Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0106643, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1166, 203} \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{-10+x^2}{4+9 x^2+2 x^4} \, dx &=-\left (3 \int \frac{1}{1+2 x^2} \, dx\right )+4 \int \frac{1}{8+2 x^2} \, dx\\ &=\tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0124791, size = 22, normalized size = 1. \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 17, normalized size = 0.8 \begin{align*} \arctan \left ({\frac{x}{2}} \right ) -{\frac{3\,\arctan \left ( x\sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64696, size = 22, normalized size = 1. \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27442, size = 65, normalized size = 2.95 \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.132663, size = 20, normalized size = 0.91 \begin{align*} \operatorname{atan}{\left (\frac{x}{2} \right )} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1067, size = 22, normalized size = 1. \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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