Integrand size = 95, antiderivative size = 27 \[ \int \frac {1}{16} e^{x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )} \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx=e^{\frac {1}{16} e^x \left (-3+3 e^{-4+x}-x\right ) x (5+x)^2} \]
Time = 5.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {1}{16} e^{x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )} \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx=e^{-\frac {1}{16} e^{-4+x} x (5+x)^2 \left (-3 e^x+e^4 (3+x)\right )} \]
Integrate[(E^(x + (E^x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^2 + 3*x^3)))/16)*(-75 - 185*x - 94*x^2 - 17*x^3 - x^4 + E^(-4 + x)*( 75 + 210*x + 69*x^2 + 6*x^3)))/16,x]
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.
\(\displaystyle \int \frac {1}{16} \left (-x^4-17 x^3-94 x^2+e^{x-4} \left (6 x^3+69 x^2+210 x+75\right )-185 x-75\right ) \exp \left (\frac {1}{16} e^x \left (-x^4-13 x^3-55 x^2+e^{x-4} \left (3 x^3+30 x^2+75 x\right )-75 x\right )+x\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \int -\exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) \left (x^4+17 x^3+94 x^2+185 x-3 e^{x-4} \left (2 x^3+23 x^2+70 x+25\right )+75\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{16} \int \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) \left (x^4+17 x^3+94 x^2+185 x-3 e^{x-4} \left (2 x^3+23 x^2+70 x+25\right )+75\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{16} \int \left (\exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) x^4+17 \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) x^3+94 \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) x^2+185 \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) x+75 \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right )-3 \exp \left (2 x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )-4\right ) \left (2 x^3+23 x^2+70 x+25\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{16} \left (-75 \int \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right )dx+75 \int \exp \left (2 x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )-4\right )dx-185 \int \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) xdx+210 \int \exp \left (2 x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )-4\right ) xdx-94 \int \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) x^2dx+69 \int \exp \left (2 x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )-4\right ) x^2dx-17 \int \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) x^3dx+6 \int \exp \left (2 x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )-4\right ) x^3dx-\int \exp \left (x-\frac {1}{16} e^x \left (x^4+13 x^3+55 x^2+75 x-3 e^{x-4} \left (x^3+10 x^2+25 x\right )\right )\right ) x^4dx\right )\) |
Int[(E^(x + (E^x*(-75*x - 55*x^2 - 13*x^3 - x^4 + E^(-4 + x)*(75*x + 30*x^ 2 + 3*x^3)))/16)*(-75 - 185*x - 94*x^2 - 17*x^3 - x^4 + E^(-4 + x)*(75 + 2 10*x + 69*x^2 + 6*x^3)))/16,x]
3.5.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
risch | \({\mathrm e}^{-\frac {x \left (5+x \right )^{2} \left (-3 \,{\mathrm e}^{x -4}+3+x \right ) {\mathrm e}^{x}}{16}}\) | \(21\) |
norman | \({\mathrm e}^{\frac {\left (\left (3 x^{3}+30 x^{2}+75 x \right ) {\mathrm e}^{x} {\mathrm e}^{-4}-x^{4}-13 x^{3}-55 x^{2}-75 x \right ) {\mathrm e}^{x}}{16}}\) | \(44\) |
parallelrisch | \({\mathrm e}^{\frac {\left (\left (3 x^{3}+30 x^{2}+75 x \right ) {\mathrm e}^{x -4}-x^{4}-13 x^{3}-55 x^{2}-75 x \right ) {\mathrm e}^{x}}{16}}\) | \(44\) |
int(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-75)*exp (x)*exp(1/16*((3*x^3+30*x^2+75*x)*exp(x-4)-x^4-13*x^3-55*x^2-75*x)*exp(x)) ,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {1}{16} e^{x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )} \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx=e^{\left (\frac {1}{16} \, {\left (16 \, x e^{4} + 3 \, {\left (x^{3} + 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{4} + 13 \, x^{3} + 55 \, x^{2} + 75 \, x\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )} - x\right )} \]
integrate(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-7 5)*exp(x)*exp(1/16*((3*x^3+30*x^2+75*x)*exp(x-4)-x^4-13*x^3-55*x^2-75*x)*e xp(x)),x, algorithm=\
e^(1/16*(16*x*e^4 + 3*(x^3 + 10*x^2 + 25*x)*e^(2*x) - (x^4 + 13*x^3 + 55*x ^2 + 75*x)*e^(x + 4))*e^(-4) - x)
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {1}{16} e^{x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )} \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx=e^{\left (- \frac {x^{4}}{16} - \frac {13 x^{3}}{16} - \frac {55 x^{2}}{16} - \frac {75 x}{16} + \frac {\left (3 x^{3} + 30 x^{2} + 75 x\right ) e^{x}}{16 e^{4}}\right ) e^{x}} \]
integrate(1/16*((6*x**3+69*x**2+210*x+75)*exp(x-4)-x**4-17*x**3-94*x**2-18 5*x-75)*exp(x)*exp(1/16*((3*x**3+30*x**2+75*x)*exp(x-4)-x**4-13*x**3-55*x* *2-75*x)*exp(x)),x)
exp((-x**4/16 - 13*x**3/16 - 55*x**2/16 - 75*x/16 + (3*x**3 + 30*x**2 + 75 *x)*exp(-4)*exp(x)/16)*exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (20) = 40\).
