3.102.97 \(\int \frac {e^{400+4 e^{6+2 e^{e^x} x}+\log ^2(x)+e^{3+e^{e^x} x} (-80+4 \log (x))} (-40+e^{6+e^x+2 e^{e^x} x} (8 x+8 e^x x^2)+2 \log (x)+e^{3+e^{e^x} x} (4+e^{e^x} (-80 x-80 e^x x^2+(4 x+4 e^x x^2) \log (x))))}{x^{41}} \, dx\)

Optimal. Leaf size=22 \[ e^{\left (2 \left (-10+e^{3+e^{e^x} x}\right )+\log (x)\right )^2} \]

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Rubi [B]  time = 3.22, antiderivative size = 209, normalized size of antiderivative = 9.50, number of steps used = 1, number of rules used = 1, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2288} \begin {gather*} \frac {\left (4 e^{2 e^{e^x} x+e^x+6} \left (e^x x^2+x\right )+2 e^{e^{e^x} x+3} \left (1-e^{e^x} \left (20 e^x x^2-\left (e^x x^2+x\right ) \log (x)+20 x\right )\right )+\log (x)\right ) \exp \left (4 e^{2 e^{e^x} x+6}+\log ^2(x)-4 e^{e^{e^x} x+3} (20-\log (x))+400\right )}{x^{41} \left (4 e^{2 e^{e^x} x+6} \left (e^{x+e^x} x+e^{e^x}\right )+\frac {2 e^{e^{e^x} x+3}}{x}-2 e^{e^{e^x} x+3} \left (e^{x+e^x} x+e^{e^x}\right ) (20-\log (x))+\frac {\log (x)}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(400 + 4*E^(6 + 2*E^E^x*x) + Log[x]^2 + E^(3 + E^E^x*x)*(-80 + 4*Log[x]))*(-40 + E^(6 + E^x + 2*E^E^x*x
)*(8*x + 8*E^x*x^2) + 2*Log[x] + E^(3 + E^E^x*x)*(4 + E^E^x*(-80*x - 80*E^x*x^2 + (4*x + 4*E^x*x^2)*Log[x]))))
/x^41,x]

[Out]

(E^(400 + 4*E^(6 + 2*E^E^x*x) - 4*E^(3 + E^E^x*x)*(20 - Log[x]) + Log[x]^2)*(4*E^(6 + E^x + 2*E^E^x*x)*(x + E^
x*x^2) + Log[x] + 2*E^(3 + E^E^x*x)*(1 - E^E^x*(20*x + 20*E^x*x^2 - (x + E^x*x^2)*Log[x]))))/(x^41*((2*E^(3 +
E^E^x*x))/x + 4*E^(6 + 2*E^E^x*x)*(E^E^x + E^(E^x + x)*x) - 2*E^(3 + E^E^x*x)*(E^E^x + E^(E^x + x)*x)*(20 - Lo
g[x]) + Log[x]/x))

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 (400+4 e^{6+2 e^{e^x} x}-4 e^{3+e^{e^x} x} (20-\log (x))+\log ^2(x)\right ) \left (4 e^{6+e^x+2 e^{e^x} x} \left (x+e^x x^2\right )+\log (x)+2 e^{3+e^{e^x} x} \left (1-e^{e^x} \left (20 x+20 e^x x^2-\left (x+e^x x^2\right ) \log (x)\right )\right )\right )}{x^{41} \left (\frac {2 e^{3+e^{e^x} x}}{x}+4 e^{6+2 e^{e^x} x} \left (e^{e^x}+e^{e^x+x} x\right )-2 e^{3+e^{e^x} x} \left (e^{e^x}+e^{e^x+x} x\right ) (20-\log (x))+\frac {\log (x)}{x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 42, normalized size = 1.91 \begin {gather*} e^{4 \left (-10+e^{3+e^{e^x} x}\right )^2+\log ^2(x)} x^{-40+4 e^{3+e^{e^x} x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(400 + 4*E^(6 + 2*E^E^x*x) + Log[x]^2 + E^(3 + E^E^x*x)*(-80 + 4*Log[x]))*(-40 + E^(6 + E^x + 2*E
^E^x*x)*(8*x + 8*E^x*x^2) + 2*Log[x] + E^(3 + E^E^x*x)*(4 + E^E^x*(-80*x - 80*E^x*x^2 + (4*x + 4*E^x*x^2)*Log[
x]))))/x^41,x]

[Out]

E^(4*(-10 + E^(3 + E^E^x*x))^2 + Log[x]^2)*x^(-40 + 4*E^(3 + E^E^x*x))

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fricas [B]  time = 0.66, size = 36, normalized size = 1.64 \begin {gather*} e^{\left (4 \, {\left (\log \relax (x) - 20\right )} e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + \log \relax (x)^{2} + 4 \, e^{\left (2 \, x e^{\left (e^{x}\right )} + 6\right )} - 40 \, \log \relax (x) + 400\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp(x)*x^2+4*x)*log(x)-80*exp(x)*x^2-80*
x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3)+2*log(x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*log(x)-80)*exp(x*exp(exp(x
))+3)+log(x)^2-40*log(x)+400)/x,x, algorithm="fricas")

