Integrand size = 34, antiderivative size = 16 \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=\frac {-4+x}{19+e^2-\frac {11 x}{2}} \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 2006, 27, 32} \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=-\frac {4 \left (3-e^2\right )}{11 \left (2 \left (19+e^2\right )-11 x\right )} \]
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Rule 12
Rule 27
Rule 32
Rule 2006
Rubi steps \begin{align*} \text {integral}& = -\left (\left (4 \left (3-e^2\right )\right ) \int \frac {1}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx\right ) \\ & = -\left (\left (4 \left (3-e^2\right )\right ) \int \frac {1}{4 \left (19+e^2\right )^2-44 \left (19+e^2\right ) x+121 x^2} \, dx\right ) \\ & = -\left (\left (4 \left (3-e^2\right )\right ) \int \frac {1}{\left (38+2 e^2-11 x\right )^2} \, dx\right ) \\ & = -\frac {4 \left (3-e^2\right )}{11 \left (2 \left (19+e^2\right )-11 x\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=\frac {4 \left (-3+e^2\right )}{418+22 e^2-121 x} \]
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Time = 0.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
method | result | size |
gosper | \(\frac {-\frac {12}{11}+\frac {4 \,{\mathrm e}^{2}}{11}}{2 \,{\mathrm e}^{2}-11 x +38}\) | \(18\) |
norman | \(\frac {-\frac {12}{11}+\frac {4 \,{\mathrm e}^{2}}{11}}{2 \,{\mathrm e}^{2}-11 x +38}\) | \(19\) |
parallelrisch | \(\frac {4 \,{\mathrm e}^{2}-12}{22 \,{\mathrm e}^{2}-121 x +418}\) | \(20\) |
risch | \(\frac {2 \,{\mathrm e}^{2}}{11 \left (19-\frac {11 x}{2}+{\mathrm e}^{2}\right )}-\frac {6}{11 \left (19-\frac {11 x}{2}+{\mathrm e}^{2}\right )}\) | \(26\) |
meijerg | \(\frac {6 x}{11 \left (-\frac {2 \,{\mathrm e}^{2}}{11}-\frac {38}{11}\right ) \left ({\mathrm e}^{2}+19\right ) \left (1-\frac {11 x}{2 \left ({\mathrm e}^{2}+19\right )}\right )}-\frac {2 \,{\mathrm e}^{2} x}{11 \left (-\frac {2 \,{\mathrm e}^{2}}{11}-\frac {38}{11}\right ) \left ({\mathrm e}^{2}+19\right ) \left (1-\frac {11 x}{2 \left ({\mathrm e}^{2}+19\right )}\right )}\) | \(64\) |
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=-\frac {4 \, {\left (e^{2} - 3\right )}}{11 \, {\left (11 \, x - 2 \, e^{2} - 38\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=- \frac {-12 + 4 e^{2}}{121 x - 418 - 22 e^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=-\frac {4 \, {\left (e^{2} - 3\right )}}{11 \, {\left (11 \, x - 2 \, e^{2} - 38\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=-\frac {4 \, {\left (e^{2} - 3\right )}}{11 \, {\left (11 \, x - 2 \, e^{2} - 38\right )}} \]
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Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {-12+4 e^2}{1444+4 e^4+e^2 (152-44 x)-836 x+121 x^2} \, dx=\frac {\frac {4\,{\mathrm {e}}^2}{11}-\frac {12}{11}}{2\,{\mathrm {e}}^2-11\,x+38} \]
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