Integrand size = 31, antiderivative size = 80 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=-42 x+\frac {5 x^3}{3}-\frac {x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {449 \arctan (x)}{8}+\frac {219 \arctan \left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1682, 1692, 1690, 1180, 209} \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=-\frac {449 \arctan (x)}{8}+\frac {219 \arctan \left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {5 x^3}{3}+\frac {\left (24-409 x^2\right ) x}{8 \left (x^4+3 x^2+2\right )}-\frac {\left (207 x^2+206\right ) x}{4 \left (x^4+3 x^2+2\right )^2}-42 x \]
[In]
[Out]
Rule 209
Rule 1180
Rule 1682
Rule 1690
Rule 1692
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {1}{8} \int \frac {-412+1230 x^2+424 x^4-216 x^6+96 x^8-40 x^{10}}{\left (2+3 x^2+x^4\right )^2} \, dx \\ & = -\frac {x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \frac {728+1500 x^2-864 x^4+160 x^6}{2+3 x^2+x^4} \, dx \\ & = -\frac {x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \left (-1344+160 x^2+\frac {4 \left (854+1303 x^2\right )}{2+3 x^2+x^4}\right ) \, dx \\ & = -42 x+\frac {5 x^3}{3}-\frac {x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{8} \int \frac {854+1303 x^2}{2+3 x^2+x^4} \, dx \\ & = -42 x+\frac {5 x^3}{3}-\frac {x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {449}{8} \int \frac {1}{1+x^2} \, dx+219 \int \frac {1}{2+x^2} \, dx \\ & = -42 x+\frac {5 x^3}{3}-\frac {x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac {x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {449}{8} \tan ^{-1}(x)+\frac {219 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {x \left (-5124-15416 x^2-16233 x^4-6755 x^6-768 x^8+40 x^{10}\right )}{24 \left (2+3 x^2+x^4\right )^2}-\frac {449 \arctan (x)}{8}+\frac {219 \arctan \left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {5 x^{3}}{3}-42 x +\frac {-\frac {409}{8} x^{7}-\frac {1203}{8} x^{5}-145 x^{3}-\frac {91}{2} x}{\left (x^{4}+3 x^{2}+2\right )^{2}}-\frac {449 \arctan \left (x \right )}{8}+\frac {219 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{2}\) | \(58\) |
default | \(\frac {-53 x^{3}-54 x}{\left (x^{2}+2\right )^{2}}+\frac {219 \arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {5 x^{3}}{3}-42 x -\frac {-\frac {15}{8} x^{3}-\frac {17}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {449 \arctan \left (x \right )}{8}\) | \(62\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {40 \, x^{11} - 768 \, x^{9} - 6755 \, x^{7} - 16233 \, x^{5} - 15416 \, x^{3} + 2628 \, \sqrt {2} {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 1347 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 5124 \, x}{24 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {5 x^{3}}{3} - 42 x + \frac {- 409 x^{7} - 1203 x^{5} - 1160 x^{3} - 364 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} - \frac {449 \operatorname {atan}{\left (x \right )}}{8} + \frac {219 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{2} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {5}{3} \, x^{3} + \frac {219}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 42 \, x - \frac {409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac {449}{8} \, \arctan \left (x\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {5}{3} \, x^{3} + \frac {219}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 42 \, x - \frac {409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac {449}{8} \, \arctan \left (x\right ) \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \frac {x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx=\frac {219\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{2}-\frac {449\,\mathrm {atan}\left (x\right )}{8}-42\,x-\frac {\frac {409\,x^7}{8}+\frac {1203\,x^5}{8}+145\,x^3+\frac {91\,x}{2}}{x^8+6\,x^6+13\,x^4+12\,x^2+4}+\frac {5\,x^3}{3} \]
[In]
[Out]