Optimal. Leaf size=70 \[ -293 x+\frac {98 x^3}{3}-\frac {27 x^5}{5}+\frac {5 x^7}{7}-\frac {x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac {9}{2} \tan ^{-1}(x)+340 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 70, 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 = {1682, 1690,
1180, 209} \begin {gather*} \frac {9 \text {ArcTan}(x)}{2}+340 \sqrt {2} \text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )+\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1180
Rule 1682
Rule 1690
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*}
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Mathematica [A]
time = 0.03, size = 71, normalized size = 1.01 \begin {gather*} -293 x+\frac {98 x^3}{3}-\frac {27 x^5}{5}+\frac {5 x^7}{7}+\frac {-206 x-207 x^3}{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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 56, normalized size = 0.80
method | result | size |
default | \(\frac {5 x^{7}}{7}-\frac {27 x^{5}}{5}+\frac {98 x^{3}}{3}-293 x +\frac {x}{2 x^{2}+2}+\frac {9 \arctan \left (x \right )}{2}-\frac {104 x}{x^{2}+2}+340 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\) | \(56\) |
risch | \(\frac {5 x^{7}}{7}-\frac {27 x^{5}}{5}+\frac {98 x^{3}}{3}-293 x +\frac {-\frac {207}{2} x^{3}-103 x}{x^{4}+3 x^{2}+2}+\frac {9 \arctan \left (x \right )}{2}+340 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 58, normalized size = 0.83 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 79, normalized size = 1.13 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 68, normalized size = 0.97 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.88, size = 58, normalized size = 0.83 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.95, size = 58, normalized size = 0.83 \begin {gather*} \frac {9\,\mathrm {atan}\left (x\right )}{2}-293\,x+340\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )-\frac {\frac {207\,x^3}{2}+103\,x}{x^4+3\,x^2+2}+\frac {98\,x^3}{3}-\frac {27\,x^5}{5}+\frac {5\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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