Integrand size = 111, antiderivative size = 33 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {16 \left (-3+e^x\right )^2 x^2}{\frac {6 x}{5}-\frac {x}{3-e^8 x}} \]
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Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(33)=66\).
Time = 0.62 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.73, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.090, Rules used = {27, 6820, 12, 6874, 697, 2230, 2225, 2207, 2208, 2209} \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=-80 e^x x+\frac {40}{3} e^{2 x} x+120 x+80 e^x-\frac {20 e^{2 x}}{3}-\frac {2600 e^{x-8}}{3 \left (13-6 e^8 x\right )}+\frac {1300 e^{2 x-8}}{9 \left (13-6 e^8 x\right )}+\frac {1300}{e^8 \left (13-6 e^8 x\right )}-\frac {20}{9} \left (5-3 e^8\right ) e^{2 x-8}+\frac {40}{3} \left (5-6 e^8\right ) e^{x-8} \]
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Rule 12
Rule 27
Rule 697
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{\left (-13+6 e^8 x\right )^2} \, dx \\ & = \int \frac {80 \left (3-e^x\right ) \left (117-78 e^8 x+18 e^{16} x^2-39 e^x (1+2 x)-6 e^{16+x} x^2 (1+2 x)+2 e^{8+x} x (13+31 x)\right )}{\left (13-6 e^8 x\right )^2} \, dx \\ & = 80 \int \frac {\left (3-e^x\right ) \left (117-78 e^8 x+18 e^{16} x^2-39 e^x (1+2 x)-6 e^{16+x} x^2 (1+2 x)+2 e^{8+x} x (13+31 x)\right )}{\left (13-6 e^8 x\right )^2} \, dx \\ & = 80 \int \left (\frac {9 \left (39-26 e^8 x+6 e^{16} x^2\right )}{\left (-13+6 e^8 x\right )^2}+\frac {6 e^x \left (-39-13 \left (3-2 e^8\right ) x+e^8 \left (31-6 e^8\right ) x^2-6 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2}+\frac {e^{2 x} \left (39+26 \left (3-e^8\right ) x-2 e^8 \left (31-3 e^8\right ) x^2+12 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2}\right ) \, dx \\ & = 80 \int \frac {e^{2 x} \left (39+26 \left (3-e^8\right ) x-2 e^8 \left (31-3 e^8\right ) x^2+12 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2} \, dx+480 \int \frac {e^x \left (-39-13 \left (3-2 e^8\right ) x+e^8 \left (31-6 e^8\right ) x^2-6 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2} \, dx+720 \int \frac {39-26 e^8 x+6 e^{16} x^2}{\left (-13+6 e^8 x\right )^2} \, dx \\ & = 80 \int \left (\frac {1}{18} e^{-8+2 x} \left (-5+3 e^8\right )+\frac {1}{3} e^{2 x} x+\frac {65 e^{2 x}}{6 \left (-13+6 e^8 x\right )^2}-\frac {65 e^{-8+2 x}}{18 \left (-13+6 e^8 x\right )}\right ) \, dx+480 \int \left (\frac {1}{36} e^{-8+x} \left (5-6 e^8\right )-\frac {e^x x}{6}-\frac {65 e^x}{6 \left (-13+6 e^8 x\right )^2}+\frac {65 e^{-8+x}}{36 \left (-13+6 e^8 x\right )}\right ) \, dx+720 \int \left (\frac {1}{6}+\frac {65}{6 \left (-13+6 e^8 x\right )^2}\right ) \, dx \\ & = 120 x+\frac {1300}{e^8 \left (13-6 e^8 x\right )}+\frac {80}{3} \int e^{2 x} x \, dx-80 \int e^x x \, dx-\frac {2600}{9} \int \frac {e^{-8+2 x}}{-13+6 e^8 x} \, dx+\frac {2600}{3} \int \frac {e^{2 x}}{\left (-13+6 e^8 x\right )^2} \, dx+\frac {2600}{3} \int \frac {e^{-8+x}}{-13+6 e^8 x} \, dx-5200 \int \frac {e^x}{\left (-13+6 e^8 x\right )^2} \, dx+\frac {1}{3} \left (40 \left (5-6 e^8\right )\right ) \int e^{-8+x} \, dx-\frac {1}{9} \left (40 \left (5-3 e^8\right )\right ) \int e^{-8+2 x} \, dx \\ & = \frac {40}{3} e^{-8+x} \left (5-6 e^8\right )-\frac {20}{9} e^{-8+2 x} \left (5-3 e^8\right )+120 x-80 e^x x+\frac {40}{3} e^{2 x} x+\frac {1300}{e^8 \left (13-6 e^8 x\right )}-\frac {2600 e^{-8+x}}{3 \left (13-6 e^8 x\right )}+\frac {1300 e^{-8+2 x}}{9 \left (13-6 e^8 x\right )}-\frac {1300}{27} e^{-16+\frac {13}{3 e^8}} \text {Ei}\left (-\frac {13-6 e^8 x}{3 e^8}\right )+\frac {1300}{9} e^{-16+\frac {13}{6 e^8}} \text {Ei}\left (-\frac {13-6 e^8 x}{6 e^8}\right )-\frac {40}{3} \int e^{2 x} \, dx+80 \int e^x \, dx+\frac {2600 \int \frac {e^{2 x}}{-13+6 e^8 x} \, dx}{9 e^8}-\frac {2600 \int \frac {e^x}{-13+6 e^8 x} \, dx}{3 e^8} \\ & = 80 e^x-\frac {20 e^{2 x}}{3}+\frac {40}{3} e^{-8+x} \left (5-6 e^8\right )-\frac {20}{9} e^{-8+2 x} \left (5-3 e^8\right )+120 x-80 e^x x+\frac {40}{3} e^{2 x} x+\frac {1300}{e^8 \left (13-6 e^8 x\right )}-\frac {2600 e^{-8+x}}{3 \left (13-6 e^8 x\right )}+\frac {1300 e^{-8+2 x}}{9 \left (13-6 e^8 x\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).
