Integrand size = 87, antiderivative size = 28 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx=e^{\frac {-5+e^{e^{-4+x}} (5-x)}{4-\frac {e}{3}+x}} \] Output:
exp((exp(exp(-4+x))*(5-x)-5)/(4+x-1/3*exp(1)))
Time = 5.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx=e^{\frac {3 \left (5+e^{e^{-4+x}} (-5+x)\right )}{e-3 (4+x)}} \] Input:
Integrate[(E^((15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x))*(45 + E^E^( -4 + x)*(-81 + 3*E + E^(-4 + x)*(180 + 9*x - 9*x^2 + E*(-15 + 3*x)))))/(14 4 + E^2 + E*(-24 - 6*x) + 72*x + 9*x^2),x]
Output:
E^((3*(5 + E^E^(-4 + x)*(-5 + x)))/(E - 3*(4 + 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.
\(\displaystyle \int \frac {e^{\frac {e^{e^{x-4}} (3 x-15)+15}{-3 x+e-12}} \left (e^{e^{x-4}} \left (e^{x-4} \left (-9 x^2+9 x+e (3 x-15)+180\right )+3 e-81\right )+45\right )}{9 x^2+72 x+e (-6 x-24)+e^2+144} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {e^{e^{x-4}} (3 x-15)+15}{-3 x+e-12}} \left (e^{e^{x-4}} \left (e^{x-4} \left (-9 x^2+9 x+e (3 x-15)+180\right )+3 e-81\right )+45\right )}{9 x^2+6 (12-e) x+(e-12)^2}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 36 \int \frac {e^{-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}} \left (15-e^{e^{x-4}} \left (-e^{x-4} \left (-3 x^2+3 x-e (5-x)+60\right )-e+27\right )\right )}{12 (3 x-e+12)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int \frac {e^{-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}} \left (15-e^{e^{x-4}} \left (-e^{x-4} \left (-3 x^2+3 x-e (5-x)+60\right )-e+27\right )\right )}{(3 x-e+12)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 3 \int \left (\frac {e^{-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}} \left (15-27 \left (1-\frac {e}{27}\right ) e^{e^{x-4}}\right )}{(3 x-e+12)^2}+\frac {\exp \left (-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}+e^{x-4}+x-4\right ) (x-5)}{-3 x+e-12}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-\frac {1}{3} \int \exp \left (-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}+e^{x-4}+x-4\right )dx-(27-e) \int \frac {\exp \left (e^{x-4}-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}\right )}{(-3 x+e-12)^2}dx-\frac {1}{3} (27-e) \int \frac {\exp \left (-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}+e^{x-4}+x-4\right )}{-3 x+e-12}dx+15 \int \frac {e^{-\frac {3 \left (5-e^{e^{x-4}} (5-x)\right )}{3 x-e+12}}}{(-3 x+e-12)^2}dx\right )\) |
Input:
Int[(E^((15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x))*(45 + E^E^(-4 + x )*(-81 + 3*E + E^(-4 + x)*(180 + 9*x - 9*x^2 + E*(-15 + 3*x)))))/(144 + E^ 2 + E*(-24 - 6*x) + 72*x + 9*x^2),x]
Output:
$Aborted
Time = 7.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (3 x -15\right ) {\mathrm e}^{{\mathrm e}^{x -4}}+15}{{\mathrm e}-3 x -12}}\) | \(25\) |
risch | \({\mathrm e}^{\frac {3 \,{\mathrm e}^{{\mathrm e}^{x -4}} x -15 \,{\mathrm e}^{{\mathrm e}^{x -4}}+15}{{\mathrm e}-3 x -12}}\) | \(29\) |
norman | \(\frac {\left ({\mathrm e}-12\right ) {\mathrm e}^{\frac {\left (3 x -15\right ) {\mathrm e}^{{\mathrm e}^{x -4}}+15}{{\mathrm e}-3 x -12}}-3 x \,{\mathrm e}^{\frac {\left (3 x -15\right ) {\mathrm e}^{{\mathrm e}^{x -4}}+15}{{\mathrm e}-3 x -12}}}{{\mathrm e}-3 x -12}\) | \(68\) |
Input:
int(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(x-4)+3*exp(1)-81)*exp(exp(x-4))+ 45)*exp(((3*x-15)*exp(exp(x-4))+15)/(exp(1)-3*x-12))/(exp(1)^2+(-6*x-24)*e xp(1)+9*x^2+72*x+144),x,method=_RETURNVERBOSE)
