3.33.86 $\int \frac{e^{e^{-2x}x}(2+e^{3x}(2-2e^2)+e^{2x}(-2+2e^2)-3x-2x^2+e^x(-1+e^2(1-2x)+2x)+e^2(-x+2x^2))}{e^{5x}(-1+3e^2-3e^4+e^6)+e^{4x}(6+e^2(-12-9x)+3x-3e^{6}x+e^4(6+9x))+e^{3x}(-12-12x-3x^2+3e^6{x}^2+e^4(-12x-9x^2)+e^2(12+24x+9x^2))+e^{2x}(8+12x+6x^2+x^3-e^6{x}^3+e^2(-12x-12x^2-3x^3)+e^4(6x^2+3x^3))}dx$

Optimal. Leaf size=29 $$\frac{e^{e^{-2x}x}}{{(-2+(1-e^2)(e^x-x))}^2}$$

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Rubi [B]  time = 5.25, antiderivative size = 232, normalized size of antiderivative = 8.00, number of steps used = 1, number of rules used = 1, integrand size = 247, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.004, Rules used = {2288} $$\begin{array}{c}-\frac{e^{e^{-2x}x}(-2x^2-e^2(x-2x^2)-3x-e^x(-e^2(1-2x)-2x+1)+2)}{(e^{-2x}-2e^{-2x}x)(3e^{3x}(-e^6{x}^2+x^2+e^4(3x^2+4x)-e^2(3x^2+8x+4)+4x+4)-e^{2x}(-e^6{x}^3+x^3+6x^2+3e^4(x^3+2x^2)-3e^2(x^3+4x^2+4x)+12x+8)-3e^{4x}(-e^{6}x+x+e^4(3x+2)-e^2(3x+4)+2)+{(1-e^2)}^3{e}^{5x})}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^(x/E^(2*x))*(2 + E^(3*x)*(2 - 2*E^2) + E^(2*x)*(-2 + 2*E^2) - 3*x - 2*x^2 + E^x*(-1 + E^2*(1 - 2*x) + 2
*x) + E^2*(-x + 2*x^2)))/(E^(5*x)*(-1 + 3*E^2 - 3*E^4 + E^6) + E^(4*x)*(6 + E^2*(-12 - 9*x) + 3*x - 3*E^6*x +
E^4*(6 + 9*x)) + E^(3*x)*(-12 - 12*x - 3*x^2 + 3*E^6*x^2 + E^4*(-12*x - 9*x^2) + E^2*(12 + 24*x + 9*x^2)) + E^
(2*x)*(8 + 12*x + 6*x^2 + x^3 - E^6*x^3 + E^2*(-12*x - 12*x^2 - 3*x^3) + E^4*(6*x^2 + 3*x^3))),x]

[Out]

-((E^(x/E^(2*x))*(2 - E^x*(1 - E^2*(1 - 2*x) - 2*x) - 3*x - 2*x^2 - E^2*(x - 2*x^2)))/((E^(-2*x) - (2*x)/E^(2*
x))*(E^(5*x)*(1 - E^2)^3 - 3*E^(4*x)*(2 + x - E^6*x + E^4*(2 + 3*x) - E^2*(4 + 3*x)) + 3*E^(3*x)*(4 + 4*x + x^
2 - E^6*x^2 + E^4*(4*x + 3*x^2) - E^2*(4 + 8*x + 3*x^2)) - E^(2*x)*(8 + 12*x + 6*x^2 + x^3 - E^6*x^3 + 3*E^4*(
2*x^2 + x^3) - 3*E^2*(4*x + 4*x^2 + x^3)))))

Rule 2288

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]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-\frac{e^{e^{-2x}x}(2-e^x(1-e^2(1-2x)-2x)-3x-2x^2-e^2(x-2x^2))}{(e^{-2x}-2e^{-2x}x)(e^{5x}{(1-e^2)}^3-3e^{4x}(2+x-e^{6}x+e^4(2+3x)-e^2(4+3x))+3e^{3x}(4+4x+x^2-e^6{x}^2+e^4(4x+3x^2)-e^2(4+8x+3x^2))-e^{2x}(8+12x+6x^2+x^3-e^6{x}^3+3e^4(2x^2+x^3)-3e^2(4x+4x^2+x^3)))}\end{array}\end{array}$$

