Integrand size = 284, antiderivative size = 29 \[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=\left (e^{e^{2 x}}+x+e^x \left (5+x-\log \left (\frac {3 (-4+x)}{x}\right )\right )\right )^2 \] Output:
((5+x-ln(3*(-4+x)/x))*exp(x)+exp(exp(x)^2)+x)^2
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=\left (e^{e^{2 x}}+x+e^x (5+x)-e^x \log \left (\frac {3 (-4+x)}{x}\right )\right )^2 \] Input:
Integrate[(-8*x^2 + 2*x^3 + E^(2*E^(2*x) + 2*x)*(-16*x + 4*x^2) + E^x*(-48 *x - 46*x^2 + 6*x^3 + 2*x^4) + E^(2*x)*(-40 - 248*x - 28*x^2 + 14*x^3 + 2* x^4) + (E^(2*x)*(8 + 88*x - 6*x^2 - 4*x^3) + E^x*(8*x + 6*x^2 - 2*x^3))*Lo g[(-12 + 3*x)/x] + E^(2*x)*(-8*x + 2*x^2)*Log[(-12 + 3*x)/x]^2 + E^E^(2*x) *(-8*x + 2*x^2 + E^x*(-8 - 48*x + 4*x^2 + 2*x^3) + E^(2*x)*(-16*x^2 + 4*x^ 3) + E^(3*x)*(-80*x + 4*x^2 + 4*x^3) + (E^(3*x)*(16*x - 4*x^2) + E^x*(8*x - 2*x^2))*Log[(-12 + 3*x)/x]))/(-4*x + x^2),x]
Output:
(E^E^(2*x) + x + E^x*(5 + x) - E^x*Log[(3*(-4 + x))/x])^2
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.
| \(\displaystyle \int \frac {2 x^3-8 x^2+e^{2 x+2 e^{2 x}} \left (4 x^2-16 x\right )+e^{2 x} \left (2 x^2-8 x\right ) \log ^2\left (\frac {3 x-12}{x}\right )+\left (e^{2 x} \left (-4 x^3-6 x^2+88 x+8\right )+e^x \left (-2 x^3+6 x^2+8 x\right )\right ) \log \left (\frac {3 x-12}{x}\right )+e^{e^{2 x}} \left (2 x^2+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {3 x-12}{x}\right )+e^x \left (2 x^3+4 x^2-48 x-8\right )+e^{2 x} \left (4 x^3-16 x^2\right )+e^{3 x} \left (4 x^3+4 x^2-80 x\right )-8 x\right )+e^x \left (2 x^4+6 x^3-46 x^2-48 x\right )+e^{2 x} \left (2 x^4+14 x^3-28 x^2-248 x-40\right )}{x^2-4 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^3-8 x^2+e^{2 x+2 e^{2 x}} \left (4 x^2-16 x\right )+e^{2 x} \left (2 x^2-8 x\right ) \log ^2\left (\frac {3 x-12}{x}\right )+\left (e^{2 x} \left (-4 x^3-6 x^2+88 x+8\right )+e^x \left (-2 x^3+6 x^2+8 x\right )\right ) \log \left (\frac {3 x-12}{x}\right )+e^{e^{2 x}} \left (2 x^2+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {3 x-12}{x}\right )+e^x \left (2 x^3+4 x^2-48 x-8\right )+e^{2 x} \left (4 x^3-16 x^2\right )+e^{3 x} \left (4 x^3+4 x^2-80 x\right )-8 x\right )+e^x \left (2 x^4+6 x^3-46 x^2-48 x\right )+e^{2 x} \left (2 x^4+14 x^3-28 x^2-248 x-40\right )}{(x-4) x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^{2 x} \left (x^4+2 e^{e^{2 x}} x^3+7 x^3-2 x^3 \log \left (\frac {3 (x-4)}{x}\right )-8 e^{e^{2 x}} x^2+2 e^{2 e^{2 x}} x^2-14 x^2+x^2 \log ^2\left (\frac {3 (x-4)}{x}\right )-3 x^2 \log \left (\frac {3 (x-4)}{x}\right )-8 e^{2 e^{2 x}} x-124 x-4 x \log ^2\left (\frac {3 (x-4)}{x}\right )+44 x \log \left (\frac {3 (x-4)}{x}\right )+4 \log \left (\frac {3 (x-4)}{x}\right )-20\right )}{(x-4) x}+\frac {2 e^x \left (x^4+e^{e^{2 x}} x^3+3 x^3-x^3 \log \left (\frac {3 (x-4)}{x}\right )+2 e^{e^{2 x}} x^2-23 x^2-e^{e^{2 x}} x^2 \log \left (\frac {3 (x-4)}{x}\right )+3 x^2 \log \left (\frac {3 (x-4)}{x}\right )-24 e^{e^{2 x}} x-24 x-4 e^{e^{2 x}}+4 e^{e^{2 x}} x \log \left (\frac {3 (x-4)}{x}\right )+4 x \log \left (\frac {3 (x-4)}{x}\right )\right )}{(x-4) x}+2 \left (x+e^{e^{2 x}}\right )+4 e^{3 x+e^{2 x}} \left (x-\log \left (3-\frac {12}{x}\right )+5\right )\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 e^{2 x} \left (x^4+2 e^{e^{2 x}} x^3+7 x^3-2 x^3 \log \left (\frac {3 (x-4)}{x}\right )-8 e^{e^{2 x}} x^2+2 e^{2 e^{2 x}} x^2-14 x^2+x^2 \log ^2\left (\frac {3 (x-4)}{x}\right )-3 x^2 \log \left (\frac {3 (x-4)}{x}\right )-8 e^{2 e^{2 x}} x-124 x-4 x \log ^2\left (\frac {3 (x-4)}{x}\right )+44 x \log \left (\frac {3 (x-4)}{x}\right )+4 \log \left (\frac {3 (x-4)}{x}\right )-20\right )}{(x-4) x}+\frac {2 e^x \left (x^4+e^{e^{2 x}} x^3+3 x^3-x^3 \log \left (\frac {3 (x-4)}{x}\right )+2 e^{e^{2 x}} x^2-23 x^2-e^{e^{2 x}} x^2 \log \left (\frac {3 (x-4)}{x}\right )+3 x^2 \log \left (\frac {3 (x-4)}{x}\right )-24 e^{e^{2 x}} x-24 x-4 e^{e^{2 x}}+4 e^{e^{2 x}} x \log \left (\frac {3 (x-4)}{x}\right )+4 x \log \left (\frac {3 (x-4)}{x}\right )\right )}{(x-4) x}+2 \left (x+e^{e^{2 x}}\right )+4 e^{3 x+e^{2 x}} \left (x-\log \left (3-\frac {12}{x}\right )+5\right )\right )dx\) |
Input:
Int[(-8*x^2 + 2*x^3 + E^(2*E^(2*x) + 2*x)*(-16*x + 4*x^2) + E^x*(-48*x - 4 6*x^2 + 6*x^3 + 2*x^4) + E^(2*x)*(-40 - 248*x - 28*x^2 + 14*x^3 + 2*x^4) + (E^(2*x)*(8 + 88*x - 6*x^2 - 4*x^3) + E^x*(8*x + 6*x^2 - 2*x^3))*Log[(-12 + 3*x)/x] + E^(2*x)*(-8*x + 2*x^2)*Log[(-12 + 3*x)/x]^2 + E^E^(2*x)*(-8*x + 2*x^2 + E^x*(-8 - 48*x + 4*x^2 + 2*x^3) + E^(2*x)*(-16*x^2 + 4*x^3) + E ^(3*x)*(-80*x + 4*x^2 + 4*x^3) + (E^(3*x)*(16*x - 4*x^2) + E^x*(8*x - 2*x^ 2))*Log[(-12 + 3*x)/x]))/(-4*x + x^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(26)=52\).