Output:
exp(((3*x-15)*exp(exp(x-4))+15)/(exp(1)-3*x-12))
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx=e^{\left (-\frac {3 \, {\left ({\left (x - 5\right )} e^{\left (e^{\left (x - 4\right )}\right )} + 5\right )}}{3 \, x - e + 12}\right )} \] Input:
integrate(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(-4+x)+3*exp(1)-81)*exp(exp (-4+x))+45)*exp(((3*x-15)*exp(exp(-4+x))+15)/(exp(1)-3*x-12))/(exp(1)^2+(- 6*x-24)*exp(1)+9*x^2+72*x+144),x, algorithm="fricas")
Output:
e^(-3*((x - 5)*e^(e^(x - 4)) + 5)/(3*x - e + 12))
Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx=e^{\frac {\left (3 x - 15\right ) e^{e^{x - 4}} + 15}{- 3 x - 12 + e}} \] Input:
integrate(((((3*x-15)*exp(1)-9*x**2+9*x+180)*exp(-4+x)+3*exp(1)-81)*exp(ex p(-4+x))+45)*exp(((3*x-15)*exp(exp(-4+x))+15)/(exp(1)-3*x-12))/(exp(1)**2+ (-6*x-24)*exp(1)+9*x**2+72*x+144),x)
Output:
exp(((3*x - 15)*exp(exp(x - 4)) + 15)/(-3*x - 12 + E))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx=e^{\left (-\frac {e^{\left (e^{\left (x - 4\right )} + 1\right )}}{3 \, x - e + 12} + \frac {27 \, e^{\left (e^{\left (x - 4\right )}\right )}}{3 \, x - e + 12} - \frac {15}{3 \, x - e + 12} - e^{\left (e^{\left (x - 4\right )}\right )}\right )} \] Input:
integrate(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(-4+x)+3*exp(1)-81)*exp(exp (-4+x))+45)*exp(((3*x-15)*exp(exp(-4+x))+15)/(exp(1)-3*x-12))/(exp(1)^2+(- 6*x-24)*exp(1)+9*x^2+72*x+144),x, algorithm="maxima")
Output:
e^(-e^(e^(x - 4) + 1)/(3*x - e + 12) + 27*e^(e^(x - 4))/(3*x - e + 12) - 1 5/(3*x - e + 12) - e^(e^(x - 4)))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx=e^{\left (-\frac {3 \, x e^{\left (e^{\left (x - 4\right )}\right )}}{3 \, x - e + 12} + \frac {15 \, e^{\left (e^{\left (x - 4\right )}\right )}}{3 \, x - e + 12} - \frac {15}{3 \, x - e + 12}\right )} \] Input:
integrate(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(-4+x)+3*exp(1)-81)*exp(exp (-4+x))+45)*exp(((3*x-15)*exp(exp(-4+x))+15)/(exp(1)-3*x-12))/(exp(1)^2+(- 6*x-24)*exp(1)+9*x^2+72*x+144),x, algorithm="giac")
Output:
e^(-3*x*e^(e^(x - 4))/(3*x - e + 12) + 15*e^(e^(x - 4))/(3*x - e + 12) - 1 5/(3*x - e + 12))
Time = 0.49 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx={\mathrm {e}}^{-\frac {15}{3\,x-\mathrm {e}+12}}\,{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^{{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}}{3\,x-\mathrm {e}+12}}\,{\mathrm {e}}^{-\frac {3\,x\,{\mathrm {e}}^{{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}}{3\,x-\mathrm {e}+12}} \] Input:
int((exp(-(exp(exp(x - 4))*(3*x - 15) + 15)/(3*x - exp(1) + 12))*(exp(exp( x - 4))*(3*exp(1) + exp(x - 4)*(9*x - 9*x^2 + exp(1)*(3*x - 15) + 180) - 8 1) + 45))/(72*x + exp(2) + 9*x^2 - exp(1)*(6*x + 24) + 144),x)
Output:
exp(-15/(3*x - exp(1) + 12))*exp((15*exp(exp(-4)*exp(x)))/(3*x - exp(1) + 12))*exp(-(3*x*exp(exp(-4)*exp(x)))/(3*x - exp(1) + 12))
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx=\frac {e^{\frac {3 e^{\frac {e^{x}}{e^{4}}} x +15}{e -3 x -12}}}{e^{\frac {15 e^{\frac {e^{x}}{e^{4}}}}{e -3 x -12}}} \] Input:
int(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(-4+x)+3*exp(1)-81)*exp(exp(-4+x) )+45)*exp(((3*x-15)*exp(exp(-4+x))+15)/(exp(1)-3*x-12))/(exp(1)^2+(-6*x-24 )*exp(1)+9*x^2+72*x+144),x)
Output:
e**((3*e**(e**x/e**4)*x + 15)/(e - 3*x - 12))/e**((15*e**(e**x/e**4))/(e - 3*x - 12))