3.69.52 $\int e^{-18-2e^6-2e^3(2-2x)+6x-2x^2}(1-e^{18+2e^6+2e^3(2-2x)-6x+2x^2}+6x+4e^{3}x-4x^2)dx$

Optimal. Leaf size=26 $$5-x+e^{-16-2{(1+e^3-x)}^2+2x}x$$

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Rubi [A]  time = 1.62, antiderivative size = 36, normalized size of antiderivative = 1.38, number of steps used = 15, number of rules used = 7, integrand size = 73, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.096, Rules used = {6, 6741, 6742, 2234, 2205, 2240, 2241} $$\begin{array}{c}x\exp (-2x^2+2(3+2e^3)x-2(9+2e^3+e^6))-x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[E^(-18 - 2*E^6 - 2*E^3*(2 - 2*x) + 6*x - 2*x^2)*(1 - E^(18 + 2*E^6 + 2*E^3*(2 - 2*x) - 6*x + 2*x^2) + 6*x
+ 4*E^3*x - 4*x^2),x]

[Out]

-x + E^(-2*(9 + 2*E^3 + E^6) + 2*(3 + 2*E^3)*x - 2*x^2)*x

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int e^{-18-2e^6-2e^3(2-2x)+6x-2x^2}(1-e^{18+2e^6+2e^3(2-2x)-6x+2x^2}+(6+4e^3)x-4x^2)dx\\ & =\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)(1-e^{18+2e^6+2e^3(2-2x)-6x+2x^2}+(6+4e^3)x-4x^2)dx\\ & =\int (-1+\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)+2\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)(3+2e^3)x-4\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)x^2)dx\\ & =-x-4\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)x^{2}dx+\left(2(3+2e^3)\right)\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)xdx+\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)dx\\ & =-\frac{1}{2}\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)(3+2e^3)-x+\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)x+e^{-\frac{27}{2}+2e^3}\int e^{-\frac{1}{8}{(2(3+2e^3)-4x)}^2}dx-\left(2(3+2e^3)\right)\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)xdx+{(3+2e^3)}^2\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)dx-\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)dx\\ & =-x+\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)x-\frac{1}{2}{e}^{-\frac{27}{2}+2e^3}\sqrt{\frac{\pi }{2}}\text{erf}\left(\frac{3+2e^3-2x}{\sqrt{2}}\right)-e^{-\frac{27}{2}+2e^3}\int e^{-\frac{1}{8}{(2(3+2e^3)-4x)}^2}dx-{(3+2e^3)}^2\int \exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)dx+\left(e^{-\frac{27}{2}+2e^3}{(3+2e^3)}^2\right)\int e^{-\frac{1}{8}{(2(3+2e^3)-4x)}^2}dx\\ & =-x+\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)x-\frac{1}{2}{e}^{-\frac{27}{2}+2e^3}{(3+2e^3)}^2\sqrt{\frac{\pi }{2}}\text{erf}\left(\frac{3+2e^3-2x}{\sqrt{2}}\right)-\left(e^{-\frac{27}{2}+2e^3}{(3+2e^3)}^2\right)\int e^{-\frac{1}{8}{(2(3+2e^3)-4x)}^2}dx\\ & =-x+\exp (-2(9+2e^3+e^6)+2(3+2e^3)x-2x^2)x\end{array}\end{array}$$

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Mathematica [A]  time = 0.32, size = 27, normalized size = 1.04 $$\begin{array}{c}(-1+e^{-2(9+e^6-2e^3(-1+x)-3x+x^2)})x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[E^(-18 - 2*E^6 - 2*E^3*(2 - 2*x) + 6*x - 2*x^2)*(1 - E^(18 + 2*E^6 + 2*E^3*(2 - 2*x) - 6*x + 2*x^2)
+ 6*x + 4*E^3*x - 4*x^2),x]

[Out]

(-1 + E^(-2*(9 + E^6 - 2*E^3*(-1 + x) - 3*x + x^2)))*x

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fricas [B]  time = 0.58, size = 52, normalized size = 2.00 $$\begin{array}{c}-(x{e}^{(2x^2-4(x-1)e^3-6x+2e^6+18)}-x)e^{(-2x^2+4(x-1)e^3+6x-2e^6-18)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(exp(3)^2+(-2*x+2)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+1)/exp(exp(3)^2+(-2*x+2)*exp(3)+x^2
-3*x+9)^2,x, algorithm="fricas")

[Out]

-(x*e^(2*x^2 - 4*(x - 1)*e^3 - 6*x + 2*e^6 + 18) - x)*e^(-2*x^2 + 4*(x - 1)*e^3 + 6*x - 2*e^6 - 18)

