Integrand size = 99, antiderivative size = 26 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\left (2+e^2\right )^2 \left (-3+x+e^{2+x-e^{-x} x} x\right ) \] Output:
(exp(x-x/exp(x)+2)*x-3+x)*(exp(2)+2)^2
Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\left (2+e^2\right )^2 \left (1+e^{2+x-e^{-x} x}\right ) x \] Input:
Integrate[(E^x*(4 + 4*E^2 + E^4) + E^((-x + E^x*(2 + x))/E^x)*(-4*x + 4*x^ 2 + E^4*(-x + x^2) + E^2*(-4*x + 4*x^2) + E^x*(4 + 4*x + E^4*(1 + x) + E^2 *(4 + 4*x))))/E^x,x]
Output:
(2 + E^2)^2*(1 + E^(2 + x - x/E^x))*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.
\(\displaystyle \int e^{-x} \left (e^{e^{-x} \left (e^x (x+2)-x\right )} \left (4 x^2+e^4 \left (x^2-x\right )+e^2 \left (4 x^2-4 x\right )-4 x+e^x \left (4 x+e^4 (x+1)+e^2 (4 x+4)+4\right )\right )+\left (4+4 e^2+e^4\right ) e^x\right ) \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (2+e^2\right )^2 e^{-e^{-x} x} \left (e^2 (x-1) x+e^{e^{-x} x}+e^{x+2} (x+1)\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (2+e^2\right )^2 \int e^{-e^{-x} x} \left (-e^2 (1-x) x+e^{e^{-x} x}+e^{x+2} (x+1)\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \left (2+e^2\right )^2 \int \left (e^{2-e^{-x} x} (x-1) x+e^{-e^{-x} x+x+2} (x+1)+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \left (2+e^2\right )^2 \left (\int e^{2-e^{-x} x} x^2dx+\int e^{-e^{-x} x+x+2}dx-\int e^{2-e^{-x} x} xdx+\int e^{-e^{-x} x+x+2} xdx+x\right )\) |
Input:
Int[(E^x*(4 + 4*E^2 + E^4) + E^((-x + E^x*(2 + x))/E^x)*(-4*x + 4*x^2 + E^ 4*(-x + x^2) + E^2*(-4*x + 4*x^2) + E^x*(4 + 4*x + E^4*(1 + x) + E^2*(4 + 4*x))))/E^x,x]
Output:
$Aborted
Time = 0.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
method | result | size |
risch | \(4 \,{\mathrm e}^{2} x +x \,{\mathrm e}^{4}+4 x +x \left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+4\right ) {\mathrm e}^{\left ({\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\) | \(42\) |
norman | \(\left (\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+4\right ) x \,{\mathrm e}^{x}+\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+4\right ) x \,{\mathrm e}^{x} {\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\) | \(51\) |
parallelrisch | \({\mathrm e}^{4} {\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}} x +x \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2} {\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}} x +4 \,{\mathrm e}^{2} x +4 \,{\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}} x +4 x\) | \(78\) |
Input:
int(((((1+x)*exp(2)^2+(4+4*x)*exp(2)+4*x+4)*exp(x)+(x^2-x)*exp(2)^2+(4*x^2 -4*x)*exp(2)+4*x^2-4*x)*exp(((2+x)*exp(x)-x)/exp(x))+(exp(2)^2+4*exp(2)+4) *exp(x))/exp(x),x,method=_RETURNVERBOSE)
Output:
4*exp(2)*x+x*exp(4)+4*x+x*(exp(4)+4*exp(2)+4)*exp((exp(x)*x+2*exp(x)-x)*ex p(-x))
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x e^{4} + 4 \, x e^{2} + {\left (x e^{4} + 4 \, x e^{2} + 4 \, x\right )} e^{\left ({\left ({\left (x + 2\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )} + 4 \, x \] Input:
