Optimal. Leaf size=23 \[ \tan ^{-1}\left (4 x+\sqrt {7}\right )-\tan ^{-1}\left (\sqrt {7}-4 x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 23, 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} \[ \tan ^{-1}\left (4 x+\sqrt {7}\right )-\tan ^{-1}\left (\sqrt {7}-4 x\right ) \]
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 {\sqrt {7} x}{2}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {\sqrt {7} x}{2}+x^2} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{4}-x^2} \, dx,x,-\frac {\sqrt {7}}{2}+2 x\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{4}-x^2} \, dx,x,\frac {\sqrt {7}}{2}+2 x\right )\\ &=-\tan ^{-1}\left (\sqrt {7}-4 x\right )+\tan ^{-1}\left (\sqrt {7}+4 x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 0.61 \[ -\tan ^{-1}\left (\frac {x}{2 x^2-1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 15, normalized size = 0.65 \[ \arctan \left (4 \, x^{3} - x\right ) + \arctan \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 42, normalized size = 1.83 \[ \arctan \left (2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x + \sqrt {14} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \arctan \left (2 \, \sqrt {2} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x - \sqrt {14} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.87 \[ \arctan \left (4 x -\sqrt {7}\right )+\arctan \left (4 x +\sqrt {7}\right ) \]
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 \[ \int \frac {2 \, x^{2} + 1}{4 \, x^{4} - 3 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.35, size = 15, normalized size = 0.65 \[ \mathrm {atan}\left (2\,x\right )-\mathrm {atan}\left (x-4\,x^3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 12, normalized size = 0.52 \[ \operatorname {atan}{\left (2 x \right )} + \operatorname {atan}{\left (4 x^{3} - x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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