Integrand size = 54, antiderivative size = 29 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx=\left (5+e^{4 e^4+3 (2-x) x}\right )^2+e^5 x (1+x) \] Output:
exp(5)*(1+x)*x+(5+exp(4*exp(4)+3*(2-x)*x))^2
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx=e^{8 e^4-6 (-2+x) x}+10 e^{4 e^4-3 (-2+x) x}+e^5 x+e^5 x^2 \] Input:
Integrate[E^(4*E^4 + 6*x - 3*x^2)*(60 - 60*x) + E^(8*E^4 + 12*x - 6*x^2)*( 12 - 12*x) + E^5*(1 + 2*x),x]
Output:
E^(8*E^4 - 6*(-2 + x)*x) + 10*E^(4*E^4 - 3*(-2 + x)*x) + E^5*x + E^5*x^2
Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, 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.
\(\displaystyle \int \left (e^{-3 x^2+6 x+4 e^4} (60-60 x)+e^{-6 x^2+12 x+8 e^4} (12-12 x)+e^5 (2 x+1)\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{-6 x^2+12 x+8 e^4}+10 e^{-3 x^2+6 x+4 e^4}+\frac {1}{4} e^5 (2 x+1)^2\) |
Input:
Int[E^(4*E^4 + 6*x - 3*x^2)*(60 - 60*x) + E^(8*E^4 + 12*x - 6*x^2)*(12 - 1 2*x) + E^5*(1 + 2*x),x]
Output:
E^(8*E^4 + 12*x - 6*x^2) + 10*E^(4*E^4 + 6*x - 3*x^2) + (E^5*(1 + 2*x)^2)/ 4
Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45
method | result | size |
risch | \({\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+x \,{\mathrm e}^{5}+x^{2} {\mathrm e}^{5}+10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}\) | \(42\) |
norman | \({\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+x \,{\mathrm e}^{5}+x^{2} {\mathrm e}^{5}+10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}\) | \(44\) |
parallelrisch | \({\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+x \,{\mathrm e}^{5}+x^{2} {\mathrm e}^{5}+10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}\) | \(44\) |
default | \({\mathrm e}^{5} \left (x^{2}+x \right )+10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} \sqrt {\pi }\, {\mathrm e}^{3} \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, x -\sqrt {3}\right )-60 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-3 x^{2}+6 x}}{6}+\frac {\sqrt {\pi }\, {\mathrm e}^{3} \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, x -\sqrt {3}\right )}{6}\right )+{\mathrm e}^{8 \,{\mathrm e}^{4}} \sqrt {\pi }\, {\mathrm e}^{6} \sqrt {6}\, \operatorname {erf}\left (\sqrt {6}\, x -\sqrt {6}\right )-12 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-6 x^{2}+12 x}}{12}+\frac {\sqrt {\pi }\, {\mathrm e}^{6} \sqrt {6}\, \operatorname {erf}\left (\sqrt {6}\, x -\sqrt {6}\right )}{12}\right )\) | \(147\) |
parts | \({\mathrm e}^{5} \left (x^{2}+x \right )+10 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} \sqrt {\pi }\, {\mathrm e}^{3} \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, x -\sqrt {3}\right )-60 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-3 x^{2}+6 x}}{6}+\frac {\sqrt {\pi }\, {\mathrm e}^{3} \sqrt {3}\, \operatorname {erf}\left (\sqrt {3}\, x -\sqrt {3}\right )}{6}\right )+{\mathrm e}^{8 \,{\mathrm e}^{4}} \sqrt {\pi }\, {\mathrm e}^{6} \sqrt {6}\, \operatorname {erf}\left (\sqrt {6}\, x -\sqrt {6}\right )-12 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-6 x^{2}+12 x}}{12}+\frac {\sqrt {\pi }\, {\mathrm e}^{6} \sqrt {6}\, \operatorname {erf}\left (\sqrt {6}\, x -\sqrt {6}\right )}{12}\right )\) | \(147\) |
