Integrand size = 99, antiderivative size = 32 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx=\left (4-\frac {1}{e^4}\right ) (-3+x) \left (-3+\left (2-e^{e^x}-x\right )^2-x^2\right ) \] Output:
(-3+x)*((-exp(exp(x))+2-x)^2-x^2-3)*(4-exp(-4))
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx=\frac {\left (-1+4 e^4\right ) \left (e^{2 e^x} (-3+x)+13 x-4 x^2+e^{e^x} \left (12-10 x+2 x^2\right )\right )}{e^4} \] Input:
Integrate[(-13 + E^4*(52 - 32*x) + 8*x + E^(2*E^x)*(-1 + 4*E^4 + E^x*(6 - 2*x + E^4*(-24 + 8*x))) + E^E^x*(10 - 4*x + E^4*(-40 + 16*x) + E^x*(-12 + 10*x - 2*x^2 + E^4*(48 - 40*x + 8*x^2))))/E^4,x]
Output:
((-1 + 4*E^4)*(E^(2*E^x)*(-3 + x) + 13*x - 4*x^2 + E^E^x*(12 - 10*x + 2*x^ 2)))/E^4
Leaf count is larger than twice the leaf count of optimal. \(78\) vs. \(2(32)=64\).
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {27, 2009}
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.
\(\displaystyle \int \frac {e^{e^x} \left (e^x \left (-2 x^2+e^4 \left (8 x^2-40 x+48\right )+10 x-12\right )-4 x+e^4 (16 x-40)+10\right )+e^4 (52-32 x)+8 x+e^{2 e^x} \left (e^x \left (-2 x+e^4 (8 x-24)+6\right )+4 e^4-1\right )-13}{e^4} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (-e^{2 e^x} \left (-2 e^x \left (-4 e^4 (3-x)-x+3\right )-4 e^4+1\right )+4 e^4 (13-8 x)+8 x+2 e^{e^x} \left (-4 e^4 (5-2 x)-2 x-e^x \left (x^2-5 x-4 e^4 \left (x^2-5 x+6\right )+6\right )+5\right )-13\right )dx}{e^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 x^2-2 e^{e^x} \left (x^2-4 e^4 \left (x^2-5 x+6\right )-5 x+6\right )-\frac {1}{4} e^4 (13-8 x)^2+e^{2 e^x} \left (-4 e^4 (3-x)-x+3\right )-13 x}{e^4}\) |
Input:
Int[(-13 + E^4*(52 - 32*x) + 8*x + E^(2*E^x)*(-1 + 4*E^4 + E^x*(6 - 2*x + E^4*(-24 + 8*x))) + E^E^x*(10 - 4*x + E^4*(-40 + 16*x) + E^x*(-12 + 10*x - 2*x^2 + E^4*(48 - 40*x + 8*x^2))))/E^4,x]
Output:
(-1/4*(E^4*(13 - 8*x)^2) + E^(2*E^x)*(3 - 4*E^4*(3 - x) - x) - 13*x + 4*x^ 2 - 2*E^E^x*(6 - 5*x + x^2 - 4*E^4*(6 - 5*x + x^2)))/E^4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(29)=58\).
Time = 0.42 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.62
method | result | size |
risch | \(-16 \,{\mathrm e}^{4} {\mathrm e}^{-4} x^{2}+52 \,{\mathrm e}^{4} {\mathrm e}^{-4} x +4 \,{\mathrm e}^{-4} x^{2}-13 \,{\mathrm e}^{-4} x +\left (4 x \,{\mathrm e}^{4}-12 \,{\mathrm e}^{4}-x +3\right ) {\mathrm e}^{-4+2 \,{\mathrm e}^{x}}+\left (8 x^{2} {\mathrm e}^{4}-40 x \,{\mathrm e}^{4}-2 x^{2}+48 \,{\mathrm e}^{4}+10 x -12\right ) {\mathrm e}^{-4+{\mathrm e}^{x}}\) | \(84\) |
default | \({\mathrm e}^{-4} \left (-13 x +4 \,{\mathrm e}^{4} \left (-4 x^{2}+13 x \right )+\left (-12 \,{\mathrm e}^{4}+3\right ) {\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (4 \,{\mathrm e}^{4}-1\right ) x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (48 \,{\mathrm e}^{4}-12\right ) {\mathrm e}^{{\mathrm e}^{x}}+\left (-40 \,{\mathrm e}^{4}+10\right ) x \,{\mathrm e}^{{\mathrm e}^{x}}+\left (8 \,{\mathrm e}^{4}-2\right ) x^{2} {\mathrm e}^{{\mathrm e}^{x}}+4 x^{2}\right )\) | \(87\) |
parallelrisch | \({\mathrm e}^{-4} \left (8 x^{2} {\mathrm e}^{{\mathrm e}^{x}} {\mathrm e}^{4}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x \,{\mathrm e}^{4}-16 x^{2} {\mathrm e}^{4}-40 \,{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{x}} x -12 \,{\mathrm e}^{4} {\mathrm e}^{2 \,{\mathrm e}^{x}}-2 x^{2} {\mathrm e}^{{\mathrm e}^{x}}-x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+52 x \,{\mathrm e}^{4}+48 \,{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{x}}+4 x^{2}+10 x \,{\mathrm e}^{{\mathrm e}^{x}}+3 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}-12 \,{\mathrm e}^{{\mathrm e}^{x}}-13 x \right )\) | \(105\) |