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Mathematica [A]  time = 0.23, size = 31, normalized size = 1.07 $$\begin{array}{c}\frac{e^{e^{-2x}x}}{{(2-e^x+e^{2+x}+x-e^{2}x)}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(x/E^(2*x))*(2 + E^(3*x)*(2 - 2*E^2) + E^(2*x)*(-2 + 2*E^2) - 3*x - 2*x^2 + E^x*(-1 + E^2*(1 - 2*
x) + 2*x) + E^2*(-x + 2*x^2)))/(E^(5*x)*(-1 + 3*E^2 - 3*E^4 + E^6) + E^(4*x)*(6 + E^2*(-12 - 9*x) + 3*x - 3*E^
6*x + E^4*(6 + 9*x)) + E^(3*x)*(-12 - 12*x - 3*x^2 + 3*E^6*x^2 + E^4*(-12*x - 9*x^2) + E^2*(12 + 24*x + 9*x^2)
) + E^(2*x)*(8 + 12*x + 6*x^2 + x^3 - E^6*x^3 + E^2*(-12*x - 12*x^2 - 3*x^3) + E^4*(6*x^2 + 3*x^3))),x]

[Out]

E^(x/E^(2*x))/(2 - E^x + E^(2 + x) + x - E^2*x)^2

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fricas [B]  time = 0.58, size = 66, normalized size = 2.28 $$\begin{array}{c}\frac{e^{\left(x{e}^{(-2x)}\right)}}{x^2{e}^4+x^2-2(x^2+2x)e^2+(e^4-2e^2+1)e^{\left(2x\right)}-2(x{e}^4-2(x+1)e^2+x+2)e^x+4x+4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(2)+2)*exp(x)^3+(2*exp(2)-2)*exp(x)^2+((1-2*x)*exp(2)+2*x-1)*exp(x)+(2*x^2-x)*exp(2)-2*x^2-3
*x+2)*exp(x/exp(x)^2)/((exp(2)^3-3*exp(2)^2+3*exp(2)-1)*exp(x)^5+(-3*x*exp(2)^3+(9*x+6)*exp(2)^2+(-9*x-12)*exp
(2)+6+3*x)*exp(x)^4+(3*x^2*exp(2)^3+(-9*x^2-12*x)*exp(2)^2+(9*x^2+24*x+12)*exp(2)-3*x^2-12*x-12)*exp(x)^3+(-x^
3*exp(2)^3+(3*x^3+6*x^2)*exp(2)^2+(-3*x^3-12*x^2-12*x)*exp(2)+x^3+6*x^2+12*x+8)*exp(x)^2),x, algorithm="fricas
")

[Out]

e^(x*e^(-2*x))/(x^2*e^4 + x^2 - 2*(x^2 + 2*x)*e^2 + (e^4 - 2*e^2 + 1)*e^(2*x) - 2*(x*e^4 - 2*(x + 1)*e^2 + x +
2)*e^x + 4*x + 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{(2x^2-(2x^2-x)e^2+2(e^2-1)e^{\left(3x\right)}-2(e^2-1)e^{\left(2x\right)}+((2x-1)e^2-2x+1)e^x+3x-2)e^{\left(x{e}^{(-2x)}\right)}}{(e^6-3e^4+3e^2-1)e^{\left(5x\right)}-3(x{e}^6-(3x+2)e^4+(3x+4)e^2-x-2)e^{\left(4x\right)}+3(x^2{e}^6-x^2-(3x^2+4x)e^4+(3x^2+8x+4)e^2-4x-4)e^{\left(3x\right)}-(x^3{e}^6-x^3-6x^2-3(x^3+2x^2)e^4+3(x^3+4x^2+4x)e^2-12x-8)e^{\left(2x\right)}}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(2)+2)*exp(x)^3+(2*exp(2)-2)*exp(x)^2+((1-2*x)*exp(2)+2*x-1)*exp(x)+(2*x^2-x)*exp(2)-2*x^2-3
*x+2)*exp(x/exp(x)^2)/((exp(2)^3-3*exp(2)^2+3*exp(2)-1)*exp(x)^5+(-3*x*exp(2)^3+(9*x+6)*exp(2)^2+(-9*x-12)*exp
(2)+6+3*x)*exp(x)^4+(3*x^2*exp(2)^3+(-9*x^2-12*x)*exp(2)^2+(9*x^2+24*x+12)*exp(2)-3*x^2-12*x-12)*exp(x)^3+(-x^
3*exp(2)^3+(3*x^3+6*x^2)*exp(2)^2+(-3*x^3-12*x^2-12*x)*exp(2)+x^3+6*x^2+12*x+8)*exp(x)^2),x, algorithm="giac")