orering | \(\frac {\left (72 x^{6}-216 x^{5}+90 x^{4}+306 x^{3}-409 x^{2}+197 x -34\right ) \left (\left (-12 x +12\right ) {\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+\left (-60 x +60\right ) {\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}+\left (1+2 x \right ) {\mathrm e}^{5}\right )}{144 x^{5}-504 x^{4}+576 x^{3}-198 x^{2}-36 x +27}+\frac {\left (72 x^{5}-144 x^{4}-20 x^{3}+210 x^{2}-168 x +41\right ) \left (-12 \,{\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}+2 \left (-12 x +12\right ) {\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x} \left (6-6 x \right )-60 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}+\left (-60 x +60\right ) \left (6-6 x \right ) {\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}+2 \,{\mathrm e}^{5}\right )}{576 x^{5}-2016 x^{4}+2304 x^{3}-792 x^{2}-144 x +108}+\frac {\left (24 x^{5}-48 x^{4}+66 x^{2}-60 x +17\right ) \left (-48 \,{\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x} \left (6-6 x \right )+4 \left (-12 x +12\right ) {\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x} \left (6-6 x \right )^{2}-12 \left (-12 x +12\right ) {\mathrm e}^{8 \,{\mathrm e}^{4}-6 x^{2}+12 x}-120 \left (6-6 x \right ) {\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}-6 \left (-60 x +60\right ) {\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}+\left (-60 x +60\right ) \left (6-6 x \right )^{2} {\mathrm e}^{4 \,{\mathrm e}^{4}-3 x^{2}+6 x}\right )}{216 \left (-1+x \right ) \left (16 x^{5}-56 x^{4}+64 x^{3}-22 x^{2}-4 x +3\right )}\) | \(458\) |
Input:
int((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x^2+6*x )+(1+2*x)*exp(5),x,method=_RETURNVERBOSE)
Output:
exp(8*exp(4)-6*x^2+12*x)+x*exp(5)+x^2*exp(5)+10*exp(4*exp(4)-3*x^2+6*x)
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx={\left (x^{2} + x\right )} e^{5} + 10 \, e^{\left (-3 \, x^{2} + 6 \, x + 4 \, e^{4}\right )} + e^{\left (-6 \, x^{2} + 12 \, x + 8 \, e^{4}\right )} \] Input:
integrate((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x ^2+6*x)+(1+2*x)*exp(5),x, algorithm="fricas")
Output:
(x^2 + x)*e^5 + 10*e^(-3*x^2 + 6*x + 4*e^4) + e^(-6*x^2 + 12*x + 8*e^4)
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx=x^{2} e^{5} + x e^{5} + e^{- 6 x^{2} + 12 x + 8 e^{4}} + 10 e^{- 3 x^{2} + 6 x + 4 e^{4}} \] Input:
integrate((-12*x+12)*exp(4*exp(4)-3*x**2+6*x)**2+(-60*x+60)*exp(4*exp(4)-3 *x**2+6*x)+(1+2*x)*exp(5),x)
Output:
x**2*exp(5) + x*exp(5) + exp(-6*x**2 + 12*x + 8*exp(4)) + 10*exp(-3*x**2 + 6*x + 4*exp(4))
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx={\left (x^{2} + x\right )} e^{5} + 10 \, e^{\left (-3 \, x^{2} + 6 \, x + 4 \, e^{4}\right )} + e^{\left (-6 \, x^{2} + 12 \, x + 8 \, e^{4}\right )} \] Input:
integrate((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x ^2+6*x)+(1+2*x)*exp(5),x, algorithm="maxima")
Output:
(x^2 + x)*e^5 + 10*e^(-3*x^2 + 6*x + 4*e^4) + e^(-6*x^2 + 12*x + 8*e^4)
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx={\left (x^{2} + x\right )} e^{5} + 10 \, e^{\left (-3 \, x^{2} + 6 \, x + 4 \, e^{4}\right )} + e^{\left (-6 \, x^{2} + 12 \, x + 8 \, e^{4}\right )} \] Input:
integrate((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x ^2+6*x)+(1+2*x)*exp(5),x, algorithm="giac")
Output:
(x^2 + x)*e^5 + 10*e^(-3*x^2 + 6*x + 4*e^4) + e^(-6*x^2 + 12*x + 8*e^4)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx=10\,{\mathrm {e}}^{-3\,x^2+6\,x+4\,{\mathrm {e}}^4}+{\mathrm {e}}^{-6\,x^2+12\,x+8\,{\mathrm {e}}^4}+x\,{\mathrm {e}}^5+x^2\,{\mathrm {e}}^5 \] Input:
int(exp(5)*(2*x + 1) - exp(6*x + 4*exp(4) - 3*x^2)*(60*x - 60) - exp(12*x + 8*exp(4) - 6*x^2)*(12*x - 12),x)
Output:
10*exp(6*x + 4*exp(4) - 3*x^2) + exp(12*x + 8*exp(4) - 6*x^2) + x*exp(5) + x^2*exp(5)
Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \left (e^{4 e^4+6 x-3 x^2} (60-60 x)+e^{8 e^4+12 x-6 x^2} (12-12 x)+e^5 (1+2 x)\right ) \, dx=\frac {e^{8 e^{4}+12 x}+10 e^{4 e^{4}+3 x^{2}+6 x}+e^{6 x^{2}} e^{5} x^{2}+e^{6 x^{2}} e^{5} x}{e^{6 x^{2}}} \] Input:
int((-12*x+12)*exp(4*exp(4)-3*x^2+6*x)^2+(-60*x+60)*exp(4*exp(4)-3*x^2+6*x )+(1+2*x)*exp(5),x)
Output:
(e**(8*e**4 + 12*x) + 10*e**(4*e**4 + 3*x**2 + 6*x) + e**(6*x**2)*e**5*x** 2 + e**(6*x**2)*e**5*x)/e**(6*x**2)