norman | \(\left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+13 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x -4 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x^{2}+12 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} {\mathrm e}^{{\mathrm e}^{x}}-3 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} {\mathrm e}^{2 \,{\mathrm e}^{x}}-10 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x \,{\mathrm e}^{{\mathrm e}^{x}}+2 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x^{2} {\mathrm e}^{{\mathrm e}^{x}}\) | \(113\) |
Input:
int(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8*x^2-40 *x+48)*exp(4)-2*x^2+10*x-12)*exp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp(x))+( -32*x+52)*exp(4)+8*x-13)/exp(4),x,method=_RETURNVERBOSE)
Output:
-16*exp(4)*exp(-4)*x^2+52*exp(4)*exp(-4)*x+4*exp(-4)*x^2-13*exp(-4)*x+(4*x *exp(4)-12*exp(4)-x+3)*exp(-4+2*exp(x))+(8*x^2*exp(4)-40*x*exp(4)-2*x^2+48 *exp(4)+10*x-12)*exp(-4+exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx={\left (4 \, x^{2} - 4 \, {\left (4 \, x^{2} - 13 \, x\right )} e^{4} + {\left (4 \, {\left (x - 3\right )} e^{4} - x + 3\right )} e^{\left (2 \, e^{x}\right )} - 2 \, {\left (x^{2} - 4 \, {\left (x^{2} - 5 \, x + 6\right )} e^{4} - 5 \, x + 6\right )} e^{\left (e^{x}\right )} - 13 \, x\right )} e^{\left (-4\right )} \] Input:
integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8* x^2-40*x+48)*exp(4)-2*x^2+10*x-12)*exp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp (x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x, algorithm="fricas")
Output:
(4*x^2 - 4*(4*x^2 - 13*x)*e^4 + (4*(x - 3)*e^4 - x + 3)*e^(2*e^x) - 2*(x^2 - 4*(x^2 - 5*x + 6)*e^4 - 5*x + 6)*e^(e^x) - 13*x)*e^(-4)
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (24) = 48\).
Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx=\frac {x^{2} \cdot \left (4 - 16 e^{4}\right )}{e^{4}} + \frac {x \left (-13 + 52 e^{4}\right )}{e^{4}} + \frac {\left (- x e^{4} + 4 x e^{8} - 12 e^{8} + 3 e^{4}\right ) e^{2 e^{x}} + \left (- 2 x^{2} e^{4} + 8 x^{2} e^{8} - 40 x e^{8} + 10 x e^{4} - 12 e^{4} + 48 e^{8}\right ) e^{e^{x}}}{e^{8}} \] Input:
integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))**2+(((8 *x**2-40*x+48)*exp(4)-2*x**2+10*x-12)*exp(x)+(16*x-40)*exp(4)+10-4*x)*exp( exp(x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x)
Output:
x**2*(4 - 16*exp(4))*exp(-4) + x*(-13 + 52*exp(4))*exp(-4) + ((-x*exp(4) + 4*x*exp(8) - 12*exp(8) + 3*exp(4))*exp(2*exp(x)) + (-2*x**2*exp(4) + 8*x* *2*exp(8) - 40*x*exp(8) + 10*x*exp(4) - 12*exp(4) + 48*exp(8))*exp(exp(x)) )*exp(-8)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx={\left (4 \, x^{2} - 4 \, {\left (4 \, x^{2} - 13 \, x\right )} e^{4} + {\left (x {\left (4 \, e^{4} - 1\right )} - 12 \, e^{4} + 3\right )} e^{\left (2 \, e^{x}\right )} + 2 \, {\left (x^{2} {\left (4 \, e^{4} - 1\right )} - 5 \, x {\left (4 \, e^{4} - 1\right )} + 24 \, e^{4} - 6\right )} e^{\left (e^{x}\right )} - 13 \, x\right )} e^{\left (-4\right )} \] Input:
integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8* x^2-40*x+48)*exp(4)-2*x^2+10*x-12)*exp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp (x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x, algorithm="maxima")
Output:
(4*x^2 - 4*(4*x^2 - 13*x)*e^4 + (x*(4*e^4 - 1) - 12*e^4 + 3)*e^(2*e^x) + 2 *(x^2*(4*e^4 - 1) - 5*x*(4*e^4 - 1) + 24*e^4 - 6)*e^(e^x) - 13*x)*e^(-4)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.72 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx={\left (4 \, x^{2} - 4 \, {\left (4 \, x^{2} - 13 \, x\right )} e^{4} + 2 \, {\left (4 \, x^{2} e^{\left (x + e^{x} + 4\right )} - x^{2} e^{\left (x + e^{x}\right )} - 20 \, x e^{\left (x + e^{x} + 4\right )} + 5 \, x e^{\left (x + e^{x}\right )} + 24 \, e^{\left (x + e^{x} + 4\right )} - 6 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} - x e^{\left (2 \, e^{x}\right )} + 4 \, x e^{\left (2 \, e^{x} + 4\right )} - 13 \, x + 3 \, e^{\left (2 \, e^{x}\right )} - 12 \, e^{\left (2 \, e^{x} + 4\right )}\right )} e^{\left (-4\right )} \] Input:
integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8* x^2-40*x+48)*exp(4)-2*x^2+10*x-12)*exp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp (x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x, algorithm="giac")
Output:
(4*x^2 - 4*(4*x^2 - 13*x)*e^4 + 2*(4*x^2*e^(x + e^x + 4) - x^2*e^(x + e^x) - 20*x*e^(x + e^x + 4) + 5*x*e^(x + e^x) + 24*e^(x + e^x + 4) - 6*e^(x + e^x))*e^(-x) - x*e^(2*e^x) + 4*x*e^(2*e^x + 4) - 13*x + 3*e^(2*e^x) - 12*e ^(2*e^x + 4))*e^(-4)
Time = 4.73 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx={\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{-4}\,\left (8\,{\mathrm {e}}^4-2\right )\,x^2-{\mathrm {e}}^{-4}\,\left (40\,{\mathrm {e}}^4-10\right )\,x+{\mathrm {e}}^{-4}\,\left (48\,{\mathrm {e}}^4-12\right )\right )-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{-4}\,\left (12\,{\mathrm {e}}^4-3\right )-x\,{\mathrm {e}}^{-4}\,\left (4\,{\mathrm {e}}^4-1\right )\right )+x\,{\mathrm {e}}^{-4}\,\left (52\,{\mathrm {e}}^4-13\right )-\frac {x^2\,{\mathrm {e}}^{-4}\,\left (32\,{\mathrm {e}}^4-8\right )}{2} \] Input:
int(exp(-4)*(8*x + exp(2*exp(x))*(4*exp(4) + exp(x)*(exp(4)*(8*x - 24) - 2 *x + 6) - 1) + exp(exp(x))*(exp(x)*(10*x + exp(4)*(8*x^2 - 40*x + 48) - 2* x^2 - 12) - 4*x + exp(4)*(16*x - 40) + 10) - exp(4)*(32*x - 52) - 13),x)
Output:
exp(exp(x))*(exp(-4)*(48*exp(4) - 12) - x*exp(-4)*(40*exp(4) - 10) + x^2*e xp(-4)*(8*exp(4) - 2)) - exp(2*exp(x))*(exp(-4)*(12*exp(4) - 3) - x*exp(-4 )*(4*exp(4) - 1)) + x*exp(-4)*(52*exp(4) - 13) - (x^2*exp(-4)*(32*exp(4) - 8))/2
Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.06 \[ \int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )}{e^4} \, dx=\frac {4 e^{2 e^{x}} e^{4} x -12 e^{2 e^{x}} e^{4}-e^{2 e^{x}} x +3 e^{2 e^{x}}+8 e^{e^{x}} e^{4} x^{2}-40 e^{e^{x}} e^{4} x +48 e^{e^{x}} e^{4}-2 e^{e^{x}} x^{2}+10 e^{e^{x}} x -12 e^{e^{x}}-16 e^{4} x^{2}+52 e^{4} x +4 x^{2}-13 x}{e^{4}} \] Input:
int(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8*x^2-40 *x+48)*exp(4)-2*x^2+10*x-12)*exp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp(x))+( -32*x+52)*exp(4)+8*x-13)/exp(4),x)
Output:
(4*e**(2*e**x)*e**4*x - 12*e**(2*e**x)*e**4 - e**(2*e**x)*x + 3*e**(2*e**x ) + 8*e**(e**x)*e**4*x**2 - 40*e**(e**x)*e**4*x + 48*e**(e**x)*e**4 - 2*e* *(e**x)*x**2 + 10*e**(e**x)*x - 12*e**(e**x) - 16*e**4*x**2 + 52*e**4*x + 4*x**2 - 13*x)/e**4