Integrand size = 34, antiderivative size = 26 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=2-e^{\left (-4+5 e^5 (4-x)-\frac {5}{x}\right ) x}+x \] Output:
2+x-exp(x*(5*(4-x)*exp(5)-5/x-4))
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=-e^{-5-4 x+20 e^5 x-5 e^5 x^2}+x \] Input:
Integrate[1 + E^(-5 - 4*x + E^5*(20*x - 5*x^2))*(4 + E^5*(-20 + 10*x)),x]
Output:
-E^(-5 - 4*x + 20*E^5*x - 5*E^5*x^2) + x
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {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.
\(\displaystyle \int \left (e^{e^5 \left (20 x-5 x^2\right )-4 x-5} \left (e^5 (10 x-20)+4\right )+1\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x-e^{-5 e^5 x^2-4 \left (1-5 e^5\right ) x-5}\) |
Input:
Int[1 + E^(-5 - 4*x + E^5*(20*x - 5*x^2))*(4 + E^5*(-20 + 10*x)),x]
Output:
-E^(-5 - 4*(1 - 5*E^5)*x - 5*E^5*x^2) + x
Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
norman | \(x -{\mathrm e}^{\left (-5 x^{2}+20 x \right ) {\mathrm e}^{5}-4 x -5}\) | \(23\) |
risch | \(x -{\mathrm e}^{-5 x^{2} {\mathrm e}^{5}+20 x \,{\mathrm e}^{5}-4 x -5}\) | \(23\) |
parallelrisch | \(x -{\mathrm e}^{\left (-5 x^{2}+20 x \right ) {\mathrm e}^{5}-4 x -5}\) | \(23\) |
default | \(x +\frac {2 \,{\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{5}-2 \,{\mathrm e}^{-5} {\mathrm e}^{5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )+10 \,{\mathrm e}^{-5} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{-5} {\mathrm e}^{-5 x^{2} {\mathrm e}^{5}+\left (20 \,{\mathrm e}^{5}-4\right ) x}}{10}+\frac {\left (20 \,{\mathrm e}^{5}-4\right ) {\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{100}\right )\) | \(202\) |
parts | \(x +\frac {2 \,{\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{5}-2 \,{\mathrm e}^{-5} {\mathrm e}^{5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )+10 \,{\mathrm e}^{-5} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{-5} {\mathrm e}^{-5 x^{2} {\mathrm e}^{5}+\left (20 \,{\mathrm e}^{5}-4\right ) x}}{10}+\frac {\left (20 \,{\mathrm e}^{5}-4\right ) {\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \operatorname {erf}\left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{100}\right )\) | \(202\) |
Input:
int(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x,method=_RETUR NVERBOSE)
Output:
x-exp((-5*x^2+20*x)*exp(5)-4*x-5)
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=x - e^{\left (-5 \, {\left (x^{2} - 4 \, x\right )} e^{5} - 4 \, x - 5\right )} \] Input:
integrate(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x, algori thm="fricas")
Output:
x - e^(-5*(x^2 - 4*x)*e^5 - 4*x - 5)
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=x - e^{- 4 x + \left (- 5 x^{2} + 20 x\right ) e^{5} - 5} \] Input:
integrate(((10*x-20)*exp(5)+4)*exp((-5*x**2+20*x)*exp(5)-4*x-5)+1,x)
Output:
x - exp(-4*x + (-5*x**2 + 20*x)*exp(5) - 5)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.18 (sec) , antiderivative size = 223, normalized size of antiderivative = 8.58 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=-2 \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (\sqrt {5} x e^{\frac {5}{2}} - \frac {2}{5} \, \sqrt {5} {\left (5 \, e^{5} - 1\right )} e^{\left (-\frac {5}{2}\right )}\right ) e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )} - \frac {5}{2}\right )} + \frac {2}{5} \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (\sqrt {5} x e^{\frac {5}{2}} - \frac {2}{5} \, \sqrt {5} {\left (5 \, e^{5} - 