3.20.31 \(\int \frac {e^{\frac {1+2 e^{e^x} x+e^{2 e^e} x^2+e^{2 e^x} x^2+e^{e^e} (2 x+2 e^{e^x} x^2)}{x^2}} (-52+2 x-x^3+e^{2 e^x+x} (52 x^3-2 x^4)+e^{e^x} (-52 x+2 x^2+e^x (52 x^2-2 x^3))+e^{e^e} (-52 x+2 x^2+e^{e^x+x} (52 x^3-2 x^4)))}{x^3} \, dx\)

Optimal. Leaf size=24 \[ e^{\left (e^{e^e}+e^{e^x}+\frac {1}{x}\right )^2} (26-x) \]

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Rubi [B]  time = 5.94, antiderivative size = 288, normalized size of antiderivative = 12.00, number of steps used = 1, number of rules used = 1, integrand size = 157, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {2288} \begin {gather*} \frac {\left (-e^{x+2 e^x} \left (26 x^3-x^4\right )+e^{e^x} \left (-x^2-e^x \left (26 x^2-x^3\right )+26 x\right )+e^{e^e} \left (-x^2-e^{x+e^x} \left (26 x^3-x^4\right )+26 x\right )-x+26\right ) \exp \left (\frac {e^{2 e^x} x^2+e^{2 e^e} x^2+2 e^{e^e} \left (e^{e^x} x^2+x\right )+2 e^{e^x} x+1}{x^2}\right )}{x^3 \left (\frac {e^{2 e^x} x^2+e^{2 e^e} x^2+2 e^{e^e} \left (e^{e^x} x^2+x\right )+2 e^{e^x} x+1}{x^3}-\frac {e^{x+2 e^x} x^2+e^{e^e} \left (e^{x+e^x} x^2+2 e^{e^x} x+1\right )+e^{2 e^x} x+e^{x+e^x} x+e^{2 e^e} x+e^{e^x}}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((1 + 2*E^E^x*x + E^(2*E^E)*x^2 + E^(2*E^x)*x^2 + E^E^E*(2*x + 2*E^E^x*x^2))/x^2)*(-52 + 2*x - x^3 + E^
(2*E^x + x)*(52*x^3 - 2*x^4) + E^E^x*(-52*x + 2*x^2 + E^x*(52*x^2 - 2*x^3)) + E^E^E*(-52*x + 2*x^2 + E^(E^x +
x)*(52*x^3 - 2*x^4))))/x^3,x]

[Out]

(E^((1 + 2*E^E^x*x + E^(2*E^E)*x^2 + E^(2*E^x)*x^2 + 2*E^E^E*(x + E^E^x*x^2))/x^2)*(26 - x - E^(2*E^x + x)*(26
*x^3 - x^4) + E^E^x*(26*x - x^2 - E^x*(26*x^2 - x^3)) + E^E^E*(26*x - x^2 - E^(E^x + x)*(26*x^3 - x^4))))/(x^3
*((1 + 2*E^E^x*x + E^(2*E^E)*x^2 + E^(2*E^x)*x^2 + 2*E^E^E*(x + E^E^x*x^2))/x^3 - (E^E^x + E^(2*E^E)*x + E^(2*
E^x)*x + E^(E^x + x)*x + E^(2*E^x + x)*x^2 + E^E^E*(1 + 2*E^E^x*x + E^(E^x + x)*x^2))/x^2))

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 {\exp \left (\frac {1+2 e^{e^x} x+e^{2 e^e} x^2+e^{2 e^x} x^2+2 e^{e^e} \left (x+e^{e^x} x^2\right )}{x^2}\right ) \left (26-x-e^{2 e^x+x} \left (26 x^3-x^4\right )+e^{e^x} \left (26 x-x^2-e^x \left (26 x^2-x^3\right )\right )+e^{e^e} \left (26 x-x^2-e^{e^x+x} \left (26 x^3-x^4\right )\right )\right )}{x^3 \left (\frac {1+2 e^{e^x} x+e^{2 e^e} x^2+e^{2 e^x} x^2+2 e^{e^e} \left (x+e^{e^x} x^2\right )}{x^3}-\frac {e^{e^x}+e^{2 e^e} x+e^{2 e^x} x+e^{e^x+x} x+e^{2 e^x+x} x^2+e^{e^e} \left (1+2 e^{e^x} x+e^{e^x+x} x^2\right )}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 29, normalized size = 1.21 \begin {gather*} -e^{\frac {\left (1+e^{e^e} x+e^{e^x} x\right )^2}{x^2}} (-26+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1 + 2*E^E^x*x + E^(2*E^E)*x^2 + E^(2*E^x)*x^2 + E^E^E*(2*x + 2*E^E^x*x^2))/x^2)*(-52 + 2*x - x^
3 + E^(2*E^x + x)*(52*x^3 - 2*x^4) + E^E^x*(-52*x + 2*x^2 + E^x*(52*x^2 - 2*x^3)) + E^E^E*(-52*x + 2*x^2 + E^(
E^x + x)*(52*x^3 - 2*x^4))))/x^3,x]