Time = 2.66 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {80 \left (65+78 e^8 x-72 e^{8+x} x+12 e^{8+2 x} x-36 e^{16} x^2+24 e^{16+x} x^2-4 e^{16+2 x} x^2\right )}{52 e^8-24 e^{16} x} \]
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Time = 0.72 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79
method | result | size |
risch | \(\frac {80 x^{2} {\mathrm e}^{2 x +8}-480 x^{2} {\mathrm e}^{x +8}+720 x^{2} {\mathrm e}^{8}-240 x \,{\mathrm e}^{2 x}+1440 \,{\mathrm e}^{x} x -1300 \,{\mathrm e}^{-8}-1560 x}{6 x \,{\mathrm e}^{8}-13}\) | \(59\) |
norman | \(\frac {-2160 x -240 x \,{\mathrm e}^{2 x}+720 x^{2} {\mathrm e}^{8}+1440 \,{\mathrm e}^{x} x -480 \,{\mathrm e}^{8} {\mathrm e}^{x} x^{2}+80 \,{\mathrm e}^{8} {\mathrm e}^{2 x} x^{2}}{6 x \,{\mathrm e}^{8}-13}\) | \(62\) |
parallelrisch | \(\frac {1040 \,{\mathrm e}^{8} {\mathrm e}^{2 x} x^{2}-6240 \,{\mathrm e}^{8} {\mathrm e}^{x} x^{2}+9360 x^{2} {\mathrm e}^{8}-3120 x \,{\mathrm e}^{2 x}+18720 \,{\mathrm e}^{x} x -28080 x}{78 x \,{\mathrm e}^{8}-169}\) | \(63\) |
parts | \(\text {Expression too large to display}\) | \(815\) |
default | \(\text {Expression too large to display}\) | \(829\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {20 \, {\left (36 \, x^{2} e^{16} - 78 \, x e^{8} + 4 \, {\left (x^{2} e^{16} - 3 \, x e^{8}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (x^{2} e^{16} - 3 \, x e^{8}\right )} e^{x} - 65\right )}}{6 \, x e^{16} - 13 \, e^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.48 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=120 x + \frac {\left (- 2880 x^{3} e^{16} + 14880 x^{2} e^{8} - 18720 x\right ) e^{x} + \left (480 x^{3} e^{16} - 2480 x^{2} e^{8} + 3120 x\right ) e^{2 x}}{36 x^{2} e^{16} - 156 x e^{8} + 169} - \frac {1300}{6 x e^{16} - 13 e^{8}} \]
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\[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\int { \frac {80 \, {\left (54 \, x^{2} e^{16} - 234 \, x e^{8} + {\left (6 \, {\left (2 \, x^{3} + x^{2}\right )} e^{16} - 2 \, {\left (31 \, x^{2} + 13 \, x\right )} e^{8} + 78 \, x + 39\right )} e^{\left (2 \, x\right )} - 6 \, {\left (6 \, {\left (x^{3} + x^{2}\right )} e^{16} - {\left (31 \, x^{2} + 26 \, x\right )} e^{8} + 39 \, x + 39\right )} e^{x} + 351\right )}}{36 \, x^{2} e^{16} - 156 \, x e^{8} + 169} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.03 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=\frac {20 \, {\left (36 \, x^{2} e^{40} + 4 \, x^{2} e^{\left (2 \, x + 40\right )} - 24 \, x^{2} e^{\left (x + 40\right )} - 78 \, x e^{32} - 12 \, x e^{\left (2 \, x + 32\right )} + 72 \, x e^{\left (x + 32\right )} - 65 \, e^{24}\right )}}{6 \, x e^{40} - 13 \, e^{32}} \]
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Time = 0.70 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{169-156 e^8 x+36 e^{16} x^2} \, dx=120\,x-\frac {1300\,{\mathrm {e}}^{-8}}{6\,x\,{\mathrm {e}}^8-13}+\frac {{\mathrm {e}}^x\,\left (240\,x\,{\mathrm {e}}^{-8}-80\,x^2\right )}{x-\frac {13\,{\mathrm {e}}^{-8}}{6}}-\frac {{\mathrm {e}}^{2\,x}\,\left (40\,x\,{\mathrm {e}}^{-8}-\frac {40\,x^2}{3}\right )}{x-\frac {13\,{\mathrm {e}}^{-8}}{6}} \]
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