Integrand size = 179, antiderivative size = 28 \[ \int \frac {-2400+160 x^2-2400 x^3-2 x^4+80 x^5+2 x^7-80 x^8+4 x^{10}+e^{-2 e^x+2 x} \left (-150+50 x-150 x^3+100 x^4+50 x^7+e^x \left (-50 x-100 x^4-50 x^7\right )\right )+e^{-e^x+x} \left (1200-200 x-40 x^2+1210 x^3-400 x^4-20 x^5+20 x^6-200 x^7+20 x^8+10 x^9+e^x \left (200 x-10 x^3+400 x^4-20 x^6+200 x^7-10 x^9\right )\right )}{x^7} \, dx=\left (\frac {1}{x^2}+x\right )^2 \left (\frac {5 \left (-4+e^{-e^x+x}\right )}{x}+x\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(28)=56\).
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.43 \[ \int \frac {-2400+160 x^2-2400 x^3-2 x^4+80 x^5+2 x^7-80 x^8+4 x^{10}+e^{-2 e^x+2 x} \left (-150+50 x-150 x^3+100 x^4+50 x^7+e^x \left (-50 x-100 x^4-50 x^7\right )\right )+e^{-e^x+x} \left (1200-200 x-40 x^2+1210 x^3-400 x^4-20 x^5+20 x^6-200 x^7+20 x^8+10 x^9+e^x \left (200 x-10 x^3+400 x^4-20 x^6+200 x^7-10 x^9\right )\right )}{x^7} \, dx=-2 \left (-\frac {200}{x^6}+\frac {20}{x^4}-\frac {400}{x^3}-\frac {1}{2 x^2}+\frac {40}{x}-x+20 x^2-\frac {x^4}{2}-\frac {25 e^{-2 e^x+2 x} \left (1+x^3\right )^2}{2 x^6}-\frac {5 e^{-e^x+x} \left (-20+x^2\right ) \left (1+x^3\right )^2}{x^6}\right ) \]
Integrate[(-2400 + 160*x^2 - 2400*x^3 - 2*x^4 + 80*x^5 + 2*x^7 - 80*x^8 + 4*x^10 + E^(-2*E^x + 2*x)*(-150 + 50*x - 150*x^3 + 100*x^4 + 50*x^7 + E^x* (-50*x - 100*x^4 - 50*x^7)) + E^(-E^x + x)*(1200 - 200*x - 40*x^2 + 1210*x ^3 - 400*x^4 - 20*x^5 + 20*x^6 - 200*x^7 + 20*x^8 + 10*x^9 + E^x*(200*x - 10*x^3 + 400*x^4 - 20*x^6 + 200*x^7 - 10*x^9)))/x^7,x]
-2*(-200/x^6 + 20/x^4 - 400/x^3 - 1/(2*x^2) + 40/x - x + 20*x^2 - x^4/2 - (25*E^(-2*E^x + 2*x)*(1 + x^3)^2)/(2*x^6) - (5*E^(-E^x + x)*(-20 + x^2)*(1 + x^3)^2)/x^6)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {4 x^{10}-80 x^8+2 x^7+80 x^5-2 x^4-2400 x^3+160 x^2+e^{2 x-2 e^x} \left (50 x^7+100 x^4-150 x^3+e^x \left (-50 x^7-100 x^4-50 x\right )+50 x-150\right )+e^{x-e^x} \left (10 x^9+20 x^8-200 x^7+20 x^6-20 x^5-400 x^4+1210 x^3-40 x^2+e^x \left (-10 x^9+200 x^7-20 x^6+400 x^4-10 x^3+200 x\right )-200 x+1200\right )-2400}{x^7} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (-\frac {50 e^{3 x-2 e^x} \left (x^3+1\right )^2}{x^6}+\frac {10 e^{-2 \left (e^x-x\right )} \left (-e^{e^x} x^6+20 e^{e^x} x^4+5 x^4-e^{e^x} x^3+20 e^{e^x} x+5 x-15\right ) \left (x^3+1\right )}{x^7}+\frac {2 \left (2 x^{10}-40 x^8+x^7+40 x^5-x^4-1200 x^3+80 x^2-1200\right )}{x^7}+\frac {10 e^{x-e^x} \left (x^9+2 x^8-20 x^7+2 x^6-2 x^5-40 x^4+121 x^3-4 x^2-20 x+120\right )}{x^7}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {50 e^{3 x-2 e^x} \left (x^3+1\right )^2}{x^6}+\frac {10 e^{-2 \left (e^x-x\right )} \left (-e^{e^x} x^6+20 e^{e^x} x^4+5 x^4-e^{e^x} x^3+20 e^{e^x} x+5 x-15\right ) \left (x^3+1\right )}{x^7}+\frac {2 \left (2 x^{10}-40 x^8+x^7+40 x^5-x^4-1200 x^3+80 x^2-1200\right )}{x^7}+\frac {10 e^{x-e^x} \left (x^9+2 x^8-20 x^7+2 x^6-2 x^5-40 x^4+121 x^3-4 x^2-20 x+120\right )}{x^7}\right )dx\) |
Int[(-2400 + 160*x^2 - 2400*x^3 - 2*x^4 + 80*x^5 + 2*x^7 - 80*x^8 + 4*x^10 + E^(-2*E^x + 2*x)*(-150 + 50*x - 150*x^3 + 100*x^4 + 50*x^7 + E^x*(-50*x - 100*x^4 - 50*x^7)) + E^(-E^x + x)*(1200 - 200*x - 40*x^2 + 1210*x^3 - 4 00*x^4 - 20*x^5 + 20*x^6 - 200*x^7 + 20*x^8 + 10*x^9 + E^x*(200*x - 10*x^3 + 400*x^4 - 20*x^6 + 200*x^7 - 10*x^9)))/x^7,x]
3.5.48.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(26)=52\).
Time = 1.63 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.43
| method | result | size |
| risch | \(x^{4}-40 x^{2}+2 x +\frac {-80 x^{5}+x^{4}+800 x^{3}-40 x^{2}+400}{x^{6}}+\frac {25 \left (x^{6}+2 x^{3}+1\right ) {\mathrm e}^{-2 \,{\mathrm e}^{x}+2 x}}{x^{6}}+\frac {10 \left (x^{8}-20 x^{6}+2 x^{5}-40 x^{3}+x^{2}-20\right ) {\mathrm e}^{x -{\mathrm e}^{x}}}{x^{6}}\) | \(96\) |
| parallelrisch | \(\frac {x^{10}+10 \,{\mathrm e}^{x -{\mathrm e}^{x}} x^{8}-40 x^{8}+25 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+2 x} x^{6}-200 \,{\mathrm e}^{x -{\mathrm e}^{x}} x^{6}+2 x^{7}+20 \,{\mathrm e}^{x -{\mathrm e}^{x}} x^{5}-80 x^{5}+50 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+2 x} x^{3}-400 \,{\mathrm e}^{x -{\mathrm e}^{x}} x^{3}+x^{4}+10 \,{\mathrm e}^{x -{\mathrm e}^{x}} x^{2}+800 x^{3}-40 x^{2}+25 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+2 x}-200 \,{\mathrm e}^{x -{\mathrm e}^{x}}+400}{x^{6}}\) | \(146\) |
