Integrand size = 58, antiderivative size = 24 \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=\frac {x}{-\frac {2 x^2}{4-x}+e^4 (4+x)} \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2019, 28, 1828, 8} \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=-\frac {4 \left (4 e^4-\left (2+e^4\right ) x\right )}{\left (2+e^4\right ) \left (16 e^4-\left (2+e^4\right ) x^2\right )} \]
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Rule 8
Rule 28
Rule 1828
Rule 2019
Rubi steps \begin{align*} \text {integral}& = \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{256 e^8-32 e^4 \left (2+e^4\right ) x^2+\left (2+e^4\right )^2 x^4} \, dx \\ & = \left (2+e^4\right )^2 \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{\left (-16 e^4 \left (2+e^4\right )+\left (2+e^4\right )^2 x^2\right )^2} \, dx \\ & = -\frac {4 \left (4 e^4-\left (2+e^4\right ) x\right )}{\left (2+e^4\right ) \left (16 e^4-\left (2+e^4\right ) x^2\right )}+\frac {\left (2+e^4\right ) \int 0 \, dx}{32 e^4} \\ & = -\frac {4 \left (4 e^4-\left (2+e^4\right ) x\right )}{\left (2+e^4\right ) \left (16 e^4-\left (2+e^4\right ) x^2\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=-\frac {4 \left (e^4 (-4+x)+2 x\right )}{\left (2+e^4\right ) \left (2 x^2+e^4 \left (-16+x^2\right )\right )} \]
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Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12
method | result | size |
norman | \(\frac {x^{2}-4 x}{x^{2} {\mathrm e}^{4}+2 x^{2}-16 \,{\mathrm e}^{4}}\) | \(27\) |
risch | \(\frac {-4 x +\frac {16 \,{\mathrm e}^{4}}{2+{\mathrm e}^{4}}}{x^{2} {\mathrm e}^{4}+2 x^{2}-16 \,{\mathrm e}^{4}}\) | \(34\) |
parallelrisch | \(-\frac {\left (-16 x^{2} {\mathrm e}^{4}+64 x \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4}}{16 \left (x^{2} {\mathrm e}^{4}+2 x^{2}-16 \,{\mathrm e}^{4}\right )}\) | \(38\) |
gosper | \(-\frac {4 \left (x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{4}+2 x \right )}{\left (x^{2} {\mathrm e}^{4}+2 x^{2}-16 \,{\mathrm e}^{4}\right ) \left (2+{\mathrm e}^{4}\right )}\) | \(39\) |
default | \(\frac {4 \left ({\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4\right ) \left (-\frac {\left (4 \,{\mathrm e}^{8}+{\mathrm e}^{4} {\mathrm e}^{8}+4 \,{\mathrm e}^{4}\right ) x}{4 \left ({\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4\right ) \left (2 \,{\mathrm e}^{4}+{\mathrm e}^{8}\right )}+\frac {{\mathrm e}^{4}}{{\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4}\right )}{x^{2} {\mathrm e}^{4}+\frac {x^{2} {\mathrm e}^{8}}{4}+x^{2}-8 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{8}}\) | \(173\) |
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=-\frac {4 \, {\left ({\left (x - 4\right )} e^{4} + 2 \, x\right )}}{4 \, x^{2} + {\left (x^{2} - 16\right )} e^{8} + 4 \, {\left (x^{2} - 8\right )} e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=\frac {x \left (- 4 e^{4} - 8\right ) + 16 e^{4}}{x^{2} \cdot \left (4 + 4 e^{4} + e^{8}\right ) - 16 e^{8} - 32 e^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=-\frac {4 \, {\left (x {\left (e^{4} + 2\right )} - 4 \, e^{4}\right )}}{x^{2} {\left (e^{8} + 4 \, e^{4} + 4\right )} - 16 \, e^{8} - 32 \, e^{4}} \]
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Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=-\frac {4 \, {\left (x e^{4} + 2 \, x - 4 \, e^{4}\right )}}{{\left (x^{2} e^{4} + 2 \, x^{2} - 16 \, e^{4}\right )} {\left (e^{4} + 2\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {8 x^2+e^4 \left (64-32 x+4 x^2\right )}{4 x^4+e^8 \left (256-32 x^2+x^4\right )+e^4 \left (-64 x^2+4 x^4\right )} \, dx=\frac {4\,x-\frac {16\,{\mathrm {e}}^4}{{\mathrm {e}}^4+2}}{16\,{\mathrm {e}}^4-x^2\,\left ({\mathrm {e}}^4+2\right )} \]
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