Time = 1.42 (sec) , antiderivative size = 151, normalized size of antiderivative = 5.21
| method | result | size |
| parallelrisch | \({\mathrm e}^{2 x} x^{2}-2 \ln \left (\frac {3 x -12}{x}\right ) {\mathrm e}^{2 x} x +{\mathrm e}^{2 x} \ln \left (\frac {3 x -12}{x}\right )^{2}+2 \,{\mathrm e}^{x} x^{2}+10 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{{\mathrm e}^{2 x}} {\mathrm e}^{x} x -2 \ln \left (\frac {3 x -12}{x}\right ) {\mathrm e}^{x} x -10 \ln \left (\frac {3 x -12}{x}\right ) {\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{2 x}} \ln \left (\frac {3 x -12}{x}\right )+x^{2}+10 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{{\mathrm e}^{2 x}} x +25 \,{\mathrm e}^{2 x}+10 \,{\mathrm e}^{{\mathrm e}^{2 x}} {\mathrm e}^{x}+{\mathrm e}^{2 \,{\mathrm e}^{2 x}}\) | \(151\) |
| risch | \(\text {Expression too large to display}\) | \(1246\) |
Input:
int(((4*x^2-16*x)*exp(x)^2*exp(exp(x)^2)^2+(((-4*x^2+16*x)*exp(x)^3+(-2*x^ 2+8*x)*exp(x))*ln((3*x-12)/x)+(4*x^3+4*x^2-80*x)*exp(x)^3+(4*x^3-16*x^2)*e xp(x)^2+(2*x^3+4*x^2-48*x-8)*exp(x)+2*x^2-8*x)*exp(exp(x)^2)+(2*x^2-8*x)*e xp(x)^2*ln((3*x-12)/x)^2+((-4*x^3-6*x^2+88*x+8)*exp(x)^2+(-2*x^3+6*x^2+8*x )*exp(x))*ln((3*x-12)/x)+(2*x^4+14*x^3-28*x^2-248*x-40)*exp(x)^2+(2*x^4+6* x^3-46*x^2-48*x)*exp(x)+2*x^3-8*x^2)/(x^2-4*x),x,method=_RETURNVERBOSE)
Output:
exp(x)^2*x^2-2*x*exp(x)^2*ln(3*(x-4)/x)+exp(x)^2*ln(3*(x-4)/x)^2+2*exp(x)* x^2+10*x*exp(x)^2+2*exp(exp(x)^2)*exp(x)*x-2*exp(x)*ln(3*(x-4)/x)*x-10*ln( 3*(x-4)/x)*exp(x)^2-2*exp(x)*exp(exp(x)^2)*ln(3*(x-4)/x)+x^2+10*exp(x)*x+2 *x*exp(exp(x)^2)+25*exp(x)^2+10*exp(exp(x)^2)*exp(x)+exp(exp(x)^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.55 \[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=e^{\left (2 \, x\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right )^{2} + x^{2} + {\left (x^{2} + 10 \, x + 25\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} + 5 \, x\right )} e^{x} + 2 \, {\left ({\left (x + 5\right )} e^{x} - e^{x} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right ) + x\right )} e^{\left (e^{\left (2 \, x\right )}\right )} - 2 \, {\left ({\left (x + 5\right )} e^{\left (2 \, x\right )} + x e^{x}\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right ) + e^{\left (2 \, e^{\left (2 \, x\right )}\right )} \] Input:
integrate(((4*x^2-16*x)*exp(x)^2*exp(exp(x)^2)^2+(((-4*x^2+16*x)*exp(x)^3+ (-2*x^2+8*x)*exp(x))*log((3*x-12)/x)+(4*x^3+4*x^2-80*x)*exp(x)^3+(4*x^3-16 *x^2)*exp(x)^2+(2*x^3+4*x^2-48*x-8)*exp(x)+2*x^2-8*x)*exp(exp(x)^2)+(2*x^2 -8*x)*exp(x)^2*log((3*x-12)/x)^2+((-4*x^3-6*x^2+88*x+8)*exp(x)^2+(-2*x^3+6 *x^2+8*x)*exp(x))*log((3*x-12)/x)+(2*x^4+14*x^3-28*x^2-248*x-40)*exp(x)^2+ (2*x^4+6*x^3-46*x^2-48*x)*exp(x)+2*x^3-8*x^2)/(x^2-4*x),x, algorithm="fric as")
Output:
e^(2*x)*log(3*(x - 4)/x)^2 + x^2 + (x^2 + 10*x + 25)*e^(2*x) + 2*(x^2 + 5* x)*e^x + 2*((x + 5)*e^x - e^x*log(3*(x - 4)/x) + x)*e^(e^(2*x)) - 2*((x + 5)*e^(2*x) + x*e^x)*log(3*(x - 4)/x) + e^(2*e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (24) = 48\).
