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

Optimal. Leaf size=29 \[ \frac {e^{e^{-2 x} x}}{\left (-2+\left (1-e^2\right ) \left (e^x-x\right )\right )^2} \]

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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 {gather*} -\frac {e^{e^{-2 x} x} \left (-2 x^2-e^2 \left (x-2 x^2\right )-3 x-e^x \left (-e^2 (1-2 x)-2 x+1\right )+2\right )}{\left (e^{-2 x}-2 e^{-2 x} x\right ) \left (3 e^{3 x} \left (-e^6 x^2+x^2+e^4 \left (3 x^2+4 x\right )-e^2 \left (3 x^2+8 x+4\right )+4 x+4\right )-e^{2 x} \left (-e^6 x^3+x^3+6 x^2+3 e^4 \left (x^3+2 x^2\right )-3 e^2 \left (x^3+4 x^2+4 x\right )+12 x+8\right )-3 e^{4 x} \left (-e^6 x+x+e^4 (3 x+2)-e^2 (3 x+4)+2\right )+\left (1-e^2\right )^3 e^{5 x}\right )} \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=-\frac {e^{e^{-2 x} x} \left (2-e^x \left (1-e^2 (1-2 x)-2 x\right )-3 x-2 x^2-e^2 \left (x-2 x^2\right )\right )}{\left (e^{-2 x}-2 e^{-2 x} x\right ) \left (e^{5 x} \left (1-e^2\right )^3-3 e^{4 x} \left (2+x-e^6 x+e^4 (2+3 x)-e^2 (4+3 x)\right )+3 e^{3 x} \left (4+4 x+x^2-e^6 x^2+e^4 \left (4 x+3 x^2\right )-e^2 \left (4+8 x+3 x^2\right )\right )-e^{2 x} \left (8+12 x+6 x^2+x^3-e^6 x^3+3 e^4 \left (2 x^2+x^3\right )-3 e^2 \left (4 x+4 x^2+x^3\right )\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 31, normalized size = 1.07 \begin {gather*} \frac {e^{e^{-2 x} x}}{\left (2-e^x+e^{2+x}+x-e^2 x\right )^2} \end {gather*}

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 {gather*} \frac {e^{\left (x e^{\left (-2 \, x\right )}\right )}}{x^{2} e^{4} + x^{2} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{2} + {\left (e^{4} - 2 \, e^{2} + 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x e^{4} - 2 \, {\left (x + 1\right )} e^{2} + x + 2\right )} e^{x} + 4 \, x + 4} \end {gather*}

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 {gather*} \int -\frac {{\left (2 \, x^{2} - {\left (2 \, x^{2} - x\right )} e^{2} + 2 \, {\left (e^{2} - 1\right )} e^{\left (3 \, x\right )} - 2 \, {\left (e^{2} - 1\right )} e^{\left (2 \, x\right )} + {\left ({\left (2 \, x - 1\right )} e^{2} - 2 \, x + 1\right )} e^{x} + 3 \, x - 2\right )} e^{\left (x e^{\left (-2 \, x\right )}\right )}}{{\left (e^{6} - 3 \, e^{4} + 3 \, e^{2} - 1\right )} e^{\left (5 \, x\right )} - 3 \, {\left (x e^{6} - {\left (3 \, x + 2\right )} e^{4} + {\left (3 \, x + 4\right )} e^{2} - x - 2\right )} e^{\left (4 \, x\right )} + 3 \, {\left (x^{2} e^{6} - x^{2} - {\left (3 \, x^{2} + 4 \, x\right )} e^{4} + {\left (3 \, x^{2} + 8 \, x + 4\right )} e^{2} - 4 \, x - 4\right )} e^{\left (3 \, x\right )} - {\left (x^{3} e^{6} - x^{3} - 6 \, x^{2} - 3 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{4} + 3 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{2} - 12 \, x - 8\right )} e^{\left (2 \, x\right )}}\,{d x} \end {gather*}

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 \[\int \frac {\left (\left (-2 \,{\mathrm e}^{2}+2\right ) {\mathrm e}^{3 x}+\left (2 \,{\mathrm e}^{2}-2\right ) {\mathrm e}^{2 x}+\left (\left (1-2 x \right ) {\mathrm e}^{2}+2 x -1\right ) {\mathrm e}^{x}+\left (2 x^{2}-x \right ) {\mathrm e}^{2}-2 x^{2}-3 x +2\right ) {\mathrm e}^{x \,{\mathrm e}^{-2 x}}}{\left ({\mathrm e}^{6}-3 \,{\mathrm e}^{4}+3 \,{\mathrm e}^{2}-1\right ) {\mathrm e}^{5 x}+\left (-3 x \,{\mathrm e}^{6}+\left (9 x +6\right ) {\mathrm e}^{4}+\left (-9 x -12\right ) {\mathrm e}^{2}+6+3 x \right ) {\mathrm e}^{4 x}+\left (3 x^{2} {\mathrm e}^{6}+\left (-9 x^{2}-12 x \right ) {\mathrm e}^{4}+\left (9 x^{2}+24 x +12\right ) {\mathrm e}^{2}-3 x^{2}-12 x -12\right ) {\mathrm e}^{3 x}+\left (-x^{3} {\mathrm e}^{6}+\left (3 x^{3}+6 x^{2}\right ) {\mathrm e}^{4}+\left (-3 x^{3}-12 x^{2}-12 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+12 x +8\right ) {\mathrm e}^{2 x}}\, dx\]

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 {gather*} \frac {e^{\left (x e^{\left (-2 \, x\right )}\right )}}{x^{2} {\left (e^{4} - 2 \, e^{2} + 1\right )} - 4 \, x {\left (e^{2} - 1\right )} + {\left (e^{4} - 2 \, e^{2} + 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x {\left (e^{4} - 2 \, e^{2} + 1\right )} - 2 \, e^{2} + 2\right )} e^{x} + 4} \end {gather*}

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 {gather*} \frac {{\mathrm {e}}^{x\,{\mathrm {e}}^{-2\,x}}}{{\left ({\mathrm {e}}^2-1\right )}^2\,\left ({\mathrm {e}}^{2\,x}+\frac {4}{{\left ({\mathrm {e}}^2-1\right )}^2}-2\,x\,{\mathrm {e}}^x+x^2-\frac {x\,\left (4\,{\mathrm {e}}^2-4\right )}{{\left ({\mathrm {e}}^2-1\right )}^2}+\frac {{\mathrm {e}}^x\,\left (4\,{\mathrm {e}}^2-4\right )}{{\left ({\mathrm {e}}^2-1\right )}^2}\right )} \end {gather*}

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 {gather*} \frac {e^{x e^{- 2 x}}}{- 2 x^{2} e^{2} + x^{2} + x^{2} e^{4} - 2 x e^{4} e^{x} - 2 x e^{x} + 4 x e^{2} e^{x} - 4 x e^{2} + 4 x - 2 e^{2} e^{2 x} + e^{2 x} + e^{4} e^{2 x} - 4 e^{x} + 4 e^{2} e^{x} + 4} \end {gather*}

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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