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giac [C]  time = 0.29, size = 134, normalized size = 5.15 $$\begin{array}{c}\frac{1}{2}\sqrt{2}\sqrt{\pi }(2e^3+3)\mathrm{erf}\left(\frac{1}{2}\sqrt{2}(2x-2e^3-3)\right)e^{(2e^3-\frac{21}{2})}-\frac{1}{2}\sqrt{2}\sqrt{\pi }(2e^6+3e^3)\mathrm{erf}\left(\frac{1}{2}\sqrt{2}(2x-2e^3-3)\right)e^{(2e^3-\frac{27}{2})}+(x+e^3)e^{(-2x^2+4x{e}^3+6x-2e^6-4e^3-18)}-x-e^{(-2x^2+4x{e}^3+6x-2e^6-4e^3-15)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(exp(3)^2+(-2*x+2)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+1)/exp(exp(3)^2+(-2*x+2)*exp(3)+x^2
-3*x+9)^2,x, algorithm="giac")

[Out]

1/2*sqrt(2)*sqrt(pi)*(2*e^3 + 3)*erf(1/2*sqrt(2)*(2*x - 2*e^3 - 3))*e^(2*e^3 - 21/2) - 1/2*sqrt(2)*sqrt(pi)*(2
*e^6 + 3*e^3)*erf(1/2*sqrt(2)*(2*x - 2*e^3 - 3))*e^(2*e^3 - 27/2) + (x + e^3)*e^(-2*x^2 + 4*x*e^3 + 6*x - 2*e^
6 - 4*e^3 - 18) - x - e^(-2*x^2 + 4*x*e^3 + 6*x - 2*e^6 - 4*e^3 - 15)

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maple [A]  time = 0.05, size = 31, normalized size = 1.19

method	result	size
risch	$-x+x{\mathrm{e}}^{4x{\mathrm{e}}^3-2x^2-4{\mathrm{e}}^3-2{\mathrm{e}}^6+6x-18}$	$31$
default	$-x-\frac{3{\mathrm{e}}^{-2x^2+(4{\mathrm{e}}^3+6)x-4{\mathrm{e}}^3-2{\mathrm{e}}^6-18}}{2}+\frac{3(4{\mathrm{e}}^3+6)\sqrt{\pi }{\mathrm{e}}^{-4{\mathrm{e}}^3-2{\mathrm{e}}^6-18+\frac{{(4{\mathrm{e}}^3+6)}^2}{8}}\sqrt{2}\mathrm{erf}(\sqrt{2}x-\frac{(4{\mathrm{e}}^3+6)\sqrt{2}}{4})}{8}+x{\mathrm{e}}^{-2x^2+(4{\mathrm{e}}^3+6)x-4{\mathrm{e}}^3-2{\mathrm{e}}^6-18}-(4{\mathrm{e}}^3+6)(-\frac{{\mathrm{e}}^{-2x^2+(4{\mathrm{e}}^3+6)x-4{\mathrm{e}}^3-2{\mathrm{e}}^6-18}}{4}+\frac{(4{\mathrm{e}}^3+6)\sqrt{\pi }{\mathrm{e}}^{-4{\mathrm{e}}^3-2{\mathrm{e}}^6-18+\frac{{(4{\mathrm{e}}^3+6)}^2}{8}}\sqrt{2}\mathrm{erf}(\sqrt{2}x-\frac{(4{\mathrm{e}}^3+6)\sqrt{2}}{4})}{16})-{\mathrm{e}}^{-2x^2+(4{\mathrm{e}}^3+6)x-4{\mathrm{e}}^3-2{\mathrm{e}}^6-15}+\frac{(4{\mathrm{e}}^3+6)\sqrt{\pi }{\mathrm{e}}^{-4{\mathrm{e}}^3-2{\mathrm{e}}^6-15+\frac{{(4{\mathrm{e}}^3+6)}^2}{8}}\sqrt{2}\mathrm{erf}(\sqrt{2}x-\frac{(4{\mathrm{e}}^3+6)\sqrt{2}}{4})}{4}$	$277$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(exp(3)^2+(-2*x+2)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+1)/exp(exp(3)^2+(-2*x+2)*exp(3)+x^2-3*x+9
)^2,x,method=_RETURNVERBOSE)

[Out]

-x+x*exp(4*x*exp(3)-2*x^2-4*exp(3)-2*exp(6)+6*x-18)