1\right )} e^{\left (-\frac {5}{2}\right )}\right ) e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )} - \frac {15}{2}\right )} + \frac {\sqrt {5} {\left (\frac {2 \, \sqrt {5} \sqrt {\frac {1}{5}} \sqrt {\pi } {\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{5}} \sqrt {{\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2}} e^{\left (-\frac {5}{2}\right )}\right ) - 1\right )} {\left (5 \, e^{5} - 1\right )} e^{\frac {5}{2}}}{\sqrt {{\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2}} \left (-e^{5}\right )^{\frac {3}{2}}} - \frac {\sqrt {5} e^{\left (-\frac {1}{5} \, {\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2} e^{\left (-5\right )} + 5\right )}}{\left (-e^{5}\right )^{\frac {3}{2}}}\right )} e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )}\right )}}{5 \, \sqrt {-e^{5}}} + x \] Input:
integrate(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x, algori thm="maxima")
Output:
-2*sqrt(5)*sqrt(pi)*erf(sqrt(5)*x*e^(5/2) - 2/5*sqrt(5)*(5*e^5 - 1)*e^(-5/ 2))*e^(4/5*(5*e^5 - 1)^2*e^(-5) - 5/2) + 2/5*sqrt(5)*sqrt(pi)*erf(sqrt(5)* x*e^(5/2) - 2/5*sqrt(5)*(5*e^5 - 1)*e^(-5/2))*e^(4/5*(5*e^5 - 1)^2*e^(-5) - 15/2) + 1/5*sqrt(5)*(2*sqrt(5)*sqrt(1/5)*sqrt(pi)*(5*x*e^5 - 10*e^5 + 2) *(erf(sqrt(1/5)*sqrt((5*x*e^5 - 10*e^5 + 2)^2)*e^(-5/2)) - 1)*(5*e^5 - 1)* e^(5/2)/(sqrt((5*x*e^5 - 10*e^5 + 2)^2)*(-e^5)^(3/2)) - sqrt(5)*e^(-1/5*(5 *x*e^5 - 10*e^5 + 2)^2*e^(-5) + 5)/(-e^5)^(3/2))*e^(4/5*(5*e^5 - 1)^2*e^(- 5))/sqrt(-e^5) + x
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.62 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=\frac {2}{5} \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{5} \, \sqrt {5} {\left (2 \, {\left (5 \, e^{5} - 1\right )} e^{\left (-5\right )} - 5 \, x\right )} e^{\frac {5}{2}}\right ) e^{\left (\frac {1}{5} \, {\left (100 \, e^{10} - 65 \, e^{5} + 4\right )} e^{\left (-5\right )} - \frac {5}{2}\right )} - \frac {1}{5} \, {\left (2 \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{5} \, \sqrt {5} {\left (2 \, {\left (5 \, e^{5} - 1\right )} e^{\left (-5\right )} - 5 \, x\right )} e^{\frac {5}{2}}\right ) e^{\left (\frac {4}{5} \, {\left (25 \, e^{10} - 10 \, e^{5} + 1\right )} e^{\left (-5\right )} - \frac {5}{2}\right )} + 5 \, e^{\left (-5 \, x^{2} e^{5} + 20 \, x e^{5} - 4 \, x\right )}\right )} e^{\left (-5\right )} + x \] Input:
integrate(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x, algori thm="giac")
Output:
2/5*sqrt(5)*sqrt(pi)*erf(-1/5*sqrt(5)*(2*(5*e^5 - 1)*e^(-5) - 5*x)*e^(5/2) )*e^(1/5*(100*e^10 - 65*e^5 + 4)*e^(-5) - 5/2) - 1/5*(2*sqrt(5)*sqrt(pi)*e rf(-1/5*sqrt(5)*(2*(5*e^5 - 1)*e^(-5) - 5*x)*e^(5/2))*e^(4/5*(25*e^10 - 10 *e^5 + 1)*e^(-5) - 5/2) + 5*e^(-5*x^2*e^5 + 20*x*e^5 - 4*x))*e^(-5) + x
Time = 2.85 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=x-{\mathrm {e}}^{-5\,x^2\,{\mathrm {e}}^5}\,{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{20\,x\,{\mathrm {e}}^5} \] Input:
int(exp(exp(5)*(20*x - 5*x^2) - 4*x - 5)*(exp(5)*(10*x - 20) + 4) + 1,x)
Output:
x - exp(-5*x^2*exp(5))*exp(-4*x)*exp(-5)*exp(20*x*exp(5))
Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \left (1+e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right )\right ) \, dx=\frac {e^{5 e^{5} x^{2}+4 x} e^{5} x -e^{20 e^{5} x}}{e^{5 e^{5} x^{2}+4 x} e^{5}} \] Input:
int(((10*x-20)*exp(5)+4)*exp((-5*x^2+20*x)*exp(5)-4*x-5)+1,x)
Output:
(e**(5*e**5*x**2 + 4*x)*e**5*x - e**(20*e**5*x))/(e**(5*e**5*x**2 + 4*x)*e **5)