Time = 0.52 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19 \[ \int \frac {1}{16} e^{x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )} \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx=e^{\left (-\frac {1}{16} \, x^{4} e^{x} + \frac {3}{16} \, x^{3} e^{\left (2 \, x - 4\right )} - \frac {13}{16} \, x^{3} e^{x} + \frac {15}{8} \, x^{2} e^{\left (2 \, x - 4\right )} - \frac {55}{16} \, x^{2} e^{x} + \frac {75}{16} \, x e^{\left (2 \, x - 4\right )} - \frac {75}{16} \, x e^{x}\right )} \]
integrate(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-7 5)*exp(x)*exp(1/16*((3*x^3+30*x^2+75*x)*exp(x-4)-x^4-13*x^3-55*x^2-75*x)*e xp(x)),x, algorithm=\
e^(-1/16*x^4*e^x + 3/16*x^3*e^(2*x - 4) - 13/16*x^3*e^x + 15/8*x^2*e^(2*x - 4) - 55/16*x^2*e^x + 75/16*x*e^(2*x - 4) - 75/16*x*e^x)
\[ \int \frac {1}{16} e^{x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )} \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx=\int { -\frac {1}{16} \, {\left (x^{4} + 17 \, x^{3} + 94 \, x^{2} - 3 \, {\left (2 \, x^{3} + 23 \, x^{2} + 70 \, x + 25\right )} e^{\left (x - 4\right )} + 185 \, x + 75\right )} e^{\left (-\frac {1}{16} \, {\left (x^{4} + 13 \, x^{3} + 55 \, x^{2} - 3 \, {\left (x^{3} + 10 \, x^{2} + 25 \, x\right )} e^{\left (x - 4\right )} + 75 \, x\right )} e^{x} + x\right )} \,d x } \]
integrate(1/16*((6*x^3+69*x^2+210*x+75)*exp(x-4)-x^4-17*x^3-94*x^2-185*x-7 5)*exp(x)*exp(1/16*((3*x^3+30*x^2+75*x)*exp(x-4)-x^4-13*x^3-55*x^2-75*x)*e xp(x)),x, algorithm=\
integrate(-1/16*(x^4 + 17*x^3 + 94*x^2 - 3*(2*x^3 + 23*x^2 + 70*x + 25)*e^ (x - 4) + 185*x + 75)*e^(-1/16*(x^4 + 13*x^3 + 55*x^2 - 3*(x^3 + 10*x^2 + 25*x)*e^(x - 4) + 75*x)*e^x + x), x)
Time = 8.96 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {1}{16} e^{x+\frac {1}{16} e^x \left (-75 x-55 x^2-13 x^3-x^4+e^{-4+x} \left (75 x+30 x^2+3 x^3\right )\right )} \left (-75-185 x-94 x^2-17 x^3-x^4+e^{-4+x} \left (75+210 x+69 x^2+6 x^3\right )\right ) \, dx={\mathrm {e}}^{-\frac {75\,x\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{\frac {3\,x^3\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4}}{16}}\,{\mathrm {e}}^{\frac {15\,x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4}}{8}}\,{\mathrm {e}}^{-\frac {x^4\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{-\frac {13\,x^3\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{-\frac {55\,x^2\,{\mathrm {e}}^x}{16}}\,{\mathrm {e}}^{\frac {75\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4}}{16}} \]
int(-(exp(-(exp(x)*(75*x - exp(x - 4)*(75*x + 30*x^2 + 3*x^3) + 55*x^2 + 1 3*x^3 + x^4))/16)*exp(x)*(185*x - exp(x - 4)*(210*x + 69*x^2 + 6*x^3 + 75) + 94*x^2 + 17*x^3 + x^4 + 75))/16,x)