Integrand size = 77, antiderivative size = 29 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=5+64 \left (-(-2+e)^2-e^{x/3}+\frac {5 e}{x}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(29)=58\).
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {6, 12, 14, 2225, 37, 2230, 2208, 2209} \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=\frac {64 \left (5 e-(2-e)^2 x\right )^2}{x^2}+128 (2-e)^2 e^{x/3}+64 e^{2 x/3}-\frac {640 e^{\frac {x}{3}+1}}{x} \]
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Rule 6
Rule 12
Rule 14
Rule 37
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^2 (-9600-7680 x)+\left (7680 e+1920 e^3\right ) x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx \\ & = \frac {1}{3} \int \frac {e^2 (-9600-7680 x)+\left (7680 e+1920 e^3\right ) x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{x^3} \, dx \\ & = \frac {1}{3} \int \left (128 e^{2 x/3}+\frac {1920 e \left (-5 e+(2-e)^2 x\right )}{x^3}+\frac {128 e^{x/3} \left (15 e-5 e x+(2-e)^2 x^2\right )}{x^2}\right ) \, dx \\ & = \frac {128}{3} \int e^{2 x/3} \, dx+\frac {128}{3} \int \frac {e^{x/3} \left (15 e-5 e x+(2-e)^2 x^2\right )}{x^2} \, dx+(640 e) \int \frac {-5 e+(2-e)^2 x}{x^3} \, dx \\ & = 64 e^{2 x/3}+\frac {64 \left (5 e-(2-e)^2 x\right )^2}{x^2}+\frac {128}{3} \int \left ((-2+e)^2 e^{x/3}+\frac {15 e^{1+\frac {x}{3}}}{x^2}-\frac {5 e^{1+\frac {x}{3}}}{x}\right ) \, dx \\ & = 64 e^{2 x/3}+\frac {64 \left (5 e-(2-e)^2 x\right )^2}{x^2}-\frac {640}{3} \int \frac {e^{1+\frac {x}{3}}}{x} \, dx+640 \int \frac {e^{1+\frac {x}{3}}}{x^2} \, dx+\frac {1}{3} \left (128 (2-e)^2\right ) \int e^{x/3} \, dx \\ & = 128 (2-e)^2 e^{x/3}+64 e^{2 x/3}-\frac {640 e^{1+\frac {x}{3}}}{x}+\frac {64 \left (5 e-(2-e)^2 x\right )^2}{x^2}-\frac {640 e \operatorname {ExpIntegralEi}\left (\frac {x}{3}\right )}{3}+\frac {640}{3} \int \frac {e^{1+\frac {x}{3}}}{x} \, dx \\ & = 128 (2-e)^2 e^{x/3}+64 e^{2 x/3}-\frac {640 e^{1+\frac {x}{3}}}{x}+\frac {64 \left (5 e-(2-e)^2 x\right )^2}{x^2} \\ \end{align*}
Time = 2.61 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=64 \left (e^{2 x/3}+2 e^{x/3} \left ((-2+e)^2-\frac {5 e}{x}\right )+\frac {25 e^2}{x^2}-\frac {10 (-2+e)^2 e}{x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03
method | result | size |
risch | \(\frac {\left (-1920 \,{\mathrm e}^{3}+7680 \,{\mathrm e}^{2}-7680 \,{\mathrm e}\right ) x +4800 \,{\mathrm e}^{2}}{3 x^{2}}+64 \,{\mathrm e}^{\frac {2 x}{3}}+\frac {128 \left ({\mathrm e}^{2} x -4 x \,{\mathrm e}-5 \,{\mathrm e}+4 x \right ) {\mathrm e}^{\frac {x}{3}}}{x}\) | \(59\) |
norman | \(\frac {\left (-640 \,{\mathrm e}^{3}+2560 \,{\mathrm e}^{2}-2560 \,{\mathrm e}\right ) x +\left (128 \,{\mathrm e}^{2}-512 \,{\mathrm e}+512\right ) x^{2} {\mathrm e}^{\frac {x}{3}}+1600 \,{\mathrm e}^{2}+64 \,{\mathrm e}^{\frac {2 x}{3}} x^{2}-640 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}} x}{x^{2}}\) | \(71\) |
parallelrisch | \(-\frac {-384 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}} x^{2}+1920 x \,{\mathrm e}^{3}+1536 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}} x^{2}-192 \,{\mathrm e}^{\frac {2 x}{3}} x^{2}-7680 \,{\mathrm e}^{2} x +1920 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}} x -1536 \,{\mathrm e}^{\frac {x}{3}} x^{2}-4800 \,{\mathrm e}^{2}+7680 x \,{\mathrm e}}{3 x^{2}}\) | \(85\) |
