Integrand size = 62, antiderivative size = 28 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx=-4 e^{\frac {1}{e^{23}}-x}+e^{\left (-e^4+\frac {x}{5}\right ) x}+x \] Output:
x-4/exp(x-exp(-23))+exp(x*(1/5*x-exp(4)))
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx=-4 e^{\frac {1}{e^{23}}-x}+e^{-e^4 x+\frac {x^2}{5}}+x \] Input:
Integrate[(20 + E^((-1 + E^23*x)/E^23)*(5 + E^((-5*E^4*x + x^2)/5)*(-5*E^4 + 2*x)))/(5*E^((-1 + E^23*x)/E^23)),x]
Output:
-4*E^(E^(-23) - x) + E^(-(E^4*x) + x^2/5) + x
Time = 0.65 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 7292, 7293, 2009}
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.
\(\displaystyle \int \frac {1}{5} e^{-\frac {e^{23} x-1}{e^{23}}} \left (e^{\frac {e^{23} x-1}{e^{23}}} \left (e^{\frac {1}{5} \left (x^2-5 e^4 x\right )} \left (2 x-5 e^4\right )+5\right )+20\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int e^{\frac {1-e^{23} x}{e^{23}}} \left (e^{-\frac {1-e^{23} x}{e^{23}}} \left (5-e^{\frac {1}{5} \left (x^2-5 e^4 x\right )} \left (5 e^4-2 x\right )\right )+20\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{5} \int e^{\frac {1}{e^{23}}-x} \left (e^{-\frac {1-e^{23} x}{e^{23}}} \left (5-e^{\frac {1}{5} \left (x^2-5 e^4 x\right )} \left (5 e^4-2 x\right )\right )+20\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{5} \int \left (2 e^{\frac {1}{5} x \left (x-5 e^4\right )} x+20 e^{\frac {1}{e^{23}}-x}-5 e^{\frac {1}{5} x \left (x-5 e^4\right )+4}+5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (5 e^{\frac {x^2}{5}-e^4 x}+5 x-20 e^{\frac {1}{e^{23}}-x}\right )\) |
Input:
Int[(20 + E^((-1 + E^23*x)/E^23)*(5 + E^((-5*E^4*x + x^2)/5)*(-5*E^4 + 2*x )))/(5*E^((-1 + E^23*x)/E^23)),x]
Output:
(-20*E^(E^(-23) - x) + 5*E^(-(E^4*x) + x^2/5) + 5*x)/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
method | result | size |
default | \(x -4 \,{\mathrm e}^{-x} {\mathrm e}^{{\mathrm e}^{-23}}+{\mathrm e}^{-x \,{\mathrm e}^{4}+\frac {x^{2}}{5}}\) | \(26\) |
risch | \(x +{\mathrm e}^{-\frac {x \left (5 \,{\mathrm e}^{4}-x \right )}{5}}-4 \,{\mathrm e}^{-\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}\) | \(28\) |
parts | \(x +{\mathrm e}^{-x \,{\mathrm e}^{4}+\frac {x^{2}}{5}}-4 \,{\mathrm e}^{-\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}\) | \(31\) |
norman | \(\left (-4+x \,{\mathrm e}^{\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}+{\mathrm e}^{-x \,{\mathrm e}^{4}+\frac {x^{2}}{5}} {\mathrm e}^{\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}\right ) {\mathrm e}^{-\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}\) | \(57\) |
parallelrisch | \(\frac {\left (-20+5 x \,{\mathrm e}^{\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}+5 \,{\mathrm e}^{-\frac {x \left (5 \,{\mathrm e}^{4}-x \right )}{5}} {\mathrm e}^{\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}\right ) {\mathrm e}^{-\left (x \,{\mathrm e}^{23}-1\right ) {\mathrm e}^{-23}}}{5}\) | \(60\) |
Input:
int(1/5*(((-5*exp(4)+2*x)*exp(-x*exp(4)+1/5*x^2)+5)*exp((x*exp(23)-1)/exp( 23))+20)/exp((x*exp(23)-1)/exp(23)),x,method=_RETURNVERBOSE)
Output:
x-4/exp(x)*exp(1/exp(23))+exp(-x*exp(4)+1/5*x^2)
