Optimal. Leaf size=15 \[ \frac{1}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}(2 x) \]
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Rubi [A] time = 0.0093769, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1163, 203} \[ \frac{1}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}(2 x) \]
Antiderivative was successfully verified.
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Rule 1163
Rule 203
Rubi steps
\begin{align*} \int \frac{1+2 x^2}{1+5 x^2+4 x^4} \, dx &=\frac{2}{3} \int \frac{1}{1+4 x^2} \, dx+\frac{4}{3} \int \frac{1}{4+4 x^2} \, dx\\ &=\frac{1}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}(2 x)\\ \end{align*}
Mathematica [A] time = 0.0075067, size = 17, normalized size = 1.13 \[ -\frac{1}{3} \tan ^{-1}\left (\frac{3 x}{2 x^2-1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 12, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ( x \right ) }{3}}+{\frac{\arctan \left ( 2\,x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45741, size = 15, normalized size = 1. \begin{align*} \frac{1}{3} \, \arctan \left (2 \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27451, size = 66, normalized size = 4.4 \begin{align*} \frac{1}{3} \, \arctan \left (\frac{4}{3} \, x^{3} + \frac{7}{3} \, x\right ) + \frac{1}{3} \, \arctan \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.104221, size = 22, normalized size = 1.47 \begin{align*} \frac{\operatorname{atan}{\left (\frac{2 x}{3} \right )}}{3} + \frac{\operatorname{atan}{\left (\frac{4 x^{3}}{3} + \frac{7 x}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13615, size = 15, normalized size = 1. \begin{align*} \frac{1}{3} \, \arctan \left (2 \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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