parts | \(64 \,{\mathrm e}^{\frac {2 x}{3}}-512 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}}+128 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}}+\frac {640 \,{\mathrm e} \left (-\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{x}-\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )\right )}{3}+\frac {640 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )}{3}+512 \,{\mathrm e}^{\frac {x}{3}}+640 \,{\mathrm e} \left (-\frac {{\mathrm e}^{2}-4 \,{\mathrm e}+4}{x}+\frac {5 \,{\mathrm e}}{2 x^{2}}\right )\) | \(89\) |
derivativedivides | \(64 \,{\mathrm e}^{\frac {2 x}{3}}+\frac {1600 \,{\mathrm e}^{2}}{x^{2}}-\frac {2560 \,{\mathrm e}}{x}+\frac {2560 \,{\mathrm e}^{2}}{x}-\frac {640 \,{\mathrm e}^{3}}{x}-512 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}}+128 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}}+\frac {640 \,{\mathrm e} \left (-\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{x}-\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )\right )}{3}+\frac {640 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )}{3}+512 \,{\mathrm e}^{\frac {x}{3}}\) | \(98\) |
default | \(64 \,{\mathrm e}^{\frac {2 x}{3}}+\frac {1600 \,{\mathrm e}^{2}}{x^{2}}-\frac {2560 \,{\mathrm e}}{x}+\frac {2560 \,{\mathrm e}^{2}}{x}-\frac {640 \,{\mathrm e}^{3}}{x}-512 \,{\mathrm e} \,{\mathrm e}^{\frac {x}{3}}+128 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}}+\frac {640 \,{\mathrm e} \left (-\frac {3 \,{\mathrm e}^{\frac {x}{3}}}{x}-\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )\right )}{3}+\frac {640 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{3}\right )}{3}+512 \,{\mathrm e}^{\frac {x}{3}}\) | \(98\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=\frac {64 \, {\left (x^{2} e^{\left (\frac {2}{3} \, x\right )} - 10 \, x e^{3} + 5 \, {\left (8 \, x + 5\right )} e^{2} - 40 \, x e + 2 \, {\left (x^{2} e^{2} + 4 \, x^{2} - {\left (4 \, x^{2} + 5 \, x\right )} e\right )} e^{\left (\frac {1}{3} \, x\right )}\right )}}{x^{2}} \]
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Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=\frac {64 x e^{\frac {2 x}{3}} + \left (- 512 e x + 512 x + 128 x e^{2} - 640 e\right ) e^{\frac {x}{3}}}{x} + \frac {x \left (- 640 e^{3} - 2560 e + 2560 e^{2}\right ) + 1600 e^{2}}{x^{2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.55 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=-\frac {640}{3} \, {\rm Ei}\left (\frac {1}{3} \, x\right ) e + \frac {640}{3} \, e \Gamma \left (-1, -\frac {1}{3} \, x\right ) - \frac {640 \, e^{3}}{x} + \frac {2560 \, e^{2}}{x} - \frac {2560 \, e}{x} + \frac {1600 \, e^{2}}{x^{2}} + 64 \, e^{\left (\frac {2}{3} \, x\right )} + 512 \, e^{\left (\frac {1}{3} \, x\right )} + 128 \, e^{\left (\frac {1}{3} \, x + 2\right )} - 512 \, e^{\left (\frac {1}{3} \, x + 1\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=\frac {64 \, {\left (x^{2} e^{\left (\frac {2}{3} \, x\right )} + 8 \, x^{2} e^{\left (\frac {1}{3} \, x\right )} + 2 \, x^{2} e^{\left (\frac {1}{3} \, x + 2\right )} - 8 \, x^{2} e^{\left (\frac {1}{3} \, x + 1\right )} - 10 \, x e^{3} + 40 \, x e^{2} - 40 \, x e - 10 \, x e^{\left (\frac {1}{3} \, x + 1\right )} + 25 \, e^{2}\right )}}{x^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {e^2 (-9600-7680 x)+7680 e x+1920 e^3 x+128 e^{2 x/3} x^3+e^{x/3} \left (512 x^3+128 e^2 x^3+e \left (1920 x-640 x^2-512 x^3\right )\right )}{3 x^3} \, dx=64\,{\mathrm {e}}^{\frac {2\,x}{3}}+\frac {1600\,{\mathrm {e}}^2-x\,\left (640\,{\mathrm {e}}^{\frac {x}{3}+1}+640\,\mathrm {e}\,{\left (\mathrm {e}-2\right )}^2\right )}{x^2}+128\,{\mathrm {e}}^{x/3}\,{\left (\mathrm {e}-2\right )}^2 \]
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