Integrand size = 31, antiderivative size = 57 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=98 x-9 x^3+x^5+\frac {x \left (102+103 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {11 \arctan (x)}{2}-118 \sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1682, 1690, 1180, 209} \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=-\frac {11 \arctan (x)}{2}-118 \sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )+x^5-9 x^3+\frac {\left (103 x^2+102\right ) x}{2 \left (x^4+3 x^2+2\right )}+98 x \]
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Rule 209
Rule 1180
Rule 1682
Rule 1690
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=98 x-9 x^3+x^5+\frac {102 x+103 x^3}{2 \left (2+3 x^2+x^4\right )}-\frac {11 \arctan (x)}{2}-118 \sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {52 x}{x^{2}+2}-118 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}+x^{5}-9 x^{3}+98 x -\frac {x}{2 \left (x^{2}+1\right )}-\frac {11 \arctan \left (x \right )}{2}\) | \(49\) |
risch | \(x^{5}-9 x^{3}+98 x +\frac {\frac {103}{2} x^{3}+51 x}{x^{4}+3 x^{2}+2}-\frac {11 \arctan \left (x \right )}{2}-118 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}\) | \(51\) |
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Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.30 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=\frac {2 \, x^{9} - 12 \, x^{7} + 146 \, x^{5} + 655 \, x^{3} - 236 \, \sqrt {2} {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 11 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) + 494 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=x^{5} - 9 x^{3} + 98 x + \frac {103 x^{3} + 102 x}{2 x^{4} + 6 x^{2} + 4} - \frac {11 \operatorname {atan}{\left (x \right )}}{2} - 118 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=x^{5} - 9 \, x^{3} - 118 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 98 \, x + \frac {103 \, x^{3} + 102 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {11}{2} \, \arctan \left (x\right ) \]
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Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=x^{5} - 9 \, x^{3} - 118 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 98 \, x + \frac {103 \, x^{3} + 102 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {11}{2} \, \arctan \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88 \[ \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=98\,x-\frac {11\,\mathrm {atan}\left (x\right )}{2}-118\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )+\frac {\frac {103\,x^3}{2}+51\,x}{x^4+3\,x^2+2}-9\,x^3+x^5 \]
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