[Out]

-(E^((1 + E^E^E*x + E^E^x*x)^2/x^2)*(-26 + x))

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fricas [B]  time = 0.76, size = 80, normalized size = 3.33 \begin {gather*} -{\left (x - 26\right )} e^{\left (\frac {{\left (x^{2} e^{\left (2 \, x + 2 \, e^{x}\right )} + x^{2} e^{\left (2 \, x + 2 \, e^{e}\right )} + 2 \, x e^{\left (2 \, x + e^{x}\right )} + 2 \, {\left (x^{2} e^{\left (2 \, x + e^{x}\right )} + x e^{\left (2 \, x\right )}\right )} e^{\left (e^{e}\right )} + e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+52*x^3)*exp(x)*exp(exp(x))+2*x^2-52*x)*exp(exp(exp(1)))+(-2*x^4+52*x^3)*exp(x)*exp(exp(x))
^2+((-2*x^3+52*x^2)*exp(x)+2*x^2-52*x)*exp(exp(x))-x^3+2*x-52)*exp((x^2*exp(exp(exp(1)))^2+(2*exp(exp(x))*x^2+
2*x)*exp(exp(exp(1)))+x^2*exp(exp(x))^2+2*x*exp(exp(x))+1)/x^2)/x^3,x, algorithm="fricas")

[Out]

-(x - 26)*e^((x^2*e^(2*x + 2*e^x) + x^2*e^(2*x + 2*e^e) + 2*x*e^(2*x + e^x) + 2*(x^2*e^(2*x + e^x) + x*e^(2*x)
)*e^(e^e) + e^(2*x))*e^(-2*x)/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{3} + 2 \, {\left (x^{4} - 26 \, x^{3}\right )} e^{\left (x + 2 \, e^{x}\right )} - 2 \, {\left (x^{2} - {\left (x^{3} - 26 \, x^{2}\right )} e^{x} - 26 \, x\right )} e^{\left (e^{x}\right )} - 2 \, {\left (x^{2} - {\left (x^{4} - 26 \, x^{3}\right )} e^{\left (x + e^{x}\right )} - 26 \, x\right )} e^{\left (e^{e}\right )} - 2 \, x + 52\right )} e^{\left (\frac {x^{2} e^{\left (2 \, e^{x}\right )} + x^{2} e^{\left (2 \, e^{e}\right )} + 2 \, x e^{\left (e^{x}\right )} + 2 \, {\left (x^{2} e^{\left (e^{x}\right )} + x\right )} e^{\left (e^{e}\right )} + 1}{x^{2}}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+52*x^3)*exp(x)*exp(exp(x))+2*x^2-52*x)*exp(exp(exp(1)))+(-2*x^4+52*x^3)*exp(x)*exp(exp(x))
^2+((-2*x^3+52*x^2)*exp(x)+2*x^2-52*x)*exp(exp(x))-x^3+2*x-52)*exp((x^2*exp(exp(exp(1)))^2+(2*exp(exp(x))*x^2+
2*x)*exp(exp(exp(1)))+x^2*exp(exp(x))^2+2*x*exp(exp(x))+1)/x^2)/x^3,x, algorithm="giac")

[Out]

integrate(-(x^3 + 2*(x^4 - 26*x^3)*e^(x + 2*e^x) - 2*(x^2 - (x^3 - 26*x^2)*e^x - 26*x)*e^(e^x) - 2*(x^2 - (x^4
- 26*x^3)*e^(x + e^x) - 26*x)*e^(e^e) - 2*x + 52)*e^((x^2*e^(2*e^x) + x^2*e^(2*e^e) + 2*x*e^(e^x) + 2*(x^2*e^
(e^x) + x)*e^(e^e) + 1)/x^2)/x^3, x)

