Integrand size = 82, antiderivative size = 26 \[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x (1+x)}} \] Output:
exp(1/3*exp(exp(x)-5)/(1+x)/x*(x^3+4))
Time = 5.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x (1+x)}} \] Input:
Integrate[(E^(-5 + E^x + (E^(-5 + E^x)*(4 + x^3))/(3*x + 3*x^2))*(-4 - 8*x + 2*x^3 + x^4 + E^x*(4*x + 4*x^2 + x^4 + x^5)))/(3*x^2 + 6*x^3 + 3*x^4),x ]
Output:
E^((E^(-5 + E^x)*(4 + x^3))/(3*x*(1 + 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.
| \(\displaystyle \int \frac {e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+e^x-5} \left (x^4+2 x^3+e^x \left (x^5+x^4+4 x^2+4 x\right )-8 x-4\right )}{3 x^4+6 x^3+3 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+e^x-5} \left (x^4+2 x^3+e^x \left (x^5+x^4+4 x^2+4 x\right )-8 x-4\right )}{x^2 \left (3 x^2+6 x+3\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+e^x-5} \left (x^4+2 x^3+e^x \left (x^5+x^4+4 x^2+4 x\right )-8 x-4\right )}{x^2 \left (\sqrt {3} x+\sqrt {3}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^3+4\right ) \exp \left (\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+x+e^x-5\right )}{3 (x+1) x}+\frac {e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+e^x-5} x^2}{3 (x+1)^2}+\frac {2 e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+e^x-5} x}{3 (x+1)^2}-\frac {8 e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+e^x-5}}{3 (x+1)^2 x}-\frac {4 e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x^2+3 x}+e^x-5}}{3 (x+1)^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{\frac {e^{e^x-5} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} \left (e^x x^5+\left (e^x+1\right ) x^4+2 x^3+4 e^x x^2+4 \left (e^x-2\right ) x-4\right )}{3 x^2 (x+1)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} \left (-e^x x^5-\left (1+e^x\right ) x^4-2 x^3-4 e^x x^2+4 \left (2-e^x\right ) x+4\right )}{x^2 (x+1)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} \left (-e^x x^5-\left (1+e^x\right ) x^4-2 x^3-4 e^x x^2+4 \left (2-e^x\right ) x+4\right )}{x^2 (x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (-\frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} x^2}{(x+1)^2}-\frac {2 e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} x}{(x+1)^2}-\frac {\exp \left (x+e^x-5+\frac {e^{-5+e^x} \left (x^3+4\right )}{3 (x+1) x}\right ) \left (x^3+4\right )}{(x+1) x}+\frac {8 e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{(x+1)^2 x}+\frac {4 e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{(x+1)^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{3} \int \frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} \left (-e^x x^5-\left (1+e^x\right ) x^4-2 x^3-4 e^x x^2-4 \left (-2+e^x\right ) x+4\right )}{x^2 (x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (-\frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} x^2}{(x+1)^2}-\frac {2 e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5} x}{(x+1)^2}-\frac {\exp \left (x+e^x-5+\frac {e^{-5+e^x} \left (x^3+4\right )}{3 (x+1) x}\right ) \left (x^3+4\right )}{(x+1) x}+\frac {8 e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{(x+1)^2 x}+\frac {4 e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{(x+1)^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (-\int \exp \left (x+e^x-5+\frac {e^{-5+e^x} \left (x^3+4\right )}{3 (x+1) x}\right )dx+4 \int \frac {\exp \left (x+e^x-5+\frac {e^{-5+e^x} \left (x^3+4\right )}{3 (x+1) x}\right )}{x}dx+\int \exp \left (x+e^x-5+\frac {e^{-5+e^x} \left (x^3+4\right )}{3 (x+1) x}\right ) xdx-3 \int \frac {\exp \left (x+e^x-5+\frac {e^{-5+e^x} \left (x^3+4\right )}{3 (x+1) x}\right )}{x+1}dx+\int e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}dx-8 \int \frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{-x-1}dx+3 \int \frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{(x+1)^2}dx-8 \int \frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{x+1}dx-4 \int \frac {e^{\frac {e^{-5+e^x} \left (x^3+4\right )}{3 x (x+1)}+e^x-5}}{x^2}dx\right )\) |