int((((-50*x^7-100*x^4-50*x)*exp(x)+50*x^7+100*x^4-150*x^3+50*x-150)*exp(x -exp(x))^2+((-10*x^9+200*x^7-20*x^6+400*x^4-10*x^3+200*x)*exp(x)+10*x^9+20 *x^8-200*x^7+20*x^6-20*x^5-400*x^4+1210*x^3-40*x^2-200*x+1200)*exp(x-exp(x ))+4*x^10-80*x^8+2*x^7+80*x^5-2*x^4-2400*x^3+160*x^2-2400)/x^7,x,method=_R ETURNVERBOSE)
x^4-40*x^2+2*x+(-80*x^5+x^4+800*x^3-40*x^2+400)/x^6+25/x^6*(x^6+2*x^3+1)*e xp(-2*exp(x)+2*x)+10/x^6*(x^8-20*x^6+2*x^5-40*x^3+x^2-20)*exp(x-exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.21 \[ \int \frac {-2400+160 x^2-2400 x^3-2 x^4+80 x^5+2 x^7-80 x^8+4 x^{10}+e^{-2 e^x+2 x} \left (-150+50 x-150 x^3+100 x^4+50 x^7+e^x \left (-50 x-100 x^4-50 x^7\right )\right )+e^{-e^x+x} \left (1200-200 x-40 x^2+1210 x^3-400 x^4-20 x^5+20 x^6-200 x^7+20 x^8+10 x^9+e^x \left (200 x-10 x^3+400 x^4-20 x^6+200 x^7-10 x^9\right )\right )}{x^7} \, dx=\frac {x^{10} - 40 \, x^{8} + 2 \, x^{7} - 80 \, x^{5} + x^{4} + 800 \, x^{3} - 40 \, x^{2} + 25 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} e^{\left (2 \, x - 2 \, e^{x}\right )} + 10 \, {\left (x^{8} - 20 \, x^{6} + 2 \, x^{5} - 40 \, x^{3} + x^{2} - 20\right )} e^{\left (x - e^{x}\right )} + 400}{x^{6}} \]
integrate((((-50*x^7-100*x^4-50*x)*exp(x)+50*x^7+100*x^4-150*x^3+50*x-150) *exp(x-exp(x))^2+((-10*x^9+200*x^7-20*x^6+400*x^4-10*x^3+200*x)*exp(x)+10* x^9+20*x^8-200*x^7+20*x^6-20*x^5-400*x^4+1210*x^3-40*x^2-200*x+1200)*exp(x -exp(x))+4*x^10-80*x^8+2*x^7+80*x^5-2*x^4-2400*x^3+160*x^2-2400)/x^7,x, al gorithm=\
(x^10 - 40*x^8 + 2*x^7 - 80*x^5 + x^4 + 800*x^3 - 40*x^2 + 25*(x^6 + 2*x^3 + 1)*e^(2*x - 2*e^x) + 10*(x^8 - 20*x^6 + 2*x^5 - 40*x^3 + x^2 - 20)*e^(x - e^x) + 400)/x^6
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.57 \[ \int \frac {-2400+160 x^2-2400 x^3-2 x^4+80 x^5+2 x^7-80 x^8+4 x^{10}+e^{-2 e^x+2 x} \left (-150+50 x-150 x^3+100 x^4+50 x^7+e^x \left (-50 x-100 x^4-50 x^7\right )\right )+e^{-e^x+x} \left (1200-200 x-40 x^2+1210 x^3-400 x^4-20 x^5+20 x^6-200 x^7+20 x^8+10 x^9+e^x \left (200 x-10 x^3+400 x^4-20 x^6+200 x^7-10 x^9\right )\right )}{x^7} \, dx=x^{4} - 40 x^{2} + 2 x + \frac {- 80 x^{5} + x^{4} + 800 x^{3} - 40 x^{2} + 400}{x^{6}} + \frac {\left (25 x^{12} + 50 x^{9} + 25 x^{6}\right ) e^{2 x - 2 e^{x}} + \left (10 x^{14} - 200 x^{12} + 20 x^{11} - 400 x^{9} + 10 x^{8} - 200 x^{6}\right ) e^{x - e^{x}}}{x^{12}} \]
