Integrand size = 11, antiderivative size = 58 \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=\frac {35 x}{16}-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}-\frac {35}{8} \arctan \left (\frac {x}{2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {294, 327, 209} \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=-\frac {35}{8} \arctan \left (\frac {x}{2}\right )-\frac {x^7}{6 \left (x^2+4\right )^3}-\frac {7 x^5}{24 \left (x^2+4\right )^2}-\frac {35 x^3}{48 \left (x^2+4\right )}+\frac {35 x}{16} \]
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Rule 209
Rule 294
Rule 327
Rubi steps \begin{align*} \text {integral}& = -\frac {x^7}{6 \left (4+x^2\right )^3}+\frac {7}{6} \int \frac {x^6}{\left (4+x^2\right )^3} \, dx \\ & = -\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}+\frac {35}{24} \int \frac {x^4}{\left (4+x^2\right )^2} \, dx \\ & = -\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}+\frac {35}{16} \int \frac {x^2}{4+x^2} \, dx \\ & = \frac {35 x}{16}-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}-\frac {35}{4} \int \frac {1}{4+x^2} \, dx \\ & = \frac {35 x}{16}-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}-\frac {35}{8} \arctan \left (\frac {x}{2}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=\frac {x \left (1680+1120 x^2+231 x^4+12 x^6\right )}{12 \left (4+x^2\right )^3}-\frac {35}{8} \arctan \left (\frac {x}{2}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.53
method | result | size |
risch | \(x +\frac {\frac {29}{4} x^{5}+\frac {136}{3} x^{3}+76 x}{\left (x^{2}+4\right )^{3}}-\frac {35 \arctan \left (\frac {x}{2}\right )}{8}\) | \(31\) |
default | \(x -\frac {16 \left (-\frac {29}{64} x^{5}-\frac {17}{6} x^{3}-\frac {19}{4} x \right )}{\left (x^{2}+4\right )^{3}}-\frac {35 \arctan \left (\frac {x}{2}\right )}{8}\) | \(32\) |
meijerg | \(\frac {x \left (\frac {9}{4} x^{6}+\frac {693}{16} x^{4}+210 x^{2}+315\right )}{144 \left (1+\frac {x^{2}}{4}\right )^{3}}-\frac {35 \arctan \left (\frac {x}{2}\right )}{8}\) | \(37\) |
parallelrisch | \(\frac {-420 i \ln \left (x +2 i\right ) x^{6}-26880 i \ln \left (x +2 i\right )+192 x^{7}-5040 i \ln \left (x +2 i\right ) x^{4}+20160 i \ln \left (x -2 i\right ) x^{2}+3696 x^{5}+5040 i \ln \left (x -2 i\right ) x^{4}+420 i \ln \left (x -2 i\right ) x^{6}+17920 x^{3}+26880 i \ln \left (x -2 i\right )-20160 i \ln \left (x +2 i\right ) x^{2}+26880 x}{192 \left (x^{2}+4\right )^{3}}\) | \(111\) |
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Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=\frac {24 \, x^{7} + 462 \, x^{5} + 2240 \, x^{3} - 105 \, {\left (x^{6} + 12 \, x^{4} + 48 \, x^{2} + 64\right )} \arctan \left (\frac {1}{2} \, x\right ) + 3360 \, x}{24 \, {\left (x^{6} + 12 \, x^{4} + 48 \, x^{2} + 64\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=x + \frac {87 x^{5} + 544 x^{3} + 912 x}{12 x^{6} + 144 x^{4} + 576 x^{2} + 768} - \frac {35 \operatorname {atan}{\left (\frac {x}{2} \right )}}{8} \]
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71 \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=x + \frac {87 \, x^{5} + 544 \, x^{3} + 912 \, x}{12 \, {\left (x^{6} + 12 \, x^{4} + 48 \, x^{2} + 64\right )}} - \frac {35}{8} \, \arctan \left (\frac {1}{2} \, x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.53 \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=x + \frac {87 \, x^{5} + 544 \, x^{3} + 912 \, x}{12 \, {\left (x^{2} + 4\right )}^{3}} - \frac {35}{8} \, \arctan \left (\frac {1}{2} \, x\right ) \]
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Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int \frac {x^8}{\left (4+x^2\right )^4} \, dx=x-\frac {35\,\mathrm {atan}\left (\frac {x}{2}\right )}{8}+\frac {\frac {29\,x^5}{4}+\frac {136\,x^3}{3}+76\,x}{x^6+12\,x^4+48\,x^2+64} \]
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