Integrand size = 50, antiderivative size = 22 \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {1}{625} e^{10 x} \left (2+e^{-e^x+x}\right )^2 \]
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Time = 0.44 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82, number of steps used = 59, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 2225, 2320, 2227, 2207} \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {4 e^{10 x}}{625}+\frac {4}{625} e^{11 x-e^x}+\frac {1}{625} e^{12 x-2 e^x} \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {1}{625} \int \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx \\ & = \frac {1}{625} \int e^{-e^x+11 x} \left (44-4 e^x\right ) \, dx+\frac {1}{625} \int e^{-2 e^x+12 x} \left (12-2 e^x\right ) \, dx+\frac {8}{125} \int e^{10 x} \, dx \\ & = \frac {4 e^{10 x}}{625}+\frac {1}{625} \text {Subst}\left (\int 4 e^{-x} (11-x) x^{10} \, dx,x,e^x\right )+\frac {1}{625} \text {Subst}\left (\int 2 e^{-2 x} (6-x) x^{11} \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}+\frac {2}{625} \text {Subst}\left (\int e^{-2 x} (6-x) x^{11} \, dx,x,e^x\right )+\frac {4}{625} \text {Subst}\left (\int e^{-x} (11-x) x^{10} \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}+\frac {2}{625} \text {Subst}\left (\int \left (6 e^{-2 x} x^{11}-e^{-2 x} x^{12}\right ) \, dx,x,e^x\right )+\frac {4}{625} \text {Subst}\left (\int \left (11 e^{-x} x^{10}-e^{-x} x^{11}\right ) \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {2}{625} \text {Subst}\left (\int e^{-2 x} x^{12} \, dx,x,e^x\right )-\frac {4}{625} \text {Subst}\left (\int e^{-x} x^{11} \, dx,x,e^x\right )+\frac {12}{625} \text {Subst}\left (\int e^{-2 x} x^{11} \, dx,x,e^x\right )+\frac {44}{625} \text {Subst}\left (\int e^{-x} x^{10} \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {44}{625} e^{-e^x+10 x}-\frac {6}{625} e^{-2 e^x+11 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {12}{625} \text {Subst}\left (\int e^{-2 x} x^{11} \, dx,x,e^x\right )-\frac {44}{625} \text {Subst}\left (\int e^{-x} x^{10} \, dx,x,e^x\right )+\frac {66}{625} \text {Subst}\left (\int e^{-2 x} x^{10} \, dx,x,e^x\right )+\frac {88}{125} \text {Subst}\left (\int e^{-x} x^9 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {88}{125} e^{-e^x+9 x}-\frac {33}{625} e^{-2 e^x+10 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {66}{625} \text {Subst}\left (\int e^{-2 x} x^{10} \, dx,x,e^x\right )+\frac {66}{125} \text {Subst}\left (\int e^{-2 x} x^9 \, dx,x,e^x\right )-\frac {88}{125} \text {Subst}\left (\int e^{-x} x^9 \, dx,x,e^x\right )+\frac {792}{125} \text {Subst}\left (\int e^{-x} x^8 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {792}{125} e^{-e^x+8 x}-\frac {33}{125} e^{-2 e^x+9 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {66}{125} \text {Subst}\left (\int e^{-2 x} x^9 \, dx,x,e^x\right )+\frac {297}{125} \text {Subst}\left (\int e^{-2 x} x^8 \, dx,x,e^x\right )-\frac {792}{125} \text {Subst}\left (\int e^{-x} x^8 \, dx,x,e^x\right )+\frac {6336}{125} \text {Subst}\left (\int e^{-x} x^7 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {6336}{125} e^{-e^x+7 x}-\frac {297}{250} e^{-2 e^x+8 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {297}{125} \text {Subst}\left (\int e^{-2 x} x^8 \, dx,x,e^x\right )+\frac {1188}{125} \text {Subst}\left (\int e^{-2 x} x^7 \, dx,x,e^x\right )-\frac {6336}{125} \text {Subst}\left (\int e^{-x} x^7 \, dx,x,e^x\right )+\frac {44352}{125} \text {Subst}\left (\int e^{-x} x^6 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {44352}{125} e^{-e^x+6 x}-\frac {594}{125} e^{-2 e^x+7 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {1188}{125} \text {Subst}\left (\int e^{-2 x} x^7 \, dx,x,e^x\right )+\frac {4158}{125} \text {Subst}\left (\int e^{-2 x} x^6 \, dx,x,e^x\right )-\frac {44352}{125} \text {Subst}\left (\int e^{-x} x^6 \, dx,x,e^x\right )+\frac {266112}{125} \text {Subst}\left (\int e^{-x} x^5 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {266112}{125} e^{-e^x+5 x}-\frac {2079}{125} e^{-2 e^x+6 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {4158}{125} \text {Subst}\left (\int e^{-2 x} x^6 \, dx,x,e^x\right )+\frac {12474}{125} \text {Subst}\left (\int e^{-2 x} x^5 \, dx,x,e^x\right )-\frac {266112}{125} \text {Subst}\left (\int e^{-x} x^5 \, dx,x,e^x\right )+\frac {266112}{25} \text {Subst}\left (\int e^{-x} x^4 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {266112}{25} e^{-e^x+4 x}-\frac {6237}{125} e^{-2 e^x+5 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {12474}{125} \text {Subst}\left (\int e^{-2 x} x^5 \, dx,x,e^x\right )+\frac {6237}{25} \text {Subst}\left (\int e^{-2 x} x^4 \, dx,x,e^x\right )-\frac {266112}{25} \text {Subst}\left (\int e^{-x} x^4 \, dx,x,e^x\right )+\frac {1064448}{25} \text {Subst}\left (\int e^{-x} x^3 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {1064448}{25} e^{-e^x+3 x}-\frac {6237}{50} e^{-2 e^x+4 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {6237}{25} \text {Subst}\left (\int e^{-2 x} x^4 \, dx,x,e^x\right )+\frac {12474}{25} \text {Subst}\left (\int e^{-2 x} x^3 \, dx,x,e^x\right )-\frac {1064448}{25} \text {Subst}\left (\int e^{-x} x^3 \, dx,x,e^x\right )+\frac {3193344}{25} \text {Subst}\left (\int e^{-x} x^2 \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {3193344}{25} e^{-e^x+2 x}-\frac {6237}{25} e^{-2 e^x+3 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {12474}{25} \text {Subst}\left (\int e^{-2 x} x^3 \, dx,x,e^x\right )+\frac {18711}{25} \text {Subst}\left (\int e^{-2 x} x^2 \, dx,x,e^x\right )-\frac {3193344}{25} \text {Subst}\left (\int e^{-x} x^2 \, dx,x,e^x\right )+\frac {6386688}{25} \text {Subst}\left (\int e^{-x} x \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}-\frac {6386688}{25} e^{-e^x+x}-\frac {18711}{50} e^{-2 e^x+2 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}+\frac {18711}{25} \text {Subst}\left (\int e^{-2 x} x \, dx,x,e^x\right )-\frac {18711}{25} \text {Subst}\left (\int e^{-2 x} x^2 \, dx,x,e^x\right )+\frac {6386688}{25} \text {Subst}\left (\int e^{-x} \, dx,x,e^x\right )-\frac {6386688}{25} \text {Subst}\left (\int e^{-x} x \, dx,x,e^x\right ) \\ & = -\frac {6386688 e^{-e^x}}{25}+\frac {4 e^{10 x}}{625}-\frac {18711}{50} e^{-2 e^x+x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}+\frac {18711}{50} \text {Subst}\left (\int e^{-2 x} \, dx,x,e^x\right )-\frac {18711}{25} \text {Subst}\left (\int e^{-2 x} x \, dx,x,e^x\right )-\frac {6386688}{25} \text {Subst}\left (\int e^{-x} \, dx,x,e^x\right ) \\ & = -\frac {18711}{100} e^{-2 e^x}+\frac {4 e^{10 x}}{625}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {18711}{50} \text {Subst}\left (\int e^{-2 x} \, dx,x,e^x\right ) \\ & = \frac {4 e^{10 x}}{625}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {1}{625} e^{-2 e^x+10 x} \left (2 e^{e^x}+e^x\right )^2 \]
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Time = 0.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {{\mathrm e}^{12 x -2 \,{\mathrm e}^{x}}}{625}+\frac {4 \,{\mathrm e}^{11 x -{\mathrm e}^{x}}}{625}+\frac {4 \,{\mathrm e}^{10 x}}{625}\) | \(30\) |
parts | \(\frac {4 \,{\mathrm e}^{10 x}}{625}+\frac {{\mathrm e}^{12 x} {\mathrm e}^{-2 \,{\mathrm e}^{x}}}{625}+\frac {4 \,{\mathrm e}^{11 x} {\mathrm e}^{-{\mathrm e}^{x}}}{625}\) | \(30\) |
default | \(\frac {4 \,{\mathrm e}^{10 x}}{625}+\frac {{\mathrm e}^{12 x} {\mathrm e}^{-2 \,{\mathrm e}^{x}}}{625}+\frac {4 \,{\mathrm e}^{11 x} {\mathrm e}^{-{\mathrm e}^{x}}}{625}\) | \(32\) |
parallelrisch | \(\frac {{\mathrm e}^{10 x} {\mathrm e}^{-2 \,{\mathrm e}^{x}+2 x}}{625}+\frac {4 \,{\mathrm e}^{10 x} {\mathrm e}^{x -{\mathrm e}^{x}}}{625}+\frac {4 \,{\mathrm e}^{10 x}}{625}\) | \(42\) |
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {1}{625} \, {\left (4 \, e^{\left (20 \, x\right )} + e^{\left (22 \, x - 2 \, e^{x}\right )} + 4 \, e^{\left (21 \, x - e^{x}\right )}\right )} e^{\left (-10 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {4 e^{10 x} e^{x - e^{x}}}{625} + \frac {e^{10 x} e^{2 x - 2 e^{x}}}{625} + \frac {4 e^{10 x}}{625} \]
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Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {4}{625} \, e^{\left (10 \, x\right )} + \frac {1}{625} \, e^{\left (12 \, x - 2 \, e^{x}\right )} + \frac {4}{625} \, e^{\left (11 \, x - e^{x}\right )} \]
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {4}{625} \, e^{\left (10 \, x\right )} + \frac {1}{625} \, e^{\left (12 \, x - 2 \, e^{x}\right )} + \frac {4}{625} \, e^{\left (11 \, x - e^{x}\right )} \]
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Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {1}{625} \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx=\frac {4\,{\mathrm {e}}^{10\,x}}{625}+\frac {4\,{\mathrm {e}}^{11\,x-{\mathrm {e}}^x}}{625}+\frac {{\mathrm {e}}^{12\,x-2\,{\mathrm {e}}^x}}{625} \]
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