Optimal. Leaf size=43 \[ \frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{3 x+1}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0348875, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1161, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{3 x+1}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1161
Rule 618
Rule 204
Rubi steps
\begin{align*} \int -\frac{1+3 x^2}{1+2 x^2+9 x^4} \, dx &=-\left (\frac{1}{6} \int \frac{1}{\frac{1}{3}-\frac{2 x}{3}+x^2} \, dx\right )-\frac{1}{6} \int \frac{1}{\frac{1}{3}+\frac{2 x}{3}+x^2} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-\frac{8}{9}-x^2} \, dx,x,-\frac{2}{3}+2 x\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-\frac{8}{9}-x^2} \, dx,x,\frac{2}{3}+2 x\right )\\ &=\frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{1+3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.100428, size = 99, normalized size = 2.3 \[ -\frac{\left (\sqrt{2}-i\right ) \tan ^{-1}\left (\frac{3 x}{\sqrt{1-2 i \sqrt{2}}}\right )}{2 \sqrt{2 \left (1-2 i \sqrt{2}\right )}}-\frac{\left (\sqrt{2}+i\right ) \tan ^{-1}\left (\frac{3 x}{\sqrt{1+2 i \sqrt{2}}}\right )}{2 \sqrt{2 \left (1+2 i \sqrt{2}\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 34, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( 6\,x-2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( 6\,x+2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45843, size = 45, normalized size = 1.05 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41658, size = 113, normalized size = 2.63 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (9 \, x^{3} + 5 \, x\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{3}{4} \, \sqrt{2} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.11981, size = 46, normalized size = 1.07 \begin{align*} - \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{4} \right )} + 2 \operatorname{atan}{\left (\frac{9 \sqrt{2} x^{3}}{4} + \frac{5 \sqrt{2} x}{4} \right )}\right )}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12498, size = 45, normalized size = 1.05 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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