integrate((((-50*x**7-100*x**4-50*x)*exp(x)+50*x**7+100*x**4-150*x**3+50*x -150)*exp(x-exp(x))**2+((-10*x**9+200*x**7-20*x**6+400*x**4-10*x**3+200*x) *exp(x)+10*x**9+20*x**8-200*x**7+20*x**6-20*x**5-400*x**4+1210*x**3-40*x** 2-200*x+1200)*exp(x-exp(x))+4*x**10-80*x**8+2*x**7+80*x**5-2*x**4-2400*x** 3+160*x**2-2400)/x**7,x)
x**4 - 40*x**2 + 2*x + (-80*x**5 + x**4 + 800*x**3 - 40*x**2 + 400)/x**6 + ((25*x**12 + 50*x**9 + 25*x**6)*exp(2*x - 2*exp(x)) + (10*x**14 - 200*x** 12 + 20*x**11 - 400*x**9 + 10*x**8 - 200*x**6)*exp(x - exp(x)))/x**12
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {-2400+160 x^2-2400 x^3-2 x^4+80 x^5+2 x^7-80 x^8+4 x^{10}+e^{-2 e^x+2 x} \left (-150+50 x-150 x^3+100 x^4+50 x^7+e^x \left (-50 x-100 x^4-50 x^7\right )\right )+e^{-e^x+x} \left (1200-200 x-40 x^2+1210 x^3-400 x^4-20 x^5+20 x^6-200 x^7+20 x^8+10 x^9+e^x \left (200 x-10 x^3+400 x^4-20 x^6+200 x^7-10 x^9\right )\right )}{x^7} \, dx=x^{4} - 40 \, x^{2} - \frac {25}{2} \, {\left (2 \, e^{x} + 1\right )} e^{\left (-2 \, e^{x}\right )} + 2 \, x - \frac {80}{x} + \frac {1}{x^{2}} + \frac {800}{x^{3}} - \frac {40}{x^{4}} + \frac {5 \, {\left (4 \, {\left (x^{8} - 20 \, x^{6} + 2 \, x^{5} - 40 \, x^{3} + x^{2} - 20\right )} e^{\left (x - e^{x}\right )} + 5 \, {\left (2 \, x^{6} e^{x} + x^{6} + 2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, e^{x}\right )}\right )}}{2 \, x^{6}} + \frac {400}{x^{6}} \]
integrate((((-50*x^7-100*x^4-50*x)*exp(x)+50*x^7+100*x^4-150*x^3+50*x-150) *exp(x-exp(x))^2+((-10*x^9+200*x^7-20*x^6+400*x^4-10*x^3+200*x)*exp(x)+10* x^9+20*x^8-200*x^7+20*x^6-20*x^5-400*x^4+1210*x^3-40*x^2-200*x+1200)*exp(x -exp(x))+4*x^10-80*x^8+2*x^7+80*x^5-2*x^4-2400*x^3+160*x^2-2400)/x^7,x, al gorithm=\
x^4 - 40*x^2 - 25/2*(2*e^x + 1)*e^(-2*e^x) + 2*x - 80/x + 1/x^2 + 800/x^3 - 40/x^4 + 5/2*(4*(x^8 - 20*x^6 + 2*x^5 - 40*x^3 + x^2 - 20)*e^(x - e^x) + 5*(2*x^6*e^x + x^6 + 2*(x^6 + 2*x^3 + 1)*e^(2*x))*e^(-2*e^x))/x^6 + 400/x ^6
