Integrand size = 42, antiderivative size = 25 \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=x^2 \left (-4 x+\frac {(1+e)^2-x}{1+x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {6, 28, 1825, 1818, 1163} \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=-4 x^3+\frac {\left ((1+e)^2 x+1\right ) x}{x^2+1}-x \]
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Rule 6
Rule 28
Rule 1163
Rule 1818
Rule 1825
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^2 x+(2+4 e) x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx \\ & = \int \frac {\left (2+4 e+2 e^2\right ) x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx \\ & = \int \frac {\left (2+4 e+2 e^2\right ) x-15 x^2-25 x^4-12 x^6}{\left (1+x^2\right )^2} \, dx \\ & = \int \frac {x \left (2+4 e+2 e^2-15 x-25 x^3-12 x^5\right )}{\left (1+x^2\right )^2} \, dx \\ & = \frac {x \left (1+(1+e)^2 x\right )}{1+x^2}-\frac {1}{2} \int \frac {2+26 x^2+24 x^4}{1+x^2} \, dx \\ & = \frac {x \left (1+(1+e)^2 x\right )}{1+x^2}-\frac {1}{2} \int \left (2+24 x^2\right ) \, dx \\ & = -x-4 x^3+\frac {x \left (1+(1+e)^2 x\right )}{1+x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=-x-4 x^3+\frac {-1-2 e-e^2+x}{1+x^2} \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
method | result | size |
default | \(-4 x^{3}-x +\frac {x -{\mathrm e}^{2}-2 \,{\mathrm e}-1}{x^{2}+1}\) | \(29\) |
risch | \(-4 x^{3}-x +\frac {x -{\mathrm e}^{2}-2 \,{\mathrm e}-1}{x^{2}+1}\) | \(29\) |
gosper | \(-\frac {4 x^{5}+5 x^{3}+{\mathrm e}^{2}+2 \,{\mathrm e}+1}{x^{2}+1}\) | \(30\) |
parallelrisch | \(-\frac {4 x^{5}+5 x^{3}+{\mathrm e}^{2}+2 \,{\mathrm e}+1}{x^{2}+1}\) | \(30\) |
norman | \(\frac {-4 x^{5}-1-5 x^{3}-{\mathrm e}^{2}-2 \,{\mathrm e}}{x^{2}+1}\) | \(31\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=-\frac {4 \, x^{5} + 5 \, x^{3} + e^{2} + 2 \, e + 1}{x^{2} + 1} \]
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Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=- 4 x^{3} - x - \frac {- x + 1 + 2 e + e^{2}}{x^{2} + 1} \]
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Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=-4 \, x^{3} - x + \frac {x - e^{2} - 2 \, e - 1}{x^{2} + 1} \]
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Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=-4 \, x^{3} - x + \frac {x - e^{2} - 2 \, e - 1}{x^{2} + 1} \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {2 x+4 e x+2 e^2 x-15 x^2-25 x^4-12 x^6}{1+2 x^2+x^4} \, dx=-\frac {4\,x^5+5\,x^3+2\,\mathrm {e}+{\mathrm {e}}^2+1}{x^2+1} \]
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