Integrand size = 47, antiderivative size = 19 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx=e^{-\left (\left (\frac {1}{2}+e^e-x\right ) x\right )} x^2 \] Output:
x^2/exp(x*(1/2-x+exp(exp(1))))
Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx=e^{-\frac {x}{2}-e^e x+x^2} x^2 \] Input:
Integrate[(E^((-x - 2*E^E*x + 2*x^2)/2)*(4*x - x^2 - 2*E^E*x^2 + 4*x^3))/2 ,x]
Output:
E^(-1/2*x - E^E*x + x^2)*x^2
Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(19)=38\).
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.63, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6, 27, 2028, 2726}
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.
\(\displaystyle \int \frac {1}{2} e^{\frac {1}{2} \left (2 x^2-2 e^e x-x\right )} \left (4 x^3-2 e^e x^2-x^2+4 x\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {1}{2} e^{\frac {1}{2} \left (2 x^2-2 e^e x-x\right )} \left (4 x^3+\left (-1-2 e^e\right ) x^2+4 x\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int e^{\frac {1}{2} \left (2 x^2-2 e^e x-x\right )} \left (4 x^3-\left (1+2 e^e\right ) x^2+4 x\right )dx\) |
\(\Big \downarrow \) 2028 |
\(\displaystyle \frac {1}{2} \int e^{\frac {1}{2} \left (2 x^2-2 e^e x-x\right )} x \left (4 x^2+\left (-1-2 e^e\right ) x+4\right )dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {e^{\frac {1}{2} \left (2 x^2-2 e^e x-x\right )} x \left (\left (1+2 e^e\right ) x-4 x^2\right )}{-4 x+2 e^e+1}\) |
Input:
Int[(E^((-x - 2*E^E*x + 2*x^2)/2)*(4*x - x^2 - 2*E^E*x^2 + 4*x^3))/2,x]
Output:
(E^((-x - 2*E^E*x + 2*x^2)/2)*x*((1 + 2*E^E)*x - 4*x^2))/(1 + 2*E^E - 4*x)
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x^{2} {\mathrm e}^{\frac {x \left (-2 \,{\mathrm e}^{{\mathrm e}}+2 x -1\right )}{2}}\) | \(19\) |
parallelrisch | \({\mathrm e}^{-\frac {x \left (2 \,{\mathrm e}^{{\mathrm e}}-2 x +1\right )}{2}} x^{2}\) | \(21\) |
gosper | \(x^{2} {\mathrm e}^{-x \,{\mathrm e}^{{\mathrm e}}+x^{2}-\frac {x}{2}}\) | \(22\) |
norman | \(x^{2} {\mathrm e}^{-x \,{\mathrm e}^{{\mathrm e}}+x^{2}-\frac {x}{2}}\) | \(22\) |
orering | \(-\frac {x \left (-2 x^{2} {\mathrm e}^{{\mathrm e}}+4 x^{3}-x^{2}+4 x \right ) {\mathrm e}^{-x \,{\mathrm e}^{{\mathrm e}}+x^{2}-\frac {x}{2}}}{2 x \,{\mathrm e}^{{\mathrm e}}-4 x^{2}+x -4}\) | \(59\) |
default | \(-\frac {x \,{\mathrm e}^{x^{2}+\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) x}}{4}+\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) \left (\frac {{\mathrm e}^{x^{2}+\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) x}}{2}+\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )^{2}}{4}} \operatorname {erf}\left (i x +\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )}{2}\right )}{4}\right )}{4}-\frac {i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )^{2}}{4}} \operatorname {erf}\left (i x +\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )}{2}\right )}{8}+x^{2} {\mathrm e}^{x^{2}+\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) x}-\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) \left (\frac {x \,{\mathrm e}^{x^{2}+\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) x}}{2}-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) \left (\frac {{\mathrm e}^{x^{2}+\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) x}}{2}+\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )^{2}}{4}} \operatorname {erf}\left (i x +\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )}{2}\right )}{4}\right )}{2}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )^{2}}{4}} \operatorname {erf}\left (i x +\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )}{2}\right )}{4}\right )-{\mathrm e}^{{\mathrm e}} \left (\frac {x \,{\mathrm e}^{x^{2}+\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) x}}{2}-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) \left (\frac {{\mathrm e}^{x^{2}+\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) x}}{2}+\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )^{2}}{4}} \operatorname {erf}\left (i x +\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )}{2}\right )}{4}\right )}{2}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )^{2}}{4}} \operatorname {erf}\left (i x +\frac {i \left (-{\mathrm e}^{{\mathrm e}}-\frac {1}{2}\right )}{2}\right )}{4}\right )\) | \(390\) |
Input:
int(1/2*(-2*x^2*exp(exp(1))+4*x^3-x^2+4*x)/exp(x*exp(exp(1))-x^2+1/2*x),x, method=_RETURNVERBOSE)
Output:
x^2*exp(1/2*x*(-2*exp(exp(1))+2*x-1))
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx=x^{2} e^{\left (x^{2} - x e^{e} - \frac {1}{2} \, x\right )} \] Input:
integrate(1/2*(-2*x^2*exp(exp(1))+4*x^3-x^2+4*x)/exp(x*exp(exp(1))-x^2+1/2 *x),x, algorithm="fricas")
Output:
x^2*e^(x^2 - x*e^e - 1/2*x)
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx=x^{2} e^{x^{2} - x e^{e} - \frac {x}{2}} \] Input:
integrate(1/2*(-2*x**2*exp(exp(1))+4*x**3-x**2+4*x)/exp(x*exp(exp(1))-x**2 +1/2*x),x)
Output:
x**2*exp(x**2 - x*exp(E) - x/2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 556, normalized size of antiderivative = 29.26 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx =\text {Too large to display} \] Input:
integrate(1/2*(-2*x^2*exp(exp(1))+4*x^3-x^2+4*x)/exp(x*exp(exp(1))-x^2+1/2 *x),x, algorithm="maxima")
Output:
1/64*(sqrt(pi)*(4*x - 2*e^e - 1)*(erf(1/4*sqrt(-(4*x - 2*e^e - 1)^2)) - 1) *(2*e^e + 1)^3/sqrt(-(4*x - 2*e^e - 1)^2) - 48*(4*x - 2*e^e - 1)^3*(2*e^e + 1)*gamma(3/2, -1/16*(4*x - 2*e^e - 1)^2)/(-(4*x - 2*e^e - 1)^2)^(3/2) + 12*(2*e^e + 1)^2*e^(1/16*(4*x - 2*e^e - 1)^2) - 64*gamma(2, -1/16*(4*x - 2 *e^e - 1)^2))*e^(-1/16*(2*e^e + 1)^2) - 1/64*(sqrt(pi)*(4*x - 2*e^e - 1)*( erf(1/4*sqrt(-(4*x - 2*e^e - 1)^2)) - 1)*(2*e^e + 1)^2/sqrt(-(4*x - 2*e^e - 1)^2) - 16*(4*x - 2*e^e - 1)^3*gamma(3/2, -1/16*(4*x - 2*e^e - 1)^2)/(-( 4*x - 2*e^e - 1)^2)^(3/2) + 8*(2*e^e + 1)*e^(1/16*(4*x - 2*e^e - 1)^2))*e^ (-1/16*(2*e^e + 1)^2) + 1/4*(sqrt(pi)*(4*x - 2*e^e - 1)*(erf(1/4*sqrt(-(4* x - 2*e^e - 1)^2)) - 1)*(2*e^e + 1)/sqrt(-(4*x - 2*e^e - 1)^2) + 4*e^(1/16 *(4*x - 2*e^e - 1)^2))*e^(-1/16*(2*e^e + 1)^2) - 1/32*(sqrt(pi)*(4*x - 2*e ^e - 1)*(erf(1/4*sqrt(-(4*x - 2*e^e - 1)^2)) - 1)*(2*e^e + 1)^2/sqrt(-(4*x - 2*e^e - 1)^2) - 16*(4*x - 2*e^e - 1)^3*gamma(3/2, -1/16*(4*x - 2*e^e - 1)^2)/(-(4*x - 2*e^e - 1)^2)^(3/2) + 8*(2*e^e + 1)*e^(1/16*(4*x - 2*e^e - 1)^2))*e^(-1/16*(2*e^e + 1)^2 + e)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 187, normalized size of antiderivative = 9.84 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx=-\frac {1}{32} i \, \sqrt {\pi } {\left (4 \, e^{\left (2 \, e\right )} + 4 \, e^{e} - 7\right )} \operatorname {erf}\left (-i \, x + \frac {1}{2} i \, e^{e} + \frac {1}{4} i\right ) e^{\left (e - \frac {1}{4} \, e^{\left (2 \, e\right )} - \frac {1}{4} \, e^{e} - \frac {1}{16}\right )} + \frac {1}{32} i \, \sqrt {\pi } {\left (4 \, e^{\left (3 \, e\right )} + 4 \, e^{\left (2 \, e\right )} - 7 \, e^{e}\right )} \operatorname {erf}\left (-i \, x + \frac {1}{2} i \, e^{e} + \frac {1}{4} i\right ) e^{\left (-\frac {1}{4} \, e^{\left (2 \, e\right )} - \frac {1}{4} \, e^{e} - \frac {1}{16}\right )} - \frac {1}{8} \, {\left (4 \, x + 2 \, e^{e} + 1\right )} e^{\left (x^{2} - x e^{e} - \frac {1}{2} \, x + e\right )} + \frac {1}{16} \, {\left ({\left (4 \, x - 2 \, e^{e} - 1\right )}^{2} + 6 \, {\left (4 \, x - 2 \, e^{e} - 1\right )} e^{e} + 8 \, x + 12 \, e^{\left (2 \, e\right )} + 4 \, e^{e} - 1\right )} e^{\left (x^{2} - x e^{e} - \frac {1}{2} \, x\right )} \] Input:
integrate(1/2*(-2*x^2*exp(exp(1))+4*x^3-x^2+4*x)/exp(x*exp(exp(1))-x^2+1/2 *x),x, algorithm="giac")
Output:
-1/32*I*sqrt(pi)*(4*e^(2*e) + 4*e^e - 7)*erf(-I*x + 1/2*I*e^e + 1/4*I)*e^( e - 1/4*e^(2*e) - 1/4*e^e - 1/16) + 1/32*I*sqrt(pi)*(4*e^(3*e) + 4*e^(2*e) - 7*e^e)*erf(-I*x + 1/2*I*e^e + 1/4*I)*e^(-1/4*e^(2*e) - 1/4*e^e - 1/16) - 1/8*(4*x + 2*e^e + 1)*e^(x^2 - x*e^e - 1/2*x + e) + 1/16*((4*x - 2*e^e - 1)^2 + 6*(4*x - 2*e^e - 1)*e^e + 8*x + 12*e^(2*e) + 4*e^e - 1)*e^(x^2 - x *e^e - 1/2*x)
Time = 3.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx=x^2\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{\mathrm {e}}}\,{\mathrm {e}}^{-\frac {x}{2}}\,{\mathrm {e}}^{x^2} \] Input:
int(exp(x^2 - x*exp(exp(1)) - x/2)*(2*x - x^2*exp(exp(1)) - x^2/2 + 2*x^3) ,x)
Output:
x^2*exp(-x*exp(exp(1)))*exp(-x/2)*exp(x^2)
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {1}{2} e^{\frac {1}{2} \left (-x-2 e^e x+2 x^2\right )} \left (4 x-x^2-2 e^e x^2+4 x^3\right ) \, dx=\frac {e^{x^{2}} x^{2}}{e^{e^{e} x +\frac {x}{2}}} \] Input:
int(1/2*(-2*x^2*exp(exp(1))+4*x^3-x^2+4*x)/exp(x*exp(exp(1))-x^2+1/2*x),x)
Output:
(e**(x**2)*x**2)/e**((2*e**e*x + x)/2)