Integrand size = 80, antiderivative size = 26 \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=\left (\frac {3}{9+\frac {e^x}{4 \left (-3+e^{25}\right )}}-x\right ) x \]
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Leaf count is larger than twice the leaf count of optimal. \(119\) vs. \(2(26)=52\).
Time = 0.37 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.58, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6873, 6874, 2215, 2221, 2317, 2438, 2216, 2222, 2320, 36, 29, 31} \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=-\frac {5 x^2}{6}-\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {x}{3}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \log \left (e^x-36 \left (3-e^{25}\right )\right ) \]
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Rule 29
Rule 31
Rule 36
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {3888+\left (1-\frac {6}{e^{25}}\right ) e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2} \, dx \\ & = \int \left (\frac {12 \left (3-e^{25}\right ) (-1+x)}{e^x-108 \left (1-\frac {e^{25}}{3}\right )}-2 x+\frac {432 \left (3-e^{25}\right )^2 x}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2}\right ) \, dx \\ & = -x^2+\left (12 \left (3-e^{25}\right )\right ) \int \frac {-1+x}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx+\left (432 \left (3-e^{25}\right )^2\right ) \int \frac {x}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2} \, dx \\ & = -\frac {1}{6} (1-x)^2-x^2+\frac {1}{3} \int \frac {e^x (-1+x)}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx+\left (12 \left (3-e^{25}\right )\right ) \int \frac {e^x x}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2} \, dx-\left (12 \left (3-e^{25}\right )\right ) \int \frac {x}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx \\ & = -\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} \int \frac {e^x x}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx-\frac {1}{3} \int \log \left (1-\frac {e^x}{108 \left (1-\frac {e^{25}}{3}\right )}\right ) \, dx+\left (12 \left (3-e^{25}\right )\right ) \int \frac {1}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx \\ & = -\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \int \log \left (1-\frac {e^x}{108 \left (1-\frac {e^{25}}{3}\right )}\right ) \, dx-\frac {1}{3} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{108 \left (1-\frac {e^{25}}{3}\right )}\right )}{x} \, dx,x,e^x\right )+\left (12 \left (3-e^{25}\right )\right ) \text {Subst}\left (\int \frac {1}{x \left (-108+36 e^{25}+x\right )} \, dx,x,e^x\right ) \\ & = -\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \operatorname {PolyLog}\left (2,\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-108+36 e^{25}+x} \, dx,x,e^x\right )+\frac {1}{3} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{108 \left (1-\frac {e^{25}}{3}\right )}\right )}{x} \, dx,x,e^x\right ) \\ & = -\frac {1}{6} (1-x)^2-\frac {x}{3}-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \log \left (e^x-36 \left (3-e^{25}\right )\right ) \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=-x \left (-\frac {12 \left (-3+e^{25}\right )}{-108+36 e^{25}+e^x}+x\right ) \]
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Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-x^{2}+\frac {12 \left ({\mathrm e}^{25}-3\right ) x}{{\mathrm e}^{x}+36 \,{\mathrm e}^{25}-108}\) | \(24\) |
norman | \(\frac {\left (-36+12 \,{\mathrm e}^{25}\right ) x +\left (-36 \,{\mathrm e}^{25}+108\right ) x^{2}-{\mathrm e}^{x} x^{2}}{{\mathrm e}^{x}+36 \,{\mathrm e}^{25}-108}\) | \(38\) |
parallelrisch | \(-\frac {36 x^{2} {\mathrm e}^{25}+{\mathrm e}^{x} x^{2}-12 x \,{\mathrm e}^{25}-108 x^{2}+36 x}{{\mathrm e}^{x}+36 \,{\mathrm e}^{25}-108}\) | \(40\) |
default | \(\text {Expression too large to display}\) | \(3240\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=-\frac {x^{2} e^{x} - 108 \, x^{2} + 12 \, {\left (3 \, x^{2} - x\right )} e^{25} + 36 \, x}{36 \, e^{25} + e^{x} - 108} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=- x^{2} + \frac {- 36 x + 12 x e^{25}}{e^{x} - 108 + 36 e^{25}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 8.54 \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=\frac {1}{3} \, {\left (\frac {x}{e^{50} - 6 \, e^{25} + 9} - \frac {\log \left (36 \, e^{25} + e^{x} - 108\right )}{e^{50} - 6 \, e^{25} + 9} + \frac {36}{{\left (e^{25} - 3\right )} e^{x} + 36 \, e^{50} - 216 \, e^{25} + 324}\right )} e^{50} - 2 \, {\left (\frac {x}{e^{50} - 6 \, e^{25} + 9} - \frac {\log \left (36 \, e^{25} + e^{x} - 108\right )}{e^{50} - 6 \, e^{25} + 9} + \frac {36}{{\left (e^{25} - 3\right )} e^{x} + 36 \, e^{50} - 216 \, e^{25} + 324}\right )} e^{25} + \frac {3 \, x}{e^{50} - 6 \, e^{25} + 9} - \frac {108 \, x^{2} {\left (e^{25} - 3\right )} + {\left (3 \, x^{2} + x\right )} e^{x} + 36 \, e^{25} - 108}{3 \, {\left (36 \, e^{25} + e^{x} - 108\right )}} - \frac {3 \, \log \left (36 \, e^{25} + e^{x} - 108\right )}{e^{50} - 6 \, e^{25} + 9} + \frac {108}{{\left (e^{25} - 3\right )} e^{x} + 36 \, e^{50} - 216 \, e^{25} + 324} + \frac {1}{3} \, \log \left (36 \, e^{25} + e^{x} - 108\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 6.12 \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=-\frac {108 \, x^{2} e^{50} - 648 \, x^{2} e^{25} + 3 \, x^{2} e^{\left (x + 25\right )} - 9 \, x^{2} e^{x} + 972 \, x^{2} - 36 \, x e^{50} + 216 \, x e^{25} + 468 \, e^{50} \log \left (36 \, e^{25} + e^{x} - 108\right ) - 1404 \, e^{25} \log \left (36 \, e^{25} + e^{x} - 108\right ) + 13 \, e^{\left (x + 25\right )} \log \left (36 \, e^{25} + e^{x} - 108\right ) - 468 \, e^{50} \log \left (-36 \, e^{25} - e^{x} + 108\right ) + 1404 \, e^{25} \log \left (-36 \, e^{25} - e^{x} + 108\right ) - 13 \, e^{\left (x + 25\right )} \log \left (-36 \, e^{25} - e^{x} + 108\right ) - 324 \, x}{3 \, {\left (36 \, e^{50} - 216 \, e^{25} + e^{\left (x + 25\right )} - 3 \, e^{x} + 324\right )}} \]
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Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {3888+e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )+e^{25} (-2592+15552 x)}{11664-7776 e^{25}+1296 e^{50}+e^{2 x}+e^x \left (-216+72 e^{25}\right )} \, dx=\frac {12\,x\,\left ({\mathrm {e}}^{25}-3\right )}{36\,{\mathrm {e}}^{25}+{\mathrm {e}}^x-108}-x^2 \]
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