Optimal. Leaf size=57 \[ x^5-9 x^3+\frac {\left (103 x^2+102\right ) x}{2 \left (x^4+3 x^2+2\right )}+98 x-\frac {11}{2} \tan ^{-1}(x)-118 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1668, 1676, 1166, 203} \begin {gather*} x^5-9 x^3+\frac {\left (103 x^2+102\right ) x}{2 \left (x^4+3 x^2+2\right )}+98 x-\frac {11}{2} \tan ^{-1}(x)-118 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1166
Rule 1668
Rule 1676
Rubi steps
\begin {align*} \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {x \left (102+103 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \frac {204+6 x^2-108 x^4+48 x^6-20 x^8}{2+3 x^2+x^4} \, dx\\ &=\frac {x \left (102+103 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \left (-392+108 x^2-20 x^4+\frac {2 \left (494+483 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=98 x-9 x^3+x^5+\frac {x \left (102+103 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \int \frac {494+483 x^2}{2+3 x^2+x^4} \, dx\\ &=98 x-9 x^3+x^5+\frac {x \left (102+103 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {11}{2} \int \frac {1}{1+x^2} \, dx-236 \int \frac {1}{2+x^2} \, dx\\ &=98 x-9 x^3+x^5+\frac {x \left (102+103 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {11}{2} \tan ^{-1}(x)-118 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 1.02 \begin {gather*} x^5-9 x^3+\frac {103 x^3+102 x}{2 \left (x^4+3 x^2+2\right )}+98 x-\frac {11}{2} \tan ^{-1}(x)-118 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.31, size = 74, normalized size = 1.30 \begin {gather*} \frac {2 \, x^{9} - 12 \, x^{7} + 146 \, x^{5} + 655 \, x^{3} - 236 \, \sqrt {2} {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 11 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \relax (x) + 494 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 51, normalized size = 0.89 \begin {gather*} x^{5} - 9 \, x^{3} - 118 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 98 \, x + \frac {103 \, x^{3} + 102 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {11}{2} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.86 \begin {gather*} x^{5}-9 x^{3}+98 x -\frac {x}{2 \left (x^{2}+1\right )}+\frac {52 x}{x^{2}+2}-\frac {11 \arctan \relax (x )}{2}-118 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 51, normalized size = 0.89 \begin {gather*} x^{5} - 9 \, x^{3} - 118 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 98 \, x + \frac {103 \, x^{3} + 102 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {11}{2} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 50, normalized size = 0.88 \begin {gather*} 98\,x-\frac {11\,\mathrm {atan}\relax (x)}{2}-118\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )+\frac {\frac {103\,x^3}{2}+51\,x}{x^4+3\,x^2+2}-9\,x^3+x^5 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 54, normalized size = 0.95 \begin {gather*} x^{5} - 9 x^{3} + 98 x + \frac {103 x^{3} + 102 x}{2 x^{4} + 6 x^{2} + 4} - \frac {11 \operatorname {atan}{\relax (x )}}{2} - 118 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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