Integrand size = 28, antiderivative size = 72 \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=-\frac {x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (335+217 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac {257 \arctan (x)}{8}+\frac {731 \arctan \left (\frac {x}{\sqrt {2}}\right )}{16 \sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1692, 1192, 1180, 209} \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=-\frac {257 \arctan (x)}{8}+\frac {731 \arctan \left (\frac {x}{\sqrt {2}}\right )}{16 \sqrt {2}}-\frac {x \left (12 x^2+11\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac {x \left (217 x^2+335\right )}{16 \left (x^4+3 x^2+2\right )} \]
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Rule 209
Rule 1180
Rule 1192
Rule 1692
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {1}{8} \int \frac {-38+80 x^2}{\left (2+3 x^2+x^4\right )^2} \, dx \\ & = -\frac {x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (335+217 x^2\right )}{16 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \frac {-594+434 x^2}{2+3 x^2+x^4} \, dx \\ & = -\frac {x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (335+217 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac {257}{8} \int \frac {1}{1+x^2} \, dx+\frac {731}{16} \int \frac {1}{2+x^2} \, dx \\ & = -\frac {x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (335+217 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac {257}{8} \tan ^{-1}(x)+\frac {731 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{16 \sqrt {2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {1}{32} \left (\frac {2 x \left (626+1391 x^2+986 x^4+217 x^6\right )}{\left (2+3 x^2+x^4\right )^2}-1028 \arctan (x)+731 \sqrt {2} \arctan \left (\frac {x}{\sqrt {2}}\right )\right ) \]
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Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {\frac {217}{16} x^{7}+\frac {493}{8} x^{5}+\frac {1391}{16} x^{3}+\frac {313}{8} x}{\left (x^{4}+3 x^{2}+2\right )^{2}}-\frac {257 \arctan \left (x \right )}{8}+\frac {731 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{32}\) | \(50\) |
default | \(\frac {\frac {155}{16} x^{3}+\frac {181}{8} x}{\left (x^{2}+2\right )^{2}}+\frac {731 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{32}-\frac {-\frac {31}{8} x^{3}-\frac {33}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {257 \arctan \left (x \right )}{8}\) | \(53\) |
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Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.38 \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {434 \, x^{7} + 1972 \, x^{5} + 2782 \, x^{3} + 731 \, \sqrt {2} {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 1028 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 1252 \, x}{32 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {217 x^{7} + 986 x^{5} + 1391 x^{3} + 626 x}{16 x^{8} + 96 x^{6} + 208 x^{4} + 192 x^{2} + 64} - \frac {257 \operatorname {atan}{\left (x \right )}}{8} + \frac {731 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{32} \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {731}{32} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {217 \, x^{7} + 986 \, x^{5} + 1391 \, x^{3} + 626 \, x}{16 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac {257}{8} \, \arctan \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {731}{32} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {217 \, x^{7} + 986 \, x^{5} + 1391 \, x^{3} + 626 \, x}{16 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac {257}{8} \, \arctan \left (x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {731\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{32}-\frac {257\,\mathrm {atan}\left (x\right )}{8}+\frac {\frac {217\,x^7}{16}+\frac {493\,x^5}{8}+\frac {1391\,x^3}{16}+\frac {313\,x}{8}}{x^8+6\,x^6+13\,x^4+12\,x^2+4} \]
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