Integrand size = 43, antiderivative size = 24 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=-1-\frac {3}{e^5}+\frac {2}{-e^x+e^{4 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {6820, 12, 6818} \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=-\frac {2}{e^x-e^{4 x^2}} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (e^x-8 e^{4 x^2} x\right )}{\left (e^x-e^{4 x^2}\right )^2} \, dx \\ & = 2 \int \frac {e^x-8 e^{4 x^2} x}{\left (e^x-e^{4 x^2}\right )^2} \, dx \\ & = -\frac {2}{e^x-e^{4 x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=-\frac {2}{e^x-e^{4 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67
method | result | size |
norman | \(-\frac {2}{{\mathrm e}^{x}-{\mathrm e}^{4 x^{2}}}\) | \(16\) |
risch | \(-\frac {2}{{\mathrm e}^{x}-{\mathrm e}^{4 x^{2}}}\) | \(16\) |
parallelrisch | \(-\frac {2}{{\mathrm e}^{x}-{\mathrm e}^{4 x^{2}}}\) | \(16\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2 \, e^{\left (4 \, x^{2}\right )}}{e^{\left (8 \, x^{2}\right )} - e^{\left (4 \, x^{2} + x\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{- e^{x} + e^{4 x^{2}}} \]
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Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{e^{\left (4 \, x^{2}\right )} - e^{x}} \]
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Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{e^{\left (4 \, x^{2}\right )} - e^{x}} \]
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Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {2 e^x-16 e^{4 x^2} x}{e^{2 x}+e^{8 x^2}-2 e^{x+4 x^2}} \, dx=\frac {2}{{\mathrm {e}}^{4\,x^2}-{\mathrm {e}}^x} \]
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