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 = 0.0352245, antiderivative size = 38, normalized size of antiderivative = 1., 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} \[ \frac{\tan ^{-1}\left (\frac{4 x+1}{\sqrt{7}}\right )}{\sqrt{7}}-\frac{\tan ^{-1}\left (\frac{1-4 x}{\sqrt{7}}\right )}{\sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 1161
Rule 618
Rule 204
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*}
Mathematica [C] time = 0.185003, size = 97, normalized size = 2.55 \[ \frac{\left (\sqrt{7}-i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (3-i \sqrt{7}\right )}}\right )}{\sqrt{42-14 i \sqrt{7}}}+\frac{\left (\sqrt{7}+i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (3+i \sqrt{7}\right )}}\right )}{\sqrt{42+14 i \sqrt{7}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 34, normalized size = 0.9 \begin{align*}{\frac{\sqrt{7}}{7}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{7}}{7}} \right ) }+{\frac{\sqrt{7}}{7}\arctan \left ({\frac{ \left ( 4\,x+1 \right ) \sqrt{7}}{7}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44026, size = 45, normalized size = 1.18 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30687, size = 112, normalized size = 2.95 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.115235, size = 44, normalized size = 1.16 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13606, size = 45, normalized size = 1.18 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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