integrate(((((1+x)*exp(2)^2+(4+4*x)*exp(2)+4*x+4)*exp(x)+(x^2-x)*exp(2)^2+ (4*x^2-4*x)*exp(2)+4*x^2-4*x)*exp(((2+x)*exp(x)-x)/exp(x))+(exp(2)^2+4*exp (2)+4)*exp(x))/exp(x),x, algorithm="fricas")
Output:
x*e^4 + 4*x*e^2 + (x*e^4 + 4*x*e^2 + 4*x)*e^(((x + 2)*e^x - x)*e^(-x)) + 4 *x
Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x \left (4 + 4 e^{2} + e^{4}\right ) + \left (4 x + 4 x e^{2} + x e^{4}\right ) e^{\left (- x + \left (x + 2\right ) e^{x}\right ) e^{- x}} \] Input:
integrate(((((1+x)*exp(2)**2+(4+4*x)*exp(2)+4*x+4)*exp(x)+(x**2-x)*exp(2)* *2+(4*x**2-4*x)*exp(2)+4*x**2-4*x)*exp(((2+x)*exp(x)-x)/exp(x))+(exp(2)**2 +4*exp(2)+4)*exp(x))/exp(x),x)
Output:
x*(4 + 4*exp(2) + exp(4)) + (4*x + 4*x*exp(2) + x*exp(4))*exp((-x + (x + 2 )*exp(x))*exp(-x))
\[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\int { {\left ({\left (4 \, x^{2} + {\left (x^{2} - x\right )} e^{4} + 4 \, {\left (x^{2} - x\right )} e^{2} + {\left ({\left (x + 1\right )} e^{4} + 4 \, {\left (x + 1\right )} e^{2} + 4 \, x + 4\right )} e^{x} - 4 \, x\right )} e^{\left ({\left ({\left (x + 2\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )} + {\left (e^{4} + 4 \, e^{2} + 4\right )} e^{x}\right )} e^{\left (-x\right )} \,d x } \] Input:
integrate(((((1+x)*exp(2)^2+(4+4*x)*exp(2)+4*x+4)*exp(x)+(x^2-x)*exp(2)^2+ (4*x^2-4*x)*exp(2)+4*x^2-4*x)*exp(((2+x)*exp(x)-x)/exp(x))+(exp(2)^2+4*exp (2)+4)*exp(x))/exp(x),x, algorithm="maxima")
Output:
x*e^4 + 4*x*e^2 + 4*x + integrate((x^2*(e^6 + 4*e^4 + 4*e^2) - x*(e^6 + 4* e^4 + 4*e^2) + (x*(e^6 + 4*e^4 + 4*e^2) + e^6 + 4*e^4 + 4*e^2)*e^x)*e^(-x* e^(-x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x e^{4} + 4 \, x e^{2} + x e^{\left (-x e^{\left (-x\right )} + x + 6\right )} + 4 \, x e^{\left (-x e^{\left (-x\right )} + x + 4\right )} + 4 \, x e^{\left (-x e^{\left (-x\right )} + x + 2\right )} + 4 \, x \] Input:
integrate(((((1+x)*exp(2)^2+(4+4*x)*exp(2)+4*x+4)*exp(x)+(x^2-x)*exp(2)^2+ (4*x^2-4*x)*exp(2)+4*x^2-4*x)*exp(((2+x)*exp(x)-x)/exp(x))+(exp(2)^2+4*exp (2)+4)*exp(x))/exp(x),x, algorithm="giac")
Output:
x*e^4 + 4*x*e^2 + x*e^(-x*e^(-x) + x + 6) + 4*x*e^(-x*e^(-x) + x + 4) + 4* x*e^(-x*e^(-x) + x + 2) + 4*x
Time = 1.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x\,\left ({\mathrm {e}}^{x-x\,{\mathrm {e}}^{-x}+2}+1\right )\,{\left ({\mathrm {e}}^2+2\right )}^2 \] Input:
int(exp(-x)*(exp(x)*(4*exp(2) + exp(4) + 4) - exp(-exp(-x)*(x - exp(x)*(x + 2)))*(4*x - exp(x)*(4*x + exp(4)*(x + 1) + exp(2)*(4*x + 4) + 4) + exp(2 )*(4*x - 4*x^2) + exp(4)*(x - x^2) - 4*x^2)),x)
Output:
x*(exp(x - x*exp(-x) + 2) + 1)*(exp(2) + 2)^2
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.88 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\frac {x \left (e^{\frac {x}{e^{x}}} e^{4}+4 e^{\frac {x}{e^{x}}} e^{2}+4 e^{\frac {x}{e^{x}}}+e^{x} e^{6}+4 e^{x} e^{4}+4 e^{x} e^{2}\right )}{e^{\frac {x}{e^{x}}}} \] Input:
int(((((1+x)*exp(2)^2+(4+4*x)*exp(2)+4*x+4)*exp(x)+(x^2-x)*exp(2)^2+(4*x^2 -4*x)*exp(2)+4*x^2-4*x)*exp(((2+x)*exp(x)-x)/exp(x))+(exp(2)^2+4*exp(2)+4) *exp(x))/exp(x),x)
Output:
(x*(e**(x/e**x)*e**4 + 4*e**(x/e**x)*e**2 + 4*e**(x/e**x) + e**x*e**6 + 4* e**x*e**4 + 4*e**x*e**2))/e**(x/e**x)