[Out]

integrate(-(2*x^2 - (2*x^2 - x)*e^2 + 2*(e^2 - 1)*e^(3*x) - 2*(e^2 - 1)*e^(2*x) + ((2*x - 1)*e^2 - 2*x + 1)*e^
x + 3*x - 2)*e^(x*e^(-2*x))/((e^6 - 3*e^4 + 3*e^2 - 1)*e^(5*x) - 3*(x*e^6 - (3*x + 2)*e^4 + (3*x + 4)*e^2 - x
- 2)*e^(4*x) + 3*(x^2*e^6 - x^2 - (3*x^2 + 4*x)*e^4 + (3*x^2 + 8*x + 4)*e^2 - 4*x - 4)*e^(3*x) - (x^3*e^6 - x^
3 - 6*x^2 - 3*(x^3 + 2*x^2)*e^4 + 3*(x^3 + 4*x^2 + 4*x)*e^2 - 12*x - 8)*e^(2*x)), x)

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 $$\int \frac{((-2{\mathrm{e}}^2+2){\mathrm{e}}^{3x}+(2{\mathrm{e}}^2-2){\mathrm{e}}^{2x}+((1-2x){\mathrm{e}}^2+2x-1){\mathrm{e}}^x+(2x^2-x){\mathrm{e}}^2-2x^2-3x+2){\mathrm{e}}^{x{\mathrm{e}}^{-2x}}}{({\mathrm{e}}^6-3{\mathrm{e}}^4+3{\mathrm{e}}^2-1){\mathrm{e}}^{5x}+(-3x{\mathrm{e}}^6+(9x+6){\mathrm{e}}^4+(-9x-12){\mathrm{e}}^2+6+3x){\mathrm{e}}^{4x}+(3x^2{\mathrm{e}}^6+(-9x^2-12x){\mathrm{e}}^4+(9x^2+24x+12){\mathrm{e}}^2-3x^2-12x-12){\mathrm{e}}^{3x}+(-x^3{\mathrm{e}}^6+(3x^3+6x^2){\mathrm{e}}^4+(-3x^3-12x^2-12x){\mathrm{e}}^2+x^3+6x^2+12x+8){\mathrm{e}}^{2x}}dx$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(2)+2)*exp(x)^3+(2*exp(2)-2)*exp(x)^2+((1-2*x)*exp(2)+2*x-1)*exp(x)+(2*x^2-x)*exp(2)-2*x^2-3*x+2)*
exp(x/exp(x)^2)/((exp(2)^3-3*exp(2)^2+3*exp(2)-1)*exp(x)^5+(-3*x*exp(2)^3+(9*x+6)*exp(2)^2+(-9*x-12)*exp(2)+6+
3*x)*exp(x)^4+(3*x^2*exp(2)^3+(-9*x^2-12*x)*exp(2)^2+(9*x^2+24*x+12)*exp(2)-3*x^2-12*x-12)*exp(x)^3+(-x^3*exp(
2)^3+(3*x^3+6*x^2)*exp(2)^2+(-3*x^3-12*x^2-12*x)*exp(2)+x^3+6*x^2+12*x+8)*exp(x)^2),x)

[Out]

int(((-2*exp(2)+2)*exp(x)^3+(2*exp(2)-2)*exp(x)^2+((1-2*x)*exp(2)+2*x-1)*exp(x)+(2*x^2-x)*exp(2)-2*x^2-3*x+2)*
exp(x/exp(x)^2)/((exp(2)^3-3*exp(2)^2+3*exp(2)-1)*exp(x)^5+(-3*x*exp(2)^3+(9*x+6)*exp(2)^2+(-9*x-12)*exp(2)+6+
3*x)*exp(x)^4+(3*x^2*exp(2)^3+(-9*x^2-12*x)*exp(2)^2+(9*x^2+24*x+12)*exp(2)-3*x^2-12*x-12)*exp(x)^3+(-x^3*exp(
2)^3+(3*x^3+6*x^2)*exp(2)^2+(-3*x^3-12*x^2-12*x)*exp(2)+x^3+6*x^2+12*x+8)*exp(x)^2),x)

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maxima [B]  time = 1.79, size = 64, normalized size = 2.21 $$\begin{array}{c}\frac{e^{\left(x{e}^{(-2x)}\right)}}{x^2(e^4-2e^2+1)-4x(e^2-1)+(e^4-2e^2+1)e^{\left(2x\right)}-2(x(e^4-2e^2+1)-2e^2+2)e^x+4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(2)+2)*exp(x)^3+(2*exp(2)-2)*exp(x)^2+((1-2*x)*exp(2)+2*x-1)*exp(x)+(2*x^2-x)*exp(2)-2*x^2-3
*x+2)*exp(x/exp(x)^2)/((exp(2)^3-3*exp(2)^2+3*exp(2)-1)*exp(x)^5+(-3*x*exp(2)^3+(9*x+6)*exp(2)^2+(-9*x-12)*exp
(2)+6+3*x)*exp(x)^4+(3*x^2*exp(2)^3+(-9*x^2-12*x)*exp(2)^2+(9*x^2+24*x+12)*exp(2)-3*x^2-12*x-12)*exp(x)^3+(-x^
3*exp(2)^3+(3*x^3+6*x^2)*exp(2)^2+(-3*x^3-12*x^2-12*x)*exp(2)+x^3+6*x^2+12*x+8)*exp(x)^2),x, algorithm="maxima
")

