10} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x + \sqrt {6} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {2}{5} \, \sqrt {10} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x - \sqrt {6} \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 {5}\, \arctan \left (\frac {\left (4 x -\sqrt {3}\right ) \sqrt {5}}{5}\right )}{5}+\frac {\sqrt {5}\, \arctan \left (\frac {\left (4 x +\sqrt {3}\right ) \sqrt {5}}{5}\right )}{5} \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 = 4.36, size = 29, normalized size = 0.63 \begin {gather*} \frac {\sqrt {5}\,\left (\mathrm {atan}\left (\frac {4\,\sqrt {5}\,x^3}{5}+\frac {3\,\sqrt {5}\,x}{5}\right )+\mathrm {atan}\left (\frac {2\,\sqrt {5}\,x}{5}\right )\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 44, normalized size = 0.96 \begin {gather*} \frac {\sqrt {5} \left (2 \operatorname {atan}{\left (\frac {2 \sqrt {5} x}{5} \right )} + 2 \operatorname {atan}{\left (\frac {4 \sqrt {5} x^{3}}{5} + \frac {3 \sqrt {5} x}{5} \right )}\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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