Integrand size = 166, antiderivative size = 26 \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=\frac {5 e^{-5 x} \left (5+e^{e^x x}\right )^4 (5-x)}{x^4} \] Output:
5*(5-x)/exp(5*x)*(exp(exp(x)*x)+5)^4/x^4
Time = 5.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=-\frac {5 e^{-5 x} \left (5+e^{e^x x}\right )^4 (-5+x)}{x^4} \] Input:
Integrate[(-62500 - 68750*x + 15625*x^2 + E^(E^x*x)*(-50000 - 55000*x + 12 500*x^2 + E^x*(12500*x + 10000*x^2 - 2500*x^3)) + E^(2*E^x*x)*(-15000 - 16 500*x + 3750*x^2 + E^x*(7500*x + 6000*x^2 - 1500*x^3)) + E^(3*E^x*x)*(-200 0 - 2200*x + 500*x^2 + E^x*(1500*x + 1200*x^2 - 300*x^3)) + E^(4*E^x*x)*(- 100 - 110*x + 25*x^2 + E^x*(100*x + 80*x^2 - 20*x^3)))/(E^(5*x)*x^5),x]
Output:
(-5*(5 + E^(E^x*x))^4*(-5 + x))/(E^(5*x)*x^4)
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.
| \(\displaystyle \int \frac {e^{-5 x} \left (15625 x^2+e^{e^x x} \left (12500 x^2+e^x \left (-2500 x^3+10000 x^2+12500 x\right )-55000 x-50000\right )+e^{2 e^x x} \left (3750 x^2+e^x \left (-1500 x^3+6000 x^2+7500 x\right )-16500 x-15000\right )+e^{3 e^x x} \left (500 x^2+e^x \left (-300 x^3+1200 x^2+1500 x\right )-2200 x-2000\right )+e^{4 e^x x} \left (25 x^2+e^x \left (-20 x^3+80 x^2+100 x\right )-110 x-100\right )-68750 x-62500\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 e^{-5 x} \left (e^{e^x x}+5\right )^3 \left (-e^{e^x x} \left (-5 x^2+22 x+20\right )-5 \left (-5 x^2+22 x+20\right )-4 e^{e^x x+x} x \left (x^2-4 x-5\right )\right )}{x^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int -\frac {e^{-5 x} \left (5+e^{e^x x}\right )^3 \left (e^{e^x x} \left (-5 x^2+22 x+20\right )+5 \left (-5 x^2+22 x+20\right )-4 e^{e^x x+x} x \left (-x^2+4 x+5\right )\right )}{x^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 \int \frac {e^{-5 x} \left (5+e^{e^x x}\right )^3 \left (e^{e^x x} \left (-5 x^2+22 x+20\right )+5 \left (-5 x^2+22 x+20\right )-4 e^{e^x x+x} x \left (-x^2+4 x+5\right )\right )}{x^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (-\frac {625 e^{-5 x} \left (5 x^2-22 x-20\right )}{x^5}+\frac {500 e^{e^x x-5 x} \left (e^x x^3-4 e^x x^2-5 x^2-5 e^x x+22 x+20\right )}{x^5}+\frac {150 e^{2 e^x x-5 x} \left (2 e^x x^3-8 e^x x^2-5 x^2-10 e^x x+22 x+20\right )}{x^5}+\frac {20 e^{3 e^x x-5 x} \left (3 e^x x^3-12 e^x x^2-5 x^2-15 e^x x+22 x+20\right )}{x^5}+\frac {e^{4 e^x x-5 x} \left (4 e^x x^3-16 e^x x^2-5 x^2-20 e^x x+22 x+20\right )}{x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \left (10000 \int \frac {e^{e^x x-5 x}}{x^5}dx+3000 \int \frac {e^{2 e^x x-5 x}}{x^5}dx+400 \int \frac {e^{3 e^x x-5 x}}{x^5}dx+20 \int \frac {e^{4 e^x x-5 x}}{x^5}dx-1500 \int \frac {e^{2 \left (-2+e^x\right ) x}}{x^4}dx-20 \int \frac {e^{4 \left (-1+e^x\right ) x}}{x^4}dx+11000 \int \frac {e^{e^x x-5 x}}{x^4}dx-2500 \int \frac {e^{e^x x-4 x}}{x^4}dx+3300 \int \frac {e^{2 e^x x-5 x}}{x^4}dx+440 \int \frac {e^{3 e^x x-5 x}}{x^4}dx-300 \int \frac {e^{3 e^x x-4 x}}{x^4}dx+22 \int \frac {e^{4 e^x x-5 x}}{x^4}dx-1200 \int \frac {e^{2 \left (-2+e^x\right ) x}}{x^3}dx-16 \int \frac {e^{4 \left (-1+e^x\right ) x}}{x^3}dx-2500 \int \frac {e^{e^x x-5 x}}{x^3}dx-2000 \int \frac {e^{e^x x-4 x}}{x^3}dx-750 \int \frac {e^{2 e^x x-5 x}}{x^3}dx-100 \int \frac {e^{3 e^x x-5 x}}{x^3}dx-240 \int \frac {e^{3 e^x x-4 x}}{x^3}dx-5 \int \frac {e^{4 e^x x-5 x}}{x^3}dx+300 \int \frac {e^{2 \left (-2+e^x\right ) x}}{x^2}dx+4 \int \frac {e^{4 \left (-1+e^x\right ) x}}{x^2}dx+500 \int \frac {e^{e^x x-4 x}}{x^2}dx+60 \int \frac {e^{3 e^x x-4 x}}{x^2}dx-\frac {3125 e^{-5 x}}{x^4}+\frac {625 e^{-5 x}}{x^3}\right )\) |