Input:
Int[(E^(-5 + E^x + (E^(-5 + E^x)*(4 + x^3))/(3*x + 3*x^2))*(-4 - 8*x + 2*x ^3 + x^4 + E^x*(4*x + 4*x^2 + x^4 + x^5)))/(3*x^2 + 6*x^3 + 3*x^4),x]
Output:
$Aborted
Time = 38.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \({\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{x}-5} \left (x^{3}+4\right )}{3 \left (1+x \right ) x}}\) | \(22\) |
| parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{x}-5} \left (x^{3}+4\right )}{3 \left (1+x \right ) x}}\) | \(22\) |
Input:
int(((x^5+x^4+4*x^2+4*x)*exp(x)+x^4+2*x^3-8*x-4)*exp(exp(x)-5)*exp((x^3+4) *exp(exp(x)-5)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x,method=_RETURNVERBOSE)
Output:
exp(1/3*exp(exp(x)-5)/(1+x)/x*(x^3+4))
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\left (-\frac {15 \, x^{2} - 3 \, {\left (x^{2} + x\right )} e^{x} - {\left (x^{3} + 4\right )} e^{\left (e^{x} - 5\right )} + 15 \, x}{3 \, {\left (x^{2} + x\right )}} - e^{x} + 5\right )} \] Input:
integrate(((x^5+x^4+4*x^2+4*x)*exp(x)+x^4+2*x^3-8*x-4)*exp(exp(x)-5)*exp(( x^3+4)*exp(exp(x)-5)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x, algorithm="fricas ")
Output:
e^(-1/3*(15*x^2 - 3*(x^2 + x)*e^x - (x^3 + 4)*e^(e^x - 5) + 15*x)/(x^2 + x ) - e^x + 5)
Time = 0.50 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\frac {\left (x^{3} + 4\right ) e^{e^{x} - 5}}{3 x^{2} + 3 x}} \] Input:
integrate(((x**5+x**4+4*x**2+4*x)*exp(x)+x**4+2*x**3-8*x-4)*exp(exp(x)-5)* exp((x**3+4)*exp(exp(x)-5)/(3*x**2+3*x))/(3*x**4+6*x**3+3*x**2),x)
Output:
exp((x**3 + 4)*exp(exp(x) - 5)/(3*x**2 + 3*x))
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\left (\frac {1}{3} \, x e^{\left (e^{x} - 5\right )} + \frac {4 \, e^{\left (e^{x} - 5\right )}}{3 \, x} - \frac {e^{\left (e^{x}\right )}}{x e^{5} + e^{5}} - \frac {1}{3} \, e^{\left (e^{x} - 5\right )}\right )} \] Input:
integrate(((x^5+x^4+4*x^2+4*x)*exp(x)+x^4+2*x^3-8*x-4)*exp(exp(x)-5)*exp(( x^3+4)*exp(exp(x)-5)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x, algorithm="maxima ")
Output:
e^(1/3*x*e^(e^x - 5) + 4/3*e^(e^x - 5)/x - e^(e^x)/(x*e^5 + e^5) - 1/3*e^( e^x - 5))
\[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=\int { \frac {{\left (x^{4} + 2 \, x^{3} + {\left (x^{5} + x^{4} + 4 \, x^{2} + 4 \, x\right )} e^{x} - 8 \, x - 4\right )} e^{\left (\frac {{\left (x^{3} + 4\right )} e^{\left (e^{x} - 5\right )}}{3 \, {\left (x^{2} + x\right )}} + e^{x} - 5\right )}}{3 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}} \,d x } \] Input:
integrate(((x^5+x^4+4*x^2+4*x)*exp(x)+x^4+2*x^3-8*x-4)*exp(exp(x)-5)*exp(( x^3+4)*exp(exp(x)-5)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x, algorithm="giac")
Output:
integrate(1/3*(x^4 + 2*x^3 + (x^5 + x^4 + 4*x^2 + 4*x)*e^x - 8*x - 4)*e^(1 /3*(x^3 + 4)*e^(e^x - 5)/(x^2 + x) + e^x - 5)/(x^4 + 2*x^3 + x^2), x)
Time = 2.88 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx={\mathrm {e}}^{\frac {x^3\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-5}}{3\,x^2+3\,x}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-5}}{3\,x^2+3\,x}} \] Input:
int((exp((exp(exp(x) - 5)*(x^3 + 4))/(3*x + 3*x^2))*exp(exp(x) - 5)*(exp(x )*(4*x + 4*x^2 + x^4 + x^5) - 8*x + 2*x^3 + x^4 - 4))/(3*x^2 + 6*x^3 + 3*x ^4),x)
Output:
exp((x^3*exp(exp(x))*exp(-5))/(3*x + 3*x^2))*exp((4*exp(exp(x))*exp(-5))/( 3*x + 3*x^2))
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-5+e^x+\frac {e^{-5+e^x} \left (4+x^3\right )}{3 x+3 x^2}} \left (-4-8 x+2 x^3+x^4+e^x \left (4 x+4 x^2+x^4+x^5\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\frac {e^{e^{x}} x^{3}+4 e^{e^{x}}}{3 e^{5} x^{2}+3 e^{5} x}} \] Input:
int(((x^5+x^4+4*x^2+4*x)*exp(x)+x^4+2*x^3-8*x-4)*exp(exp(x)-5)*exp((x^3+4) *exp(exp(x)-5)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x)
Output:
e**((e**(e**x)*x**3 + 4*e**(e**x))/(3*e**5*x**2 + 3*e**5*x))