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 = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 203
Rule 1166
Rule 1678
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 1.08, size = 59, normalized size = 1.23 \[ -\frac {24 \, x^{3} + 19 \, \sqrt {2} {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 34 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \relax (x) + 22 \, x}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 40, normalized size = 0.83 \[ -\frac {19}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {12 \, x^{3} + 11 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac {17}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.79 \[ \frac {x}{2 x^{2}+2}-\frac {13 x}{2 \left (x^{2}+2\right )}+\frac {17 \arctan \relax (x )}{2}-\frac {19 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 40, normalized size = 0.83 \[ -\frac {19}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {12 \, x^{3} + 11 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac {17}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 40, normalized size = 0.83 \[ \frac {17\,\mathrm {atan}\relax (x)}{2}-\frac {19\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{4}-\frac {6\,x^3+\frac {11\,x}{2}}{x^4+3\,x^2+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 46, normalized size = 0.96 \[ \frac {- 12 x^{3} - 11 x}{2 x^{4} + 6 x^{2} + 4} + \frac {17 \operatorname {atan}{\relax (x )}}{2} - \frac {19 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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