Integrand size = 30, antiderivative size = 17 \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=\frac {\frac {3}{2401 e^4}+x}{e^4+x} \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 27, 32} \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=\frac {3-2401 e^8}{2401 e^4 \left (x+e^4\right )} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (3-2401 e^8\right ) \int \frac {1}{2401 e^8+4802 e^4 x+2401 x^2} \, dx}{e^4} \\ & = -\frac {\left (3-2401 e^8\right ) \int \frac {1}{2401 \left (e^4+x\right )^2} \, dx}{e^4} \\ & = -\frac {\left (3-2401 e^8\right ) \int \frac {1}{\left (e^4+x\right )^2} \, dx}{2401 e^4} \\ & = \frac {3-2401 e^8}{2401 e^4 \left (e^4+x\right )} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=-\frac {-3+2401 e^8}{2401 e^4 \left (e^4+x\right )} \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41
method | result | size |
risch | \(-\frac {{\mathrm e}^{-4} {\mathrm e}^{8}}{x +{\mathrm e}^{4}}+\frac {3 \,{\mathrm e}^{-4}}{2401 \left (x +{\mathrm e}^{4}\right )}\) | \(24\) |
gosper | \(-\frac {\left (2401 \left ({\mathrm e}^{4}\right )^{2}-3\right ) {\mathrm e}^{-4}}{2401 \left (x +{\mathrm e}^{4}\right )}\) | \(25\) |
norman | \(-\frac {\left (2401 \left ({\mathrm e}^{4}\right )^{2}-3\right ) {\mathrm e}^{-4}}{2401 \left (x +{\mathrm e}^{4}\right )}\) | \(25\) |
parallelrisch | \(-\frac {\left (2401 \left ({\mathrm e}^{4}\right )^{2}-3\right ) {\mathrm e}^{-4}}{2401 \left (x +{\mathrm e}^{4}\right )}\) | \(25\) |
meijerg | \(-\frac {3 \,{\mathrm e}^{-12} x}{2401 \left (1+x \,{\mathrm e}^{-4}\right )}+\frac {{\mathrm e}^{-4} x}{1+x \,{\mathrm e}^{-4}}\) | \(27\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=-\frac {2401 \, e^{8} - 3}{2401 \, {\left (x e^{4} + e^{8}\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=- \frac {-3 + 2401 e^{8}}{2401 x e^{4} + 2401 e^{8}} \]
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Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=-\frac {{\left (2401 \, e^{8} - 3\right )} e^{\left (-4\right )}}{2401 \, {\left (x + e^{4}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=-\frac {{\left (2401 \, e^{8} - 3\right )} e^{\left (-4\right )}}{2401 \, {\left (x + e^{4}\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-3+2401 e^8}{e^4 \left (2401 e^8+4802 e^4 x+2401 x^2\right )} \, dx=-\frac {{\mathrm {e}}^8-\frac {3}{2401}}{{\mathrm {e}}^8+x\,{\mathrm {e}}^4} \]
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