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maxima [C]  time = 0.58, size = 397, normalized size = 15.27 $$\begin{array}{c}\frac{1}{4}\sqrt{2}\sqrt{\pi }\mathrm{erf}(\sqrt{2}x-\frac{1}{2}\sqrt{2}(2e^3+3))e^{(\frac{1}{2}{(2e^3+3)}^2-2e^6-4e^3-18)}-\frac{1}{2}i\sqrt{2}(\frac{i\sqrt{\pi }(2x-2e^3-3)(\mathrm{erf}\left(\sqrt{\frac{1}{2}}\sqrt{{(2x-2e^3-3)}^2}\right)-1)(2e^3+3)}{\sqrt{{(2x-2e^3-3)}^2}}-i\sqrt{2}{e}^{(-\frac{1}{2}{(2x-2e^3-3)}^2)})e^{(\frac{1}{2}{(2e^3+3)}^2-2e^6-4e^3-15)}+\frac{1}{4}i\sqrt{2}(\frac{i\sqrt{\pi }(2x-2e^3-3)(\mathrm{erf}\left(\sqrt{\frac{1}{2}}\sqrt{{(2x-2e^3-3)}^2}\right)-1){(2e^3+3)}^2}{\sqrt{{(2x-2e^3-3)}^2}}-2i\sqrt{2}(2e^3+3)e^{(-\frac{1}{2}{(2x-2e^3-3)}^2)}-\frac{2i{(2x-2e^3-3)}^3\mathrm{\Gamma }(\frac{3}{2},\frac{1}{2}{(2x-2e^3-3)}^2)}{{\left({(2x-2e^3-3)}^2\right)}^{\frac{3}{2}}})e^{(\frac{1}{2}{(2e^3+3)}^2-2e^6-4e^3-18)}-\frac{3}{4}i\sqrt{2}(\frac{i\sqrt{\pi }(2x-2e^3-3)(\mathrm{erf}\left(\sqrt{\frac{1}{2}}\sqrt{{(2x-2e^3-3)}^2}\right)-1)(2e^3+3)}{\sqrt{{(2x-2e^3-3)}^2}}-i\sqrt{2}{e}^{(-\frac{1}{2}{(2x-2e^3-3)}^2)})e^{(\frac{1}{2}{(2e^3+3)}^2-2e^6-4e^3-18)}-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(exp(3)^2+(-2*x+2)*exp(3)+x^2-3*x+9)^2+4*x*exp(3)-4*x^2+6*x+1)/exp(exp(3)^2+(-2*x+2)*exp(3)+x^2
-3*x+9)^2,x, algorithm="maxima")

[Out]

1/4*sqrt(2)*sqrt(pi)*erf(sqrt(2)*x - 1/2*sqrt(2)*(2*e^3 + 3))*e^(1/2*(2*e^3 + 3)^2 - 2*e^6 - 4*e^3 - 18) - 1/2
*I*sqrt(2)*(I*sqrt(pi)*(2*x - 2*e^3 - 3)*(erf(sqrt(1/2)*sqrt((2*x - 2*e^3 - 3)^2)) - 1)*(2*e^3 + 3)/sqrt((2*x
- 2*e^3 - 3)^2) - I*sqrt(2)*e^(-1/2*(2*x - 2*e^3 - 3)^2))*e^(1/2*(2*e^3 + 3)^2 - 2*e^6 - 4*e^3 - 15) + 1/4*I*s
qrt(2)*(I*sqrt(pi)*(2*x - 2*e^3 - 3)*(erf(sqrt(1/2)*sqrt((2*x - 2*e^3 - 3)^2)) - 1)*(2*e^3 + 3)^2/sqrt((2*x -
2*e^3 - 3)^2) - 2*I*sqrt(2)*(2*e^3 + 3)*e^(-1/2*(2*x - 2*e^3 - 3)^2) - 2*I*(2*x - 2*e^3 - 3)^3*gamma(3/2, 1/2*
(2*x - 2*e^3 - 3)^2)/((2*x - 2*e^3 - 3)^2)^(3/2))*e^(1/2*(2*e^3 + 3)^2 - 2*e^6 - 4*e^3 - 18) - 3/4*I*sqrt(2)*(
I*sqrt(pi)*(2*x - 2*e^3 - 3)*(erf(sqrt(1/2)*sqrt((2*x - 2*e^3 - 3)^2)) - 1)*(2*e^3 + 3)/sqrt((2*x - 2*e^3 - 3)
^2) - I*sqrt(2)*e^(-1/2*(2*x - 2*e^3 - 3)^2))*e^(1/2*(2*e^3 + 3)^2 - 2*e^6 - 4*e^3 - 18) - x

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mupad [B]  time = 0.21, size = 34, normalized size = 1.31 $$\begin{array}{c}x{\mathrm{e}}^{-4{\mathrm{e}}^3}{\mathrm{e}}^{-2{\mathrm{e}}^6}{\mathrm{e}}^{6x}{\mathrm{e}}^{-18}{\mathrm{e}}^{-2x^2}{\mathrm{e}}^{4x{\mathrm{e}}^3}-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(6*x - 2*exp(6) - 2*x^2 + 2*exp(3)*(2*x - 2) - 18)*(6*x - exp(2*exp(6) - 6*x + 2*x^2 - 2*exp(3)*(2*x -
2) + 18) + 4*x*exp(3) - 4*x^2 + 1),x)

[Out]

x*exp(-4*exp(3))*exp(-2*exp(6))*exp(6*x)*exp(-18)*exp(-2*x^2)*exp(4*x*exp(3)) - x

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sympy [A]  time = 0.17, size = 29, normalized size = 1.12 $$\begin{array}{c}x{e}^{-2x^2+6x-2(2-2x)e^3-2e^6-18}-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(exp(3)**2+(-2*x+2)*exp(3)+x**2-3*x+9)**2+4*x*exp(3)-4*x**2+6*x+1)/exp(exp(3)**2+(-2*x+2)*exp(3
)+x**2-3*x+9)**2,x)

[Out]

x*exp(-2*x**2 + 6*x - 2*(2 - 2*x)*exp(3) - 2*exp(6) - 18) - x

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