Optimal. Leaf size=70 \[ \frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}-\frac{\left (207 x^2+206\right ) x}{2 \left (x^4+3 x^2+2\right )}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0845314, antiderivative size = 70, normalized size of antiderivative = 1., 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} \[ \frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}-\frac{\left (207 x^2+206\right ) x}{2 \left (x^4+3 x^2+2\right )}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1676
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-412-6 x^2+212 x^4-108 x^6+48 x^8-20 x^{10}}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (1172-392 x^2+108 x^4-20 x^6-\frac{2 \left (1378+1369 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=-293 x+\frac{98 x^3}{3}-\frac{27 x^5}{5}+\frac{5 x^7}{7}-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{1}{2} \int \frac{1378+1369 x^2}{2+3 x^2+x^4} \, dx\\ &=-293 x+\frac{98 x^3}{3}-\frac{27 x^5}{5}+\frac{5 x^7}{7}-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{9}{2} \int \frac{1}{1+x^2} \, dx+680 \int \frac{1}{2+x^2} \, dx\\ &=-293 x+\frac{98 x^3}{3}-\frac{27 x^5}{5}+\frac{5 x^7}{7}-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.045937, size = 71, normalized size = 1.01 \[ \frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}+\frac{-207 x^3-206 x}{2 \left (x^4+3 x^2+2\right )}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 56, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{7}}{7}}-{\frac{27\,{x}^{5}}{5}}+{\frac{98\,{x}^{3}}{3}}-293\,x-104\,{\frac{x}{{x}^{2}+2}}+340\,\arctan \left ( 1/2\,x\sqrt{2} \right ) \sqrt{2}+{\frac{x}{2\,{x}^{2}+2}}+{\frac{9\,\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46827, size = 78, normalized size = 1.11 \begin{align*} \frac{5}{7} \, x^{7} - \frac{27}{5} \, x^{5} + \frac{98}{3} \, x^{3} + 340 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 293 \, x - \frac{207 \, x^{3} + 206 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{9}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84306, size = 247, normalized size = 3.53 \begin{align*} \frac{150 \, x^{11} - 684 \, x^{9} + 3758 \, x^{7} - 43218 \, x^{5} - 192605 \, x^{3} + 71400 \, \sqrt{2}{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 945 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) - 144690 \, x}{210 \,{\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 = 0.186366, size = 66, normalized size = 0.94 \begin{align*} \frac{5 x^{7}}{7} - \frac{27 x^{5}}{5} + \frac{98 x^{3}}{3} - 293 x - \frac{207 x^{3} + 206 x}{2 x^{4} + 6 x^{2} + 4} + \frac{9 \operatorname{atan}{\left (x \right )}}{2} + 340 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08255, size = 78, normalized size = 1.11 \begin{align*} \frac{5}{7} \, x^{7} - \frac{27}{5} \, x^{5} + \frac{98}{3} \, x^{3} + 340 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 293 \, x - \frac{207 \, x^{3} + 206 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{9}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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