Input:
Int[(-62500 - 68750*x + 15625*x^2 + E^(E^x*x)*(-50000 - 55000*x + 12500*x^ 2 + E^x*(12500*x + 10000*x^2 - 2500*x^3)) + E^(2*E^x*x)*(-15000 - 16500*x + 3750*x^2 + E^x*(7500*x + 6000*x^2 - 1500*x^3)) + E^(3*E^x*x)*(-2000 - 22 00*x + 500*x^2 + E^x*(1500*x + 1200*x^2 - 300*x^3)) + E^(4*E^x*x)*(-100 - 110*x + 25*x^2 + E^x*(100*x + 80*x^2 - 20*x^3)))/(E^(5*x)*x^5),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(25)=50\).
Time = 0.66 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08
| method | result | size |
| risch | \(-\frac {3125 \left (-5+x \right ) {\mathrm e}^{-5 x}}{x^{4}}-\frac {5 \left (-5+x \right ) {\mathrm e}^{x \left (4 \,{\mathrm e}^{x}-5\right )}}{x^{4}}-\frac {100 \left (-5+x \right ) {\mathrm e}^{x \left (3 \,{\mathrm e}^{x}-5\right )}}{x^{4}}-\frac {750 \left (-5+x \right ) {\mathrm e}^{x \left (2 \,{\mathrm e}^{x}-5\right )}}{x^{4}}-\frac {2500 \left (-5+x \right ) {\mathrm e}^{x \left ({\mathrm e}^{x}-5\right )}}{x^{4}}\) | \(80\) |
| parallelrisch | \(-\frac {\left (-78125+25 \,{\mathrm e}^{4 \,{\mathrm e}^{x} x} x +500 \,{\mathrm e}^{3 \,{\mathrm e}^{x} x} x -125 \,{\mathrm e}^{4 \,{\mathrm e}^{x} x}+3750 \,{\mathrm e}^{2 \,{\mathrm e}^{x} x} x -2500 \,{\mathrm e}^{3 \,{\mathrm e}^{x} x}+12500 \,{\mathrm e}^{{\mathrm e}^{x} x} x -18750 \,{\mathrm e}^{2 \,{\mathrm e}^{x} x}+15625 x -62500 \,{\mathrm e}^{{\mathrm e}^{x} x}\right ) {\mathrm e}^{-5 x}}{5 x^{4}}\) | \(89\) |
Input:
int((((-20*x^3+80*x^2+100*x)*exp(x)+25*x^2-110*x-100)*exp(exp(x)*x)^4+((-3 00*x^3+1200*x^2+1500*x)*exp(x)+500*x^2-2200*x-2000)*exp(exp(x)*x)^3+((-150 0*x^3+6000*x^2+7500*x)*exp(x)+3750*x^2-16500*x-15000)*exp(exp(x)*x)^2+((-2 500*x^3+10000*x^2+12500*x)*exp(x)+12500*x^2-55000*x-50000)*exp(exp(x)*x)+1 5625*x^2-68750*x-62500)/x^5/exp(5*x),x,method=_RETURNVERBOSE)
Output:
-3125*(-5+x)/x^4*exp(-5*x)-5*(-5+x)/x^4*exp(x*(4*exp(x)-5))-100*(-5+x)/x^4 *exp(x*(3*exp(x)-5))-750*(-5+x)/x^4*exp(x*(2*exp(x)-5))-2500*(-5+x)/x^4*ex p(x*(exp(x)-5))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=-\frac {5 \, {\left ({\left (x - 5\right )} e^{\left (4 \, x e^{x}\right )} + 20 \, {\left (x - 5\right )} e^{\left (3 \, x e^{x}\right )} + 150 \, {\left (x - 5\right )} e^{\left (2 \, x e^{x}\right )} + 500 \, {\left (x - 5\right )} e^{\left (x e^{x}\right )} + 625 \, x - 3125\right )} e^{\left (-5 \, x\right )}}{x^{4}} \] Input:
integrate((((-20*x^3+80*x^2+100*x)*exp(x)+25*x^2-110*x-100)*exp(exp(x)*x)^ 4+((-300*x^3+1200*x^2+1500*x)*exp(x)+500*x^2-2200*x-2000)*exp(exp(x)*x)^3+ ((-1500*x^3+6000*x^2+7500*x)*exp(x)+3750*x^2-16500*x-15000)*exp(exp(x)*x)^ 2+((-2500*x^3+10000*x^2+12500*x)*exp(x)+12500*x^2-55000*x-50000)*exp(exp(x )*x)+15625*x^2-68750*x-62500)/x^5/exp(5*x),x, algorithm="fricas")
Output:
-5*((x - 5)*e^(4*x*e^x) + 20*(x - 5)*e^(3*x*e^x) + 150*(x - 5)*e^(2*x*e^x) + 500*(x - 5)*e^(x*e^x) + 625*x - 3125)*e^(-5*x)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (22) = 44\).
Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.96 \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=\frac {\left (15625 - 3125 x\right ) e^{- 5 x}}{x^{4}} + \frac {\left (- 2500 x^{13} e^{- 5 x} + 12500 x^{12} e^{- 5 x}\right ) e^{x e^{x}} + \left (- 750 x^{13} e^{- 5 x} + 3750 x^{12} e^{- 5 x}\right ) e^{2 x e^{x}} + \left (- 100 x^{13} e^{- 5 x} + 500 x^{12} e^{- 5 x}\right ) e^{3 x e^{x}} + \left (- 5 x^{13} e^{- 5 x} + 25 x^{12} e^{- 5 x}\right ) e^{4 x e^{x}}}{x^{16}} \] Input:
integrate((((-20*x**3+80*x**2+100*x)*exp(x)+25*x**2-110*x-100)*exp(exp(x)* x)**4+((-300*x**3+1200*x**2+1500*x)*exp(x)+500*x**2-2200*x-2000)*exp(exp(x )*x)**3+((-1500*x**3+6000*x**2+7500*x)*exp(x)+3750*x**2-16500*x-15000)*exp (exp(x)*x)**2+((-2500*x**3+10000*x**2+12500*x)*exp(x)+12500*x**2-55000*x-5 0000)*exp(exp(x)*x)+15625*x**2-68750*x-62500)/x**5/exp(5*x),x)
Output:
(15625 - 3125*x)*exp(-5*x)/x**4 + ((-2500*x**13*exp(-5*x) + 12500*x**12*ex p(-5*x))*exp(x*exp(x)) + (-750*x**13*exp(-5*x) + 3750*x**12*exp(-5*x))*exp (2*x*exp(x)) + (-100*x**13*exp(-5*x) + 500*x**12*exp(-5*x))*exp(3*x*exp(x) ) + (-5*x**13*exp(-5*x) + 25*x**12*exp(-5*x))*exp(4*x*exp(x)))/x**16
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=-\frac {5 \, {\left ({\left (x - 5\right )} e^{\left (4 \, x e^{x}\right )} + 20 \, {\left (x - 5\right )} e^{\left (3 \, x e^{x}\right )} + 150 \, {\left (x - 5\right )} e^{\left (2 \, x e^{x}\right )} + 500 \, {\left (x - 5\right )} e^{\left (x e^{x}\right )}\right )} e^{\left (-5 \, x\right )}}{x^{4}} - 390625 \, \Gamma \left (-2, 5 \, x\right ) + 8593750 \, \Gamma \left (-3, 5 \, x\right ) + 39062500 \, \Gamma \left (-4, 5 \, x\right ) \] Input:
integrate((((-20*x^3+80*x^2+100*x)*exp(x)+25*x^2-110*x-100)*exp(exp(x)*x)^ 4+((-300*x^3+1200*x^2+1500*x)*exp(x)+500*x^2-2200*x-2000)*exp(exp(x)*x)^3+ ((-1500*x^3+6000*x^2+7500*x)*exp(x)+3750*x^2-16500*x-15000)*exp(exp(x)*x)^ 2+((-2500*x^3+10000*x^2+12500*x)*exp(x)+12500*x^2-55000*x-50000)*exp(exp(x )*x)+15625*x^2-68750*x-62500)/x^5/exp(5*x),x, algorithm="maxima")
Output:
-5*((x - 5)*e^(4*x*e^x) + 20*(x - 5)*e^(3*x*e^x) + 150*(x - 5)*e^(2*x*e^x) + 500*(x - 5)*e^(x*e^x))*e^(-5*x)/x^4 - 390625*gamma(-2, 5*x) + 8593750*g amma(-3, 5*x) + 39062500*gamma(-4, 5*x)
Timed out. \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=\text {Timed out} \] Input:
integrate((((-20*x^3+80*x^2+100*x)*exp(x)+25*x^2-110*x-100)*exp(exp(x)*x)^ 4+((-300*x^3+1200*x^2+1500*x)*exp(x)+500*x^2-2200*x-2000)*exp(exp(x)*x)^3+ ((-1500*x^3+6000*x^2+7500*x)*exp(x)+3750*x^2-16500*x-15000)*exp(exp(x)*x)^ 2+((-2500*x^3+10000*x^2+12500*x)*exp(x)+12500*x^2-55000*x-50000)*exp(exp(x )*x)+15625*x^2-68750*x-62500)/x^5/exp(5*x),x, algorithm="giac")
