Optimal. Leaf size=38 \[ \frac {\tan ^{-1}\left (\frac {4 x+1}{\sqrt {7}}\right )}{\sqrt {7}}-\frac {\tan ^{-1}\left (\frac {1-4 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 38, 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+1}{\sqrt {7}}\right )}{\sqrt {7}}-\frac {\tan ^{-1}\left (\frac {1-4 x}{\sqrt {7}}\right )}{\sqrt {7}} \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+3 x^2+4 x^4} \, dx &=\frac {1}{4} \int \frac {1}{\frac {1}{2}-\frac {x}{2}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {x}{2}+x^2} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {7}{4}-x^2} \, dx,x,-\frac {1}{2}+2 x\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {7}{4}-x^2} \, dx,x,\frac {1}{2}+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1-4 x}{\sqrt {7}}\right )}{\sqrt {7}}+\frac {\tan ^{-1}\left (\frac {1+4 x}{\sqrt {7}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 97, normalized size = 2.55 \begin {gather*} \frac {\left (\sqrt {7}-i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (3-i \sqrt {7}\right )}}\right )}{\sqrt {42-14 i \sqrt {7}}}+\frac {\left (\sqrt {7}+i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (3+i \sqrt {7}\right )}}\right )}{\sqrt {42+14 i \sqrt {7}}} \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+3 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 33, normalized size = 0.87 \begin {gather*} \frac {1}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x^{3} + 5 \, x\right )}\right ) + \frac {1}{7} \, \sqrt {7} \arctan \left (\frac {2}{7} \, \sqrt {7} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 33, normalized size = 0.87 \begin {gather*} \frac {1}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x + 1\right )}\right ) + \frac {1}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x - 1\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 {7}\, \arctan \left (\frac {\left (4 x +1\right ) \sqrt {7}}{7}\right )}{7}+\frac {\sqrt {7}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {7}}{7}\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 33, normalized size = 0.87 \begin {gather*} \frac {1}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x + 1\right )}\right ) + \frac {1}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 29, normalized size = 0.76 \begin {gather*} \frac {\sqrt {7}\,\left (\mathrm {atan}\left (\frac {4\,\sqrt {7}\,x^3}{7}+\frac {5\,\sqrt {7}\,x}{7}\right )+\mathrm {atan}\left (\frac {2\,\sqrt {7}\,x}{7}\right )\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 44, normalized size = 1.16 \begin {gather*} \frac {\sqrt {7} \left (2 \operatorname {atan}{\left (\frac {2 \sqrt {7} x}{7} \right )} + 2 \operatorname {atan}{\left (\frac {4 \sqrt {7} x^{3}}{7} + \frac {5 \sqrt {7} x}{7} \right )}\right )}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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