Optimal. Leaf size=81 \[ 214 x-14 x^3+x^5+\frac {x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {477}{8} \tan ^{-1}(x)-351 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1682, 1692,
1690, 1180, 209} \begin {gather*} \frac {477 \text {ArcTan}(x)}{8}-351 \sqrt {2} \text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )+x^5-14 x^3+\frac {\left (1669 x^2+824\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac {\left (415 x^2+414\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+214 x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1180
Rule 1682
Rule 1690
Rule 1692
Rubi steps
\begin {align*} \int \frac {x^{10} \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=\frac {x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {1}{8} \int \frac {828-2478 x^2-840 x^4+424 x^6-216 x^8+96 x^{10}-40 x^{12}}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac {x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \frac {-4952-2700 x^2+3136 x^4-864 x^6+160 x^8}{2+3 x^2+x^4} \, dx\\ &=\frac {x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \left (6848-1344 x^2+160 x^4-\frac {36 \left (518+571 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=214 x-14 x^3+x^5+\frac {x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {9}{8} \int \frac {518+571 x^2}{2+3 x^2+x^4} \, dx\\ &=214 x-14 x^3+x^5+\frac {x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {477}{8} \int \frac {1}{1+x^2} \, dx-702 \int \frac {1}{2+x^2} \, dx\\ &=214 x-14 x^3+x^5+\frac {x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {477}{8} \tan ^{-1}(x)-351 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 71, normalized size = 0.88 \begin {gather*} \frac {x \left (9324+26736 x^2+26775 x^4+10581 x^6+1144 x^8-64 x^{10}+8 x^{12}\right )}{8 \left (2+3 x^2+x^4\right )^2}+\frac {477}{8} \tan ^{-1}(x)-351 \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 = 64, normalized size = 0.79
method | result | size |
risch | \(x^{5}-14 x^{3}+214 x +\frac {\frac {1669}{8} x^{7}+\frac {5831}{8} x^{5}+830 x^{3}+\frac {619}{2} x}{\left (x^{4}+3 x^{2}+2\right )^{2}}-351 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\frac {477 \arctan \left (x \right )}{8}\) | \(61\) |
default | \(x^{5}-14 x^{3}+214 x +\frac {-\frac {11}{8} x^{3}-\frac {13}{8} x}{\left (x^{2}+1\right )^{2}}+\frac {477 \arctan \left (x \right )}{8}-\frac {16 \left (-\frac {105}{8} x^{3}-\frac {79}{4} x \right )}{\left (x^{2}+2\right )^{2}}-351 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 71, normalized size = 0.88 \begin {gather*} x^{5} - 14 \, x^{3} - 351 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 214 \, x + \frac {1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac {477}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 114, normalized size = 1.41 \begin {gather*} \frac {8 \, x^{13} - 64 \, x^{11} + 1144 \, x^{9} + 10581 \, x^{7} + 26775 \, x^{5} + 26736 \, x^{3} - 2808 \, \sqrt {2} {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 477 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 9324 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 75, normalized size = 0.93 \begin {gather*} x^{5} - 14 x^{3} + 214 x + \frac {1669 x^{7} + 5831 x^{5} + 6640 x^{3} + 2476 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac {477 \operatorname {atan}{\left (x \right )}}{8} - 351 \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 = 4.45, size = 61, normalized size = 0.75 \begin {gather*} x^{5} - 14 \, x^{3} - 351 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 214 \, x + \frac {1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac {477}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 70, normalized size = 0.86 \begin {gather*} 214\,x+\frac {477\,\mathrm {atan}\left (x\right )}{8}-351\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )+\frac {\frac {1669\,x^7}{8}+\frac {5831\,x^5}{8}+830\,x^3+\frac {619\,x}{2}}{x^8+6\,x^6+13\,x^4+12\,x^2+4}-14\,x^3+x^5 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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