Integrand size = 74, antiderivative size = 26 \[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=\frac {e^{e^{-4+4 x^2}}}{x}+\frac {\log (3)}{4-2 x} \]
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Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1608, 27, 12, 6820, 2326} \[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=\frac {e^{e^{4 x^2-4}}}{x}+\frac {\log (3)}{2 (2-x)} \]
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Rule 12
Rule 27
Rule 1608
Rule 2326
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{x^2 \left (8-8 x+2 x^2\right )} \, dx \\ & = \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{2 (-2+x)^2 x^2} \, dx \\ & = \frac {1}{2} \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{(-2+x)^2 x^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {2 e^{-4+e^{-4+4 x^2}} \left (e^4-8 e^{4 x^2} x^2\right )}{x^2}+\frac {\log (3)}{(-2+x)^2}\right ) \, dx \\ & = \frac {\log (3)}{2 (2-x)}-\int \frac {e^{-4+e^{-4+4 x^2}} \left (e^4-8 e^{4 x^2} x^2\right )}{x^2} \, dx \\ & = \frac {e^{e^{-4+4 x^2}}}{x}+\frac {\log (3)}{2 (2-x)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=\frac {\frac {2 e^{4+e^{-4+4 x^2}}}{x}+\frac {e^4 \log (3)}{2-x}}{2 e^4} \]
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Time = 1.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {\ln \left (3\right )}{2 \left (-2+x \right )}+\frac {{\mathrm e}^{{\mathrm e}^{4 \left (-1+x \right ) \left (1+x \right )}}}{x}\) | \(25\) |
parallelrisch | \(-\frac {x \ln \left (3\right )-2 \,{\mathrm e}^{{\mathrm e}^{4 x^{2}-4}} x +4 \,{\mathrm e}^{{\mathrm e}^{4 x^{2}-4}}}{2 x \left (-2+x \right )}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=\frac {2 \, {\left (x - 2\right )} e^{\left (e^{\left (4 \, x^{2} - 4\right )}\right )} - x \log \left (3\right )}{2 \, {\left (x^{2} - 2 \, x\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=- \frac {\log {\left (3 \right )}}{2 x - 4} + \frac {e^{e^{4 x^{2} - 4}}}{x} \]
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Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=\frac {e^{\left (e^{\left (4 \, x^{2} - 4\right )}\right )}}{x} - \frac {\log \left (3\right )}{2 \, {\left (x - 2\right )}} \]
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\[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=\int { \frac {x^{2} \log \left (3\right ) - 2 \, {\left (x^{2} - 8 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (4 \, x^{2} - 4\right )} - 4 \, x + 4\right )} e^{\left (e^{\left (4 \, x^{2} - 4\right )}\right )}}{2 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}} \,d x } \]
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Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{-4+4 x^2}} \left (-8+8 x-2 x^2+e^{-4+4 x^2} \left (64 x^2-64 x^3+16 x^4\right )\right )+x^2 \log (3)}{8 x^2-8 x^3+2 x^4} \, dx=\frac {{\mathrm {e}}^{{\mathrm {e}}^{-4}\,{\mathrm {e}}^{4\,x^2}}}{x}-\frac {\ln \left (3\right )}{2\,x-4} \]
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