Optimal. Leaf size=48 \[ -\frac{x \left (12 x^2+11\right )}{2 \left (x^4+3 x^2+2\right )}+\frac{17}{2} \tan ^{-1}(x)-\frac{19 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0280516, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {1678, 1166, 203} \[ -\frac{x \left (12 x^2+11\right )}{2 \left (x^4+3 x^2+2\right )}+\frac{17}{2} \tan ^{-1}(x)-\frac{19 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1678
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^2} \, dx &=-\frac{x \left (11+12 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-30+4 x^2}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (11+12 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{17}{2} \int \frac{1}{1+x^2} \, dx-\frac{19}{2} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{x \left (11+12 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{17}{2} \tan ^{-1}(x)-\frac{19 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0405299, size = 46, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 x \left (12 x^2+11\right )}{x^4+3 x^2+2}+34 \tan ^{-1}(x)-19 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 38, normalized size = 0.8 \begin{align*} -{\frac{13\,x}{2\,{x}^{2}+4}}-{\frac{19\,\sqrt{2}}{4}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{x}{2\,{x}^{2}+2}}+{\frac{17\,\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49892, size = 54, normalized size = 1.12 \begin{align*} -\frac{19}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{12 \, x^{3} + 11 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{17}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05986, size = 170, normalized size = 3.54 \begin{align*} -\frac{24 \, x^{3} + 19 \, \sqrt{2}{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 34 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) + 22 \, x}{4 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.181602, size = 44, normalized size = 0.92 \begin{align*} - \frac{12 x^{3} + 11 x}{2 x^{4} + 6 x^{2} + 4} + \frac{17 \operatorname{atan}{\left (x \right )}}{2} - \frac{19 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08092, size = 54, normalized size = 1.12 \begin{align*} -\frac{19}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{12 \, x^{3} + 11 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{17}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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