Time = 48.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.00 \[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=x^{2} + \left (2 x^{2} - 2 x \log {\left (\frac {3 x - 12}{x} \right )} + 10 x\right ) e^{x} + \left (2 x e^{x} + 2 x - 2 e^{x} \log {\left (\frac {3 x - 12}{x} \right )} + 10 e^{x}\right ) e^{e^{2 x}} + \left (x^{2} - 2 x \log {\left (\frac {3 x - 12}{x} \right )} + 10 x + \log {\left (\frac {3 x - 12}{x} \right )}^{2} - 10 \log {\left (\frac {3 x - 12}{x} \right )} + 25\right ) e^{2 x} + e^{2 e^{2 x}} \] Input:
integrate(((4*x**2-16*x)*exp(x)**2*exp(exp(x)**2)**2+(((-4*x**2+16*x)*exp( x)**3+(-2*x**2+8*x)*exp(x))*ln((3*x-12)/x)+(4*x**3+4*x**2-80*x)*exp(x)**3+ (4*x**3-16*x**2)*exp(x)**2+(2*x**3+4*x**2-48*x-8)*exp(x)+2*x**2-8*x)*exp(e xp(x)**2)+(2*x**2-8*x)*exp(x)**2*ln((3*x-12)/x)**2+((-4*x**3-6*x**2+88*x+8 )*exp(x)**2+(-2*x**3+6*x**2+8*x)*exp(x))*ln((3*x-12)/x)+(2*x**4+14*x**3-28 *x**2-248*x-40)*exp(x)**2+(2*x**4+6*x**3-46*x**2-48*x)*exp(x)+2*x**3-8*x** 2)/(x**2-4*x),x)
Output:
x**2 + (2*x**2 - 2*x*log((3*x - 12)/x) + 10*x)*exp(x) + (2*x*exp(x) + 2*x - 2*exp(x)*log((3*x - 12)/x) + 10*exp(x))*exp(exp(2*x)) + (x**2 - 2*x*log( (3*x - 12)/x) + 10*x + log((3*x - 12)/x)**2 - 10*log((3*x - 12)/x) + 25)*e xp(2*x) + exp(2*exp(2*x))
\[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=\int { \frac {2 \, {\left ({\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right )^{2} + x^{3} - 4 \, x^{2} + {\left (x^{4} + 7 \, x^{3} - 14 \, x^{2} - 124 \, x - 20\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} + {\left (x^{4} + 3 \, x^{3} - 23 \, x^{2} - 24 \, x\right )} e^{x} + {\left (x^{2} + 2 \, {\left (x^{3} + x^{2} - 20 \, x\right )} e^{\left (3 \, x\right )} + 2 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + 2 \, x^{2} - 24 \, x - 4\right )} e^{x} - {\left (2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (3 \, x\right )} + {\left (x^{2} - 4 \, x\right )} e^{x}\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right ) - 4 \, x\right )} e^{\left (e^{\left (2 \, x\right )}\right )} - {\left ({\left (2 \, x^{3} + 3 \, x^{2} - 44 \, x - 4\right )} e^{\left (2 \, x\right )} + {\left (x^{3} - 3 \, x^{2} - 4 \, x\right )} e^{x}\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right )\right )}}{x^{2} - 4 \, x} \,d x } \] Input:
integrate(((4*x^2-16*x)*exp(x)^2*exp(exp(x)^2)^2+(((-4*x^2+16*x)*exp(x)^3+ (-2*x^2+8*x)*exp(x))*log((3*x-12)/x)+(4*x^3+4*x^2-80*x)*exp(x)^3+(4*x^3-16 *x^2)*exp(x)^2+(2*x^3+4*x^2-48*x-8)*exp(x)+2*x^2-8*x)*exp(exp(x)^2)+(2*x^2 -8*x)*exp(x)^2*log((3*x-12)/x)^2+((-4*x^3-6*x^2+88*x+8)*exp(x)^2+(-2*x^3+6 *x^2+8*x)*exp(x))*log((3*x-12)/x)+(2*x^4+14*x^3-28*x^2-248*x-40)*exp(x)^2+ (2*x^4+6*x^3-46*x^2-48*x)*exp(x)+2*x^3-8*x^2)/(x^2-4*x),x, algorithm="maxi ma")
Output:
e^(2*x)*log(x - 4)^2 + 2*x*e^x*log(x) + x^2 + (x^2 - 2*x*(log(3) - 5) + lo g(3)^2 + 2*(x - log(3) + 5)*log(x) + log(x)^2 - 10*log(3) + 25)*e^(2*x) + 2*((x - log(3) + log(x) + 5)*e^x - e^x*log(x - 4) + x)*e^(e^(2*x)) + 48*e^ 4*exp_integral_e(1, -x + 4) - 2*((x - log(3) + log(x) + 5)*e^(2*x) + x*e^x )*log(x - 4) + e^(2*e^(2*x)) + 2*integrate((x^3 - x^2*(log(3) - 3) + x*(3* log(3) - 23) + 4*log(3) + 4)*e^x/(x - 4), x)
\[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=\int { \frac {2 \, {\left ({\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right )^{2} + x^{3} - 4 \, x^{2} + {\left (x^{4} + 7 \, x^{3} - 14 \, x^{2} - 124 \, x - 20\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} + {\left (x^{4} + 3 \, x^{3} - 23 \, x^{2} - 24 \, x\right )} e^{x} + {\left (x^{2} + 2 \, {\left (x^{3} + x^{2} - 20 \, x\right )} e^{\left (3 \, x\right )} + 2 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + 2 \, x^{2} - 24 \, x - 4\right )} e^{x} - {\left (2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (3 \, x\right )} + {\left (x^{2} - 4 \, x\right )} e^{x}\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right ) - 4 \, x\right )} e^{\left (e^{\left (2 \, x\right )}\right )} - {\left ({\left (2 \, x^{3} + 3 \, x^{2} - 44 \, x - 4\right )} e^{\left (2 \, x\right )} + {\left (x^{3} - 3 \, x^{2} - 4 \, x\right )} e^{x}\right )} \log \left (\frac {3 \, {\left (x - 4\right )}}{x}\right )\right )}}{x^{2} - 4 \, x} \,d x } \] Input:
integrate(((4*x^2-16*x)*exp(x)^2*exp(exp(x)^2)^2+(((-4*x^2+16*x)*exp(x)^3+ (-2*x^2+8*x)*exp(x))*log((3*x-12)/x)+(4*x^3+4*x^2-80*x)*exp(x)^3+(4*x^3-16 *x^2)*exp(x)^2+(2*x^3+4*x^2-48*x-8)*exp(x)+2*x^2-8*x)*exp(exp(x)^2)+(2*x^2 -8*x)*exp(x)^2*log((3*x-12)/x)^2+((-4*x^3-6*x^2+88*x+8)*exp(x)^2+(-2*x^3+6 *x^2+8*x)*exp(x))*log((3*x-12)/x)+(2*x^4+14*x^3-28*x^2-248*x-40)*exp(x)^2+ (2*x^4+6*x^3-46*x^2-48*x)*exp(x)+2*x^3-8*x^2)/(x^2-4*x),x, algorithm="giac ")