\[ \int \frac {-2400+160 x^2-2400 x^3-2 x^4+80 x^5+2 x^7-80 x^8+4 x^{10}+e^{-2 e^x+2 x} \left (-150+50 x-150 x^3+100 x^4+50 x^7+e^x \left (-50 x-100 x^4-50 x^7\right )\right )+e^{-e^x+x} \left (1200-200 x-40 x^2+1210 x^3-400 x^4-20 x^5+20 x^6-200 x^7+20 x^8+10 x^9+e^x \left (200 x-10 x^3+400 x^4-20 x^6+200 x^7-10 x^9\right )\right )}{x^7} \, dx=\int { \frac {2 \, {\left (2 \, x^{10} - 40 \, x^{8} + x^{7} + 40 \, x^{5} - x^{4} - 1200 \, x^{3} + 80 \, x^{2} + 25 \, {\left (x^{7} + 2 \, x^{4} - 3 \, x^{3} - {\left (x^{7} + 2 \, x^{4} + x\right )} e^{x} + x - 3\right )} e^{\left (2 \, x - 2 \, e^{x}\right )} + 5 \, {\left (x^{9} + 2 \, x^{8} - 20 \, x^{7} + 2 \, x^{6} - 2 \, x^{5} - 40 \, x^{4} + 121 \, x^{3} - 4 \, x^{2} - {\left (x^{9} - 20 \, x^{7} + 2 \, x^{6} - 40 \, x^{4} + x^{3} - 20 \, x\right )} e^{x} - 20 \, x + 120\right )} e^{\left (x - e^{x}\right )} - 1200\right )}}{x^{7}} \,d x } \]
integrate((((-50*x^7-100*x^4-50*x)*exp(x)+50*x^7+100*x^4-150*x^3+50*x-150) *exp(x-exp(x))^2+((-10*x^9+200*x^7-20*x^6+400*x^4-10*x^3+200*x)*exp(x)+10* x^9+20*x^8-200*x^7+20*x^6-20*x^5-400*x^4+1210*x^3-40*x^2-200*x+1200)*exp(x -exp(x))+4*x^10-80*x^8+2*x^7+80*x^5-2*x^4-2400*x^3+160*x^2-2400)/x^7,x, al gorithm=\
integrate(2*(2*x^10 - 40*x^8 + x^7 + 40*x^5 - x^4 - 1200*x^3 + 80*x^2 + 25 *(x^7 + 2*x^4 - 3*x^3 - (x^7 + 2*x^4 + x)*e^x + x - 3)*e^(2*x - 2*e^x) + 5 *(x^9 + 2*x^8 - 20*x^7 + 2*x^6 - 2*x^5 - 40*x^4 + 121*x^3 - 4*x^2 - (x^9 - 20*x^7 + 2*x^6 - 40*x^4 + x^3 - 20*x)*e^x - 20*x + 120)*e^(x - e^x) - 120 0)/x^7, x)
Time = 9.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.54 \[ \int \frac {-2400+160 x^2-2400 x^3-2 x^4+80 x^5+2 x^7-80 x^8+4 x^{10}+e^{-2 e^x+2 x} \left (-150+50 x-150 x^3+100 x^4+50 x^7+e^x \left (-50 x-100 x^4-50 x^7\right )\right )+e^{-e^x+x} \left (1200-200 x-40 x^2+1210 x^3-400 x^4-20 x^5+20 x^6-200 x^7+20 x^8+10 x^9+e^x \left (200 x-10 x^3+400 x^4-20 x^6+200 x^7-10 x^9\right )\right )}{x^7} \, dx=2\,x+\frac {-80\,x^5+x^4+800\,x^3-40\,x^2+400}{x^6}-40\,x^2+x^4+\frac {{\mathrm {e}}^{x-{\mathrm {e}}^x}\,\left (10\,x^8-200\,x^6+20\,x^5-400\,x^3+10\,x^2-200\right )}{x^6}+\frac {{\mathrm {e}}^{2\,x-2\,{\mathrm {e}}^x}\,\left (25\,x^6+50\,x^3+25\right )}{x^6} \]
int((exp(x - exp(x))*(exp(x)*(200*x - 10*x^3 + 400*x^4 - 20*x^6 + 200*x^7 - 10*x^9) - 200*x - 40*x^2 + 1210*x^3 - 400*x^4 - 20*x^5 + 20*x^6 - 200*x^ 7 + 20*x^8 + 10*x^9 + 1200) + 160*x^2 - 2400*x^3 - 2*x^4 + 80*x^5 + 2*x^7 - 80*x^8 + 4*x^10 + exp(2*x - 2*exp(x))*(50*x - 150*x^3 + 100*x^4 + 50*x^7 - exp(x)*(50*x + 100*x^4 + 50*x^7) - 150) - 2400)/x^7,x)