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx={\left ({\left (x + e^{\left (\frac {1}{5} \, x^{2} - x e^{4}\right )}\right )} e^{\left ({\left (x e^{23} - 1\right )} e^{\left (-23\right )}\right )} - 4\right )} e^{\left (-{\left (x e^{23} - 1\right )} e^{\left (-23\right )}\right )} \] Input:
integrate(1/5*(((-5*exp(4)+2*x)*exp(-x*exp(4)+1/5*x^2)+5)*exp((x*exp(23)-1 )/exp(23))+20)/exp((x*exp(23)-1)/exp(23)),x, algorithm="fricas")
Output:
((x + e^(1/5*x^2 - x*e^4))*e^((x*e^23 - 1)*e^(-23)) - 4)*e^(-(x*e^23 - 1)* e^(-23))
Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx=x + e^{\frac {x^{2}}{5} - x e^{4}} - 4 e^{- \frac {x e^{23} - 1}{e^{23}}} \] Input:
integrate(1/5*(((-5*exp(4)+2*x)*exp(-x*exp(4)+1/5*x**2)+5)*exp((x*exp(23)- 1)/exp(23))+20)/exp((x*exp(23)-1)/exp(23)),x)
Output:
x + exp(x**2/5 - x*exp(4)) - 4*exp(-(x*exp(23) - 1)*exp(-23))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.54 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx=\frac {1}{2} i \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{5} i \, \sqrt {5} x - \frac {1}{2} i \, \sqrt {5} e^{4}\right ) e^{\left (-\frac {5}{4} \, e^{8} + 4\right )} + \frac {1}{10} \, \sqrt {5} {\left (\frac {5 \, \sqrt {5} \sqrt {\frac {1}{5}} \sqrt {\pi } {\left (2 \, x - 5 \, e^{4}\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {1}{5}} \sqrt {-{\left (2 \, x - 5 \, e^{4}\right )}^{2}}\right ) - 1\right )} e^{4}}{\sqrt {-{\left (2 \, x - 5 \, e^{4}\right )}^{2}}} + 2 \, \sqrt {5} e^{\left (\frac {1}{20} \, {\left (2 \, x - 5 \, e^{4}\right )}^{2}\right )}\right )} e^{\left (-\frac {5}{4} \, e^{8}\right )} + x - 4 \, e^{\left (-x + e^{\left (-23\right )}\right )} \] Input:
integrate(1/5*(((-5*exp(4)+2*x)*exp(-x*exp(4)+1/5*x^2)+5)*exp((x*exp(23)-1 )/exp(23))+20)/exp((x*exp(23)-1)/exp(23)),x, algorithm="maxima")
Output:
1/2*I*sqrt(5)*sqrt(pi)*erf(1/5*I*sqrt(5)*x - 1/2*I*sqrt(5)*e^4)*e^(-5/4*e^ 8 + 4) + 1/10*sqrt(5)*(5*sqrt(5)*sqrt(1/5)*sqrt(pi)*(2*x - 5*e^4)*(erf(1/2 *sqrt(1/5)*sqrt(-(2*x - 5*e^4)^2)) - 1)*e^4/sqrt(-(2*x - 5*e^4)^2) + 2*sqr t(5)*e^(1/20*(2*x - 5*e^4)^2))*e^(-5/4*e^8) + x - 4*e^(-x + e^(-23))
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx=x + e^{\left (\frac {1}{5} \, x^{2} - x e^{4}\right )} - 4 \, e^{\left (-{\left (x e^{23} - 1\right )} e^{\left (-23\right )}\right )} \] Input:
integrate(1/5*(((-5*exp(4)+2*x)*exp(-x*exp(4)+1/5*x^2)+5)*exp((x*exp(23)-1 )/exp(23))+20)/exp((x*exp(23)-1)/exp(23)),x, algorithm="giac")
Output:
x + e^(1/5*x^2 - x*e^4) - 4*e^(-(x*e^23 - 1)*e^(-23))
Time = 3.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx=x-4\,{\mathrm {e}}^{{\mathrm {e}}^{-23}-x}+{\mathrm {e}}^{\frac {x^2}{5}-x\,{\mathrm {e}}^4} \] Input:
int(exp(-exp(-23)*(x*exp(23) - 1))*((exp(exp(-23)*(x*exp(23) - 1))*(exp(x^ 2/5 - x*exp(4))*(2*x - 5*exp(4)) + 5))/5 + 4),x)
Output:
x - 4*exp(exp(-23) - x) + exp(x^2/5 - x*exp(4))
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {1}{5} e^{-\frac {-1+e^{23} x}{e^{23}}} \left (20+e^{\frac {-1+e^{23} x}{e^{23}}} \left (5+e^{\frac {1}{5} \left (-5 e^4 x+x^2\right )} \left (-5 e^4+2 x\right )\right )\right ) \, dx=\frac {e^{\frac {1}{5} x^{2}+x}-4 e^{\frac {e^{27} x +1}{e^{23}}}+e^{e^{4} x +x} x}{e^{e^{4} x +x}} \] Input:
int(1/5*(((-5*exp(4)+2*x)*exp(-x*exp(4)+1/5*x^2)+5)*exp((x*exp(23)-1)/exp( 23))+20)/exp((x*exp(23)-1)/exp(23)),x)
Output:
(e**((x**2 + 5*x)/5) - 4*e**((e**27*x + 1)/e**23) + e**(e**4*x + x)*x)/e** (e**4*x + x)