[Out]

e^(x*e^(-2*x))/(x^2*(e^4 - 2*e^2 + 1) - 4*x*(e^2 - 1) + (e^4 - 2*e^2 + 1)*e^(2*x) - 2*(x*(e^4 - 2*e^2 + 1) - 2
*e^2 + 2)*e^x + 4)

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mupad [B]  time = 3.39, size = 67, normalized size = 2.31 $$\begin{array}{c}\frac{{\mathrm{e}}^{x{\mathrm{e}}^{-2x}}}{{({\mathrm{e}}^2-1)}^2({\mathrm{e}}^{2x}+\frac{4}{{({\mathrm{e}}^2-1)}^2}-2x{\mathrm{e}}^x+x^2-\frac{x(4{\mathrm{e}}^2-4)}{{({\mathrm{e}}^2-1)}^2}+\frac{{\mathrm{e}}^x(4{\mathrm{e}}^2-4)}{{({\mathrm{e}}^2-1)}^2})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x*exp(-2*x))*(3*x + exp(x)*(exp(2)*(2*x - 1) - 2*x + 1) - exp(2*x)*(2*exp(2) - 2) + exp(3*x)*(2*exp(
2) - 2) + exp(2)*(x - 2*x^2) + 2*x^2 - 2))/(exp(4*x)*(3*x - 3*x*exp(6) + exp(4)*(9*x + 6) - exp(2)*(9*x + 12)
+ 6) - exp(3*x)*(12*x + exp(4)*(12*x + 9*x^2) - exp(2)*(24*x + 9*x^2 + 12) - 3*x^2*exp(6) + 3*x^2 + 12) + exp(
2*x)*(12*x - exp(2)*(12*x + 12*x^2 + 3*x^3) + exp(4)*(6*x^2 + 3*x^3) - x^3*exp(6) + 6*x^2 + x^3 + 8) + exp(5*x
)*(3*exp(2) - 3*exp(4) + exp(6) - 1)),x)

[Out]

exp(x*exp(-2*x))/((exp(2) - 1)^2*(exp(2*x) + 4/(exp(2) - 1)^2 - 2*x*exp(x) + x^2 - (x*(4*exp(2) - 4))/(exp(2)
- 1)^2 + (exp(x)*(4*exp(2) - 4))/(exp(2) - 1)^2))

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sympy [B]  time = 1.66, size = 102, normalized size = 3.52 $$\begin{array}{c}\frac{e^{x{e}^{-2x}}}{-2x^2{e}^2+x^2+x^2{e}^4-2x{e}^4{e}^x-2x{e}^x+4x{e}^2{e}^x-4x{e}^2+4x-2e^2{e}^{2x}+e^{2x}+e^4{e}^{2x}-4e^x+4e^2{e}^x+4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(2)+2)*exp(x)**3+(2*exp(2)-2)*exp(x)**2+((1-2*x)*exp(2)+2*x-1)*exp(x)+(2*x**2-x)*exp(2)-2*x*
*2-3*x+2)*exp(x/exp(x)**2)/((exp(2)**3-3*exp(2)**2+3*exp(2)-1)*exp(x)**5+(-3*x*exp(2)**3+(9*x+6)*exp(2)**2+(-9
*x-12)*exp(2)+6+3*x)*exp(x)**4+(3*x**2*exp(2)**3+(-9*x**2-12*x)*exp(2)**2+(9*x**2+24*x+12)*exp(2)-3*x**2-12*x-
12)*exp(x)**3+(-x**3*exp(2)**3+(3*x**3+6*x**2)*exp(2)**2+(-3*x**3-12*x**2-12*x)*exp(2)+x**3+6*x**2+12*x+8)*exp
(x)**2),x)

[Out]

exp(x*exp(-2*x))/(-2*x**2*exp(2) + x**2 + x**2*exp(4) - 2*x*exp(4)*exp(x) - 2*x*exp(x) + 4*x*exp(2)*exp(x) - 4
*x*exp(2) + 4*x - 2*exp(2)*exp(2*x) + exp(2*x) + exp(4)*exp(2*x) - 4*exp(x) + 4*exp(2)*exp(x) + 4)

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