Output:
Timed out
Time = 2.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.62 \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=-\frac {{\mathrm {e}}^{-5\,x}\,\left (3125\,x-15625\right )}{x^4}-\frac {{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x-5\,x}\,\left (5\,x-25\right )}{x^4}-\frac {{\mathrm {e}}^{3\,x\,{\mathrm {e}}^x-5\,x}\,\left (100\,x-500\right )}{x^4}-\frac {{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x-5\,x}\,\left (750\,x-3750\right )}{x^4}-\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^x-5\,x}\,\left (2500\,x-12500\right )}{x^4} \] Input:
int(-(exp(-5*x)*(68750*x + exp(4*x*exp(x))*(110*x - 25*x^2 - exp(x)*(100*x + 80*x^2 - 20*x^3) + 100) + exp(3*x*exp(x))*(2200*x - 500*x^2 - exp(x)*(1 500*x + 1200*x^2 - 300*x^3) + 2000) + exp(2*x*exp(x))*(16500*x - 3750*x^2 - exp(x)*(7500*x + 6000*x^2 - 1500*x^3) + 15000) + exp(x*exp(x))*(55000*x - 12500*x^2 - exp(x)*(12500*x + 10000*x^2 - 2500*x^3) + 50000) - 15625*x^2 + 62500))/x^5,x)
Output:
- (exp(-5*x)*(3125*x - 15625))/x^4 - (exp(4*x*exp(x) - 5*x)*(5*x - 25))/x^ 4 - (exp(3*x*exp(x) - 5*x)*(100*x - 500))/x^4 - (exp(2*x*exp(x) - 5*x)*(75 0*x - 3750))/x^4 - (exp(x*exp(x) - 5*x)*(2500*x - 12500))/x^4
Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.81 \[ \int \frac {e^{-5 x} \left (-62500-68750 x+15625 x^2+e^{e^x x} \left (-50000-55000 x+12500 x^2+e^x \left (12500 x+10000 x^2-2500 x^3\right )\right )+e^{2 e^x x} \left (-15000-16500 x+3750 x^2+e^x \left (7500 x+6000 x^2-1500 x^3\right )\right )+e^{3 e^x x} \left (-2000-2200 x+500 x^2+e^x \left (1500 x+1200 x^2-300 x^3\right )\right )+e^{4 e^x x} \left (-100-110 x+25 x^2+e^x \left (100 x+80 x^2-20 x^3\right )\right )\right )}{x^5} \, dx=\frac {-5 e^{4 e^{x} x} x +25 e^{4 e^{x} x}-100 e^{3 e^{x} x} x +500 e^{3 e^{x} x}-750 e^{2 e^{x} x} x +3750 e^{2 e^{x} x}-2500 e^{e^{x} x} x +12500 e^{e^{x} x}-3125 x +15625}{e^{5 x} x^{4}} \] Input:
int((((-20*x^3+80*x^2+100*x)*exp(x)+25*x^2-110*x-100)*exp(exp(x)*x)^4+((-3 00*x^3+1200*x^2+1500*x)*exp(x)+500*x^2-2200*x-2000)*exp(exp(x)*x)^3+((-150 0*x^3+6000*x^2+7500*x)*exp(x)+3750*x^2-16500*x-15000)*exp(exp(x)*x)^2+((-2 500*x^3+10000*x^2+12500*x)*exp(x)+12500*x^2-55000*x-50000)*exp(exp(x)*x)+1 5625*x^2-68750*x-62500)/x^5/exp(5*x),x)
Output:
(5*( - e**(4*e**x*x)*x + 5*e**(4*e**x*x) - 20*e**(3*e**x*x)*x + 100*e**(3* e**x*x) - 150*e**(2*e**x*x)*x + 750*e**(2*e**x*x) - 500*e**(e**x*x)*x + 25 00*e**(e**x*x) - 625*x + 3125))/(e**(5*x)*x**4)