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maple [B]  time = 0.12, size = 58, normalized size = 2.42

method	result	size
risch	\(\left (26-x \right ) {\mathrm e}^{\frac {2 x^{2} {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{{\mathrm e}}}+x^{2} {\mathrm e}^{2 \,{\mathrm e}^{x}}+x^{2} {\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}}}+2 x \,{\mathrm e}^{{\mathrm e}^{x}}+2 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}}} x +1}{x^{2}}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^4+52*x^3)*exp(x)*exp(exp(x))+2*x^2-52*x)*exp(exp(exp(1)))+(-2*x^4+52*x^3)*exp(x)*exp(exp(x))^2+((-
2*x^3+52*x^2)*exp(x)+2*x^2-52*x)*exp(exp(x))-x^3+2*x-52)*exp((x^2*exp(exp(exp(1)))^2+(2*exp(exp(x))*x^2+2*x)*e
xp(exp(exp(1)))+x^2*exp(exp(x))^2+2*x*exp(exp(x))+1)/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

(26-x)*exp((2*x^2*exp(exp(x)+exp(exp(1)))+x^2*exp(2*exp(x))+x^2*exp(2*exp(exp(1)))+2*x*exp(exp(x))+2*exp(exp(e
xp(1)))*x+1)/x^2)

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maxima [B]  time = 0.86, size = 57, normalized size = 2.38 \begin {gather*} -{\left (x e^{\left (e^{\left (2 \, e^{e}\right )}\right )} - 26 \, e^{\left (e^{\left (2 \, e^{e}\right )}\right )}\right )} e^{\left (\frac {2 \, e^{\left (e^{x}\right )}}{x} + \frac {2 \, e^{\left (e^{e}\right )}}{x} + \frac {1}{x^{2}} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + e^{e}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+52*x^3)*exp(x)*exp(exp(x))+2*x^2-52*x)*exp(exp(exp(1)))+(-2*x^4+52*x^3)*exp(x)*exp(exp(x))
^2+((-2*x^3+52*x^2)*exp(x)+2*x^2-52*x)*exp(exp(x))-x^3+2*x-52)*exp((x^2*exp(exp(exp(1)))^2+(2*exp(exp(x))*x^2+
2*x)*exp(exp(exp(1)))+x^2*exp(exp(x))^2+2*x*exp(exp(x))+1)/x^2)/x^3,x, algorithm="maxima")

[Out]

-(x*e^(e^(2*e^e)) - 26*e^(e^(2*e^e)))*e^(2*e^(e^x)/x + 2*e^(e^e)/x + 1/x^2 + e^(2*e^x) + 2*e^(e^x + e^e))

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mupad [B]  time = 1.42, size = 51, normalized size = 2.12 \begin {gather*} -{\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {1}{x^2}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}}{x}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{x}}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^{\mathrm {e}}}}\,\left (x-26\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x*exp(exp(x)) + exp(exp(exp(1)))*(2*x + 2*x^2*exp(exp(x))) + x^2*exp(2*exp(exp(1))) + x^2*exp(2*ex
p(x)) + 1)/x^2)*(2*x + exp(exp(x))*(exp(x)*(52*x^2 - 2*x^3) - 52*x + 2*x^2) + exp(exp(exp(1)))*(2*x^2 - 52*x +
exp(exp(x))*exp(x)*(52*x^3 - 2*x^4)) - x^3 + exp(2*exp(x))*exp(x)*(52*x^3 - 2*x^4) - 52))/x^3,x)

[Out]

-exp(exp(2*exp(x)))*exp(2*exp(exp(exp(1)))*exp(exp(x)))*exp(1/x^2)*exp((2*exp(exp(exp(1))))/x)*exp((2*exp(exp(
x)))/x)*exp(exp(2*exp(exp(1))))*(x - 26)

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sympy [B]  time = 20.16, size = 60, normalized size = 2.50 \begin {gather*} \left (26 - x\right ) e^{\frac {x^{2} e^{2 e^{x}} + x^{2} e^{2 e^{e}} + 2 x e^{e^{x}} + \left (2 x^{2} e^{e^{x}} + 2 x\right ) e^{e^{e}} + 1}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**4+52*x**3)*exp(x)*exp(exp(x))+2*x**2-52*x)*exp(exp(exp(1)))+(-2*x**4+52*x**3)*exp(x)*exp(ex
p(x))**2+((-2*x**3+52*x**2)*exp(x)+2*x**2-52*x)*exp(exp(x))-x**3+2*x-52)*exp((x**2*exp(exp(exp(1)))**2+(2*exp(
exp(x))*x**2+2*x)*exp(exp(exp(1)))+x**2*exp(exp(x))**2+2*x*exp(exp(x))+1)/x**2)/x**3,x)

[Out]

(26 - x)*exp((x**2*exp(2*exp(x)) + x**2*exp(2*exp(E)) + 2*x*exp(exp(x)) + (2*x**2*exp(exp(x)) + 2*x)*exp(exp(E
)) + 1)/x**2)

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