Optimal. Leaf size=53 \[ \frac {x \left (11 x^2+9\right )}{4 \left (x^4+3 x^2+2\right )}-\frac {1}{x}-\frac {19}{2} \tan ^{-1}(x)+\frac {45 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1669, 1664, 203} \[ \frac {x \left (11 x^2+9\right )}{4 \left (x^4+3 x^2+2\right )}-\frac {1}{x}-\frac {19}{2} \tan ^{-1}(x)+\frac {45 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 1664
Rule 1669
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (2+3 x^2+x^4\right )^2} \, dx &=\frac {x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \frac {-8+19 x^2-11 x^4}{x^2 \left (2+3 x^2+x^4\right )} \, dx\\ &=\frac {x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \left (-\frac {4}{x^2}+\frac {38}{1+x^2}-\frac {45}{2+x^2}\right ) \, dx\\ &=-\frac {1}{x}+\frac {x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac {19}{2} \int \frac {1}{1+x^2} \, dx+\frac {45}{4} \int \frac {1}{2+x^2} \, dx\\ &=-\frac {1}{x}+\frac {x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac {19}{2} \tan ^{-1}(x)+\frac {45 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 51, normalized size = 0.96 \[ \frac {1}{8} \left (\frac {2 x \left (11 x^2+9\right )}{x^4+3 x^2+2}-\frac {8}{x}-76 \tan ^{-1}(x)+45 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 68, normalized size = 1.28 \[ \frac {14 \, x^{4} + 45 \, \sqrt {2} {\left (x^{5} + 3 \, x^{3} + 2 \, x\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 6 \, x^{2} - 76 \, {\left (x^{5} + 3 \, x^{3} + 2 \, x\right )} \arctan \relax (x) - 16}{8 \, {\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 45, normalized size = 0.85 \[ \frac {45}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {7 \, x^{4} - 3 \, x^{2} - 8}{4 \, {\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} - \frac {19}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.81 \[ -\frac {x}{2 \left (x^{2}+1\right )}+\frac {13 x}{4 \left (x^{2}+2\right )}-\frac {19 \arctan \relax (x )}{2}+\frac {45 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{8}-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 45, normalized size = 0.85 \[ \frac {45}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {7 \, x^{4} - 3 \, x^{2} - 8}{4 \, {\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} - \frac {19}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 45, normalized size = 0.85 \[ \frac {45\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{8}-\frac {19\,\mathrm {atan}\relax (x)}{2}-\frac {-\frac {7\,x^4}{4}+\frac {3\,x^2}{4}+2}{x^5+3\,x^3+2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 49, normalized size = 0.92 \[ \frac {7 x^{4} - 3 x^{2} - 8}{4 x^{5} + 12 x^{3} + 8 x} - \frac {19 \operatorname {atan}{\relax (x )}}{2} + \frac {45 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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