[Out]

e^(4*(log(x) - 20)*e^(x*e^(e^x) + 3) + log(x)^2 + 4*e^(2*x*e^(e^x) + 6) - 40*log(x) + 400)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (4 \, {\left (x^{2} e^{x} + x\right )} e^{\left (2 \, x e^{\left (e^{x}\right )} + e^{x} + 6\right )} - 2 \, {\left ({\left (20 \, x^{2} e^{x} - {\left (x^{2} e^{x} + x\right )} \log \relax (x) + 20 \, x\right )} e^{\left (e^{x}\right )} - 1\right )} e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + \log \relax (x) - 20\right )} e^{\left (4 \, {\left (\log \relax (x) - 20\right )} e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + \log \relax (x)^{2} + 4 \, e^{\left (2 \, x e^{\left (e^{x}\right )} + 6\right )} - 40 \, \log \relax (x) + 400\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp(x)*x^2+4*x)*log(x)-80*exp(x)*x^2-80*
x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3)+2*log(x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*log(x)-80)*exp(x*exp(exp(x
))+3)+log(x)^2-40*log(x)+400)/x,x, algorithm="giac")

[Out]

integrate(2*(4*(x^2*e^x + x)*e^(2*x*e^(e^x) + e^x + 6) - 2*((20*x^2*e^x - (x^2*e^x + x)*log(x) + 20*x)*e^(e^x)
- 1)*e^(x*e^(e^x) + 3) + log(x) - 20)*e^(4*(log(x) - 20)*e^(x*e^(e^x) + 3) + log(x)^2 + 4*e^(2*x*e^(e^x) + 6)
- 40*log(x) + 400)/x, x)

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maple [B]  time = 0.13, size = 45, normalized size = 2.05

method	result	size
risch	\(\frac {x^{4 \,{\mathrm e}^{x \,{\mathrm e}^{{\mathrm e}^{x}}+3}} {\mathrm e}^{\ln \relax (x )^{2}+400+4 \,{\mathrm e}^{2 x \,{\mathrm e}^{{\mathrm e}^{x}}+6}-80 \,{\mathrm e}^{x \,{\mathrm e}^{{\mathrm e}^{x}}+3}}}{x^{40}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp(x)*x^2+4*x)*ln(x)-80*exp(x)*x^2-80*x)*exp(
exp(x))+4)*exp(x*exp(exp(x))+3)+2*ln(x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*ln(x)-80)*exp(x*exp(exp(x))+3)+ln(
x)^2-40*ln(x)+400)/x,x,method=_RETURNVERBOSE)

[Out]

1/x^40*x^(4*exp(x*exp(exp(x))+3))*exp(ln(x)^2+400+4*exp(2*x*exp(exp(x))+6)-80*exp(x*exp(exp(x))+3))

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maxima [B]  time = 0.64, size = 44, normalized size = 2.00 \begin {gather*} \frac {e^{\left (4 \, e^{\left (x e^{\left (e^{x}\right )} + 3\right )} \log \relax (x) + \log \relax (x)^{2} + 4 \, e^{\left (2 \, x e^{\left (e^{x}\right )} + 6\right )} - 80 \, e^{\left (x e^{\left (e^{x}\right )} + 3\right )} + 400\right )}}{x^{40}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)*x^2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)^2+(((4*exp(x)*x^2+4*x)*log(x)-80*exp(x)*x^2-80*
x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3)+2*log(x)-40)*exp(4*exp(x*exp(exp(x))+3)^2+(4*log(x)-80)*exp(x*exp(exp(x
))+3)+log(x)^2-40*log(x)+400)/x,x, algorithm="maxima")

[Out]

e^(4*e^(x*e^(e^x) + 3)*log(x) + log(x)^2 + 4*e^(2*x*e^(e^x) + 6) - 80*e^(x*e^(e^x) + 3) + 400)/x^40

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mupad [B]  time = 7.37, size = 46, normalized size = 2.09 \begin {gather*} \frac {x^{4\,{\mathrm {e}}^3\,{\mathrm {e}}^{x\,{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^{400}\,{\mathrm {e}}^{{\ln \relax (x)}^2}\,{\mathrm {e}}^{4\,{\mathrm {e}}^6\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^{-80\,{\mathrm {e}}^3\,{\mathrm {e}}^{x\,{\mathrm {e}}^{{\mathrm {e}}^x}}}}{x^{40}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*exp(2*x*exp(exp(x)) + 6) - 40*log(x) + log(x)^2 + exp(x*exp(exp(x)) + 3)*(4*log(x) - 80) + 400)*(2*
log(x) - exp(x*exp(exp(x)) + 3)*(exp(exp(x))*(80*x + 80*x^2*exp(x) - log(x)*(4*x + 4*x^2*exp(x))) - 4) + exp(e
xp(x))*exp(2*x*exp(exp(x)) + 6)*(8*x + 8*x^2*exp(x)) - 40))/x,x)

[Out]

(x^(4*exp(3)*exp(x*exp(exp(x))))*exp(400)*exp(log(x)^2)*exp(4*exp(6)*exp(2*x*exp(exp(x))))*exp(-80*exp(3)*exp(
x*exp(exp(x)))))/x^40

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sympy [B]  time = 25.63, size = 41, normalized size = 1.86 \begin {gather*} \frac {e^{\left (4 \log {\relax (x )} - 80\right ) e^{x e^{e^{x}} + 3} + 4 e^{2 x e^{e^{x}} + 6} + \log {\relax (x )}^{2} + 400}}{x^{40}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)*x**2+8*x)*exp(exp(x))*exp(x*exp(exp(x))+3)**2+(((4*exp(x)*x**2+4*x)*ln(x)-80*exp(x)*x**2-
80*x)*exp(exp(x))+4)*exp(x*exp(exp(x))+3)+2*ln(x)-40)*exp(4*exp(x*exp(exp(x))+3)**2+(4*ln(x)-80)*exp(x*exp(exp
(x))+3)+ln(x)**2-40*ln(x)+400)/x,x)

[Out]

exp((4*log(x) - 80)*exp(x*exp(exp(x)) + 3) + 4*exp(2*x*exp(exp(x)) + 6