Output:
integrate(2*((x^2 - 4*x)*e^(2*x)*log(3*(x - 4)/x)^2 + x^3 - 4*x^2 + (x^4 + 7*x^3 - 14*x^2 - 124*x - 20)*e^(2*x) + 2*(x^2 - 4*x)*e^(2*x + 2*e^(2*x)) + (x^4 + 3*x^3 - 23*x^2 - 24*x)*e^x + (x^2 + 2*(x^3 + x^2 - 20*x)*e^(3*x) + 2*(x^3 - 4*x^2)*e^(2*x) + (x^3 + 2*x^2 - 24*x - 4)*e^x - (2*(x^2 - 4*x)* e^(3*x) + (x^2 - 4*x)*e^x)*log(3*(x - 4)/x) - 4*x)*e^(e^(2*x)) - ((2*x^3 + 3*x^2 - 44*x - 4)*e^(2*x) + (x^3 - 3*x^2 - 4*x)*e^x)*log(3*(x - 4)/x))/(x ^2 - 4*x), x)
Timed out. \[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=\int \frac {{\mathrm {e}}^x\,\left (-2\,x^4-6\,x^3+46\,x^2+48\,x\right )+{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\left (8\,x-{\mathrm {e}}^{3\,x}\,\left (4\,x^3+4\,x^2-80\,x\right )-\ln \left (\frac {3\,x-12}{x}\right )\,\left ({\mathrm {e}}^{3\,x}\,\left (16\,x-4\,x^2\right )+{\mathrm {e}}^x\,\left (8\,x-2\,x^2\right )\right )+{\mathrm {e}}^{2\,x}\,\left (16\,x^2-4\,x^3\right )-2\,x^2+{\mathrm {e}}^x\,\left (-2\,x^3-4\,x^2+48\,x+8\right )\right )+{\mathrm {e}}^{2\,x}\,\left (-2\,x^4-14\,x^3+28\,x^2+248\,x+40\right )-\ln \left (\frac {3\,x-12}{x}\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (-4\,x^3-6\,x^2+88\,x+8\right )+{\mathrm {e}}^x\,\left (-2\,x^3+6\,x^2+8\,x\right )\right )+8\,x^2-2\,x^3+{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{2\,x}\,\left (16\,x-4\,x^2\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (\frac {3\,x-12}{x}\right )}^2\,\left (8\,x-2\,x^2\right )}{4\,x-x^2} \,d x \] Input:
int((exp(x)*(48*x + 46*x^2 - 6*x^3 - 2*x^4) + exp(exp(2*x))*(8*x - exp(3*x )*(4*x^2 - 80*x + 4*x^3) - log((3*x - 12)/x)*(exp(3*x)*(16*x - 4*x^2) + ex p(x)*(8*x - 2*x^2)) + exp(2*x)*(16*x^2 - 4*x^3) - 2*x^2 + exp(x)*(48*x - 4 *x^2 - 2*x^3 + 8)) + exp(2*x)*(248*x + 28*x^2 - 14*x^3 - 2*x^4 + 40) - log ((3*x - 12)/x)*(exp(2*x)*(88*x - 6*x^2 - 4*x^3 + 8) + exp(x)*(8*x + 6*x^2 - 2*x^3)) + 8*x^2 - 2*x^3 + exp(2*exp(2*x))*exp(2*x)*(16*x - 4*x^2) + exp( 2*x)*log((3*x - 12)/x)^2*(8*x - 2*x^2))/(4*x - x^2),x)
Output:
