Integrand size = 140, antiderivative size = 33 \[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=5-\frac {x^2}{e^{\frac {23 x+\log (4)}{e^2-x}}-x^2} \]
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Time = 2.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6, 1608, 6820, 6843, 32} \[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=\frac {1}{1-\frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}}{x^2}} \]
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Rule 6
Rule 32
Rule 1608
Rule 6820
Rule 6843
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x-2 x^3+x^2 \left (27 e^2+\log (4)\right )\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx \\ & = \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} x \left (-2 e^4-2 x^2+x \left (27 e^2+\log (4)\right )\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx \\ & = \int \frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}} x \left (-2 e^4-2 x^2+x \left (27 e^2+\log (4)\right )\right )}{\left (e^2-x\right )^2 \left (4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}-x^2\right )^2} \, dx \\ & = \text {Subst}\left (\int \frac {1}{(-1+x)^2} \, dx,x,\frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}}{x^2}\right ) \\ & = \frac {1}{1-\frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}}{x^2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=-\frac {e^{23} x^2}{4^{-\frac {1}{-e^2+x}} e^{-\frac {23 e^2}{-e^2+x}}-e^{23} x^2} \]
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Time = 15.88 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {x^{2}}{x^{2}-{\mathrm e}^{\frac {2 \ln \left (2\right )+23 x}{{\mathrm e}^{2}-x}}}\) | \(31\) |
parallelrisch | \(\frac {x^{2}}{x^{2}-{\mathrm e}^{\frac {2 \ln \left (2\right )+23 x}{{\mathrm e}^{2}-x}}}\) | \(31\) |
norman | \(\frac {-x \,{\mathrm e}^{\frac {2 \ln \left (2\right )+23 x}{{\mathrm e}^{2}-x}}+{\mathrm e}^{2} {\mathrm e}^{\frac {2 \ln \left (2\right )+23 x}{{\mathrm e}^{2}-x}}}{\left (x^{2}-{\mathrm e}^{\frac {2 \ln \left (2\right )+23 x}{{\mathrm e}^{2}-x}}\right ) \left ({\mathrm e}^{2}-x \right )}\) | \(79\) |
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=\frac {x^{2}}{x^{2} - e^{\left (-\frac {23 \, x + 2 \, \log \left (2\right )}{x - e^{2}}\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=- \frac {x^{2}}{- x^{2} + e^{\frac {23 x + 2 \log {\left (2 \right )}}{- x + e^{2}}}} \]
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Time = 0.41 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=\frac {1}{x^{2} e^{\left (\frac {23 \, e^{2}}{x - e^{2}} + \frac {2 \, \log \left (2\right )}{x - e^{2}} + 23\right )} - 1} \]
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\[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=\int { -\frac {{\left (2 \, x^{3} - 27 \, x^{2} e^{2} - 2 \, x^{2} \log \left (2\right ) + 2 \, x e^{4}\right )} e^{\left (-\frac {23 \, x + 2 \, \log \left (2\right )}{x - e^{2}}\right )}}{x^{6} - 2 \, x^{5} e^{2} + x^{4} e^{4} - 2 \, {\left (x^{4} - 2 \, x^{3} e^{2} + x^{2} e^{4}\right )} e^{\left (-\frac {23 \, x + 2 \, \log \left (2\right )}{x - e^{2}}\right )} + {\left (x^{2} - 2 \, x e^{2} + e^{4}\right )} e^{\left (-\frac {2 \, {\left (23 \, x + 2 \, \log \left (2\right )\right )}}{x - e^{2}}\right )}} \,d x } \]
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Time = 9.73 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.97 \[ \int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x+27 e^2 x^2-2 x^3+x^2 \log (4)\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx=-\frac {x^3\,{\left (x^2-2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4\right )}^2\,\left (2\,{\mathrm {e}}^4-27\,x\,{\mathrm {e}}^2-2\,x\,\ln \left (2\right )+2\,x^2\right )}{\left (\frac {{\mathrm {e}}^{-\frac {23\,x}{x-{\mathrm {e}}^2}}}{2^{\frac {2}{x-{\mathrm {e}}^2}}}-x^2\right )\,\left (2\,x\,{\mathrm {e}}^{12}-35\,x^6\,{\mathrm {e}}^2+122\,x^5\,{\mathrm {e}}^4-178\,x^4\,{\mathrm {e}}^6+122\,x^3\,{\mathrm {e}}^8-35\,x^2\,{\mathrm {e}}^{10}-x^6\,\ln \left (4\right )+2\,x^7+4\,x^5\,{\mathrm {e}}^2\,\ln \left (4\right )-6\,x^4\,{\mathrm {e}}^4\,\ln \left (4\right )+4\,x^3\,{\mathrm {e}}^6\,\ln \left (4\right )-x^2\,{\mathrm {e}}^8\,\ln \left (4\right )\right )} \]
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