int((exp(x)*(48*x + 46*x^2 - 6*x^3 - 2*x^4) + exp(exp(2*x))*(8*x - exp(3*x )*(4*x^2 - 80*x + 4*x^3) - log((3*x - 12)/x)*(exp(3*x)*(16*x - 4*x^2) + ex p(x)*(8*x - 2*x^2)) + exp(2*x)*(16*x^2 - 4*x^3) - 2*x^2 + exp(x)*(48*x - 4 *x^2 - 2*x^3 + 8)) + exp(2*x)*(248*x + 28*x^2 - 14*x^3 - 2*x^4 + 40) - log ((3*x - 12)/x)*(exp(2*x)*(88*x - 6*x^2 - 4*x^3 + 8) + exp(x)*(8*x + 6*x^2 - 2*x^3)) + 8*x^2 - 2*x^3 + exp(2*exp(2*x))*exp(2*x)*(16*x - 4*x^2) + exp( 2*x)*log((3*x - 12)/x)^2*(8*x - 2*x^2))/(4*x - x^2), x)
Time = 0.26 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.00 \[ \int \frac {-8 x^2+2 x^3+e^{2 e^{2 x}+2 x} \left (-16 x+4 x^2\right )+e^x \left (-48 x-46 x^2+6 x^3+2 x^4\right )+e^{2 x} \left (-40-248 x-28 x^2+14 x^3+2 x^4\right )+\left (e^{2 x} \left (8+88 x-6 x^2-4 x^3\right )+e^x \left (8 x+6 x^2-2 x^3\right )\right ) \log \left (\frac {-12+3 x}{x}\right )+e^{2 x} \left (-8 x+2 x^2\right ) \log ^2\left (\frac {-12+3 x}{x}\right )+e^{e^{2 x}} \left (-8 x+2 x^2+e^x \left (-8-48 x+4 x^2+2 x^3\right )+e^{2 x} \left (-16 x^2+4 x^3\right )+e^{3 x} \left (-80 x+4 x^2+4 x^3\right )+\left (e^{3 x} \left (16 x-4 x^2\right )+e^x \left (8 x-2 x^2\right )\right ) \log \left (\frac {-12+3 x}{x}\right )\right )}{-4 x+x^2} \, dx=e^{2 e^{2 x}}-2 e^{e^{2 x}+x} \mathrm {log}\left (\frac {3 x -12}{x}\right )+2 e^{e^{2 x}+x} x +10 e^{e^{2 x}+x}+2 e^{e^{2 x}} x +e^{2 x} \mathrm {log}\left (\frac {3 x -12}{x}\right )^{2}-2 e^{2 x} \mathrm {log}\left (\frac {3 x -12}{x}\right ) x -10 e^{2 x} \mathrm {log}\left (\frac {3 x -12}{x}\right )+e^{2 x} x^{2}+10 e^{2 x} x +25 e^{2 x}-2 e^{x} \mathrm {log}\left (\frac {3 x -12}{x}\right ) x +2 e^{x} x^{2}+10 e^{x} x +x^{2} \] Input:
int(((4*x^2-16*x)*exp(x)^2*exp(exp(x)^2)^2+(((-4*x^2+16*x)*exp(x)^3+(-2*x^ 2+8*x)*exp(x))*log((3*x-12)/x)+(4*x^3+4*x^2-80*x)*exp(x)^3+(4*x^3-16*x^2)* exp(x)^2+(2*x^3+4*x^2-48*x-8)*exp(x)+2*x^2-8*x)*exp(exp(x)^2)+(2*x^2-8*x)* exp(x)^2*log((3*x-12)/x)^2+((-4*x^3-6*x^2+88*x+8)*exp(x)^2+(-2*x^3+6*x^2+8 *x)*exp(x))*log((3*x-12)/x)+(2*x^4+14*x^3-28*x^2-248*x-40)*exp(x)^2+(2*x^4 +6*x^3-46*x^2-48*x)*exp(x)+2*x^3-8*x^2)/(x^2-4*x),x)
Output:
e**(2*e**(2*x)) - 2*e**(e**(2*x) + x)*log((3*x - 12)/x) + 2*e**(e**(2*x) + x)*x + 10*e**(e**(2*x) + x) + 2*e**(e**(2*x))*x + e**(2*x)*log((3*x - 12) /x)**2 - 2*e**(2*x)*log((3*x - 12)/x)*x - 10*e**(2*x)*log((3*x - 12)/x) + e**(2*x)*x**2 + 10*e**(2*x)*x + 25*e**(2*x) - 2*e**x*log((3*x - 12)/x)*x + 2*e**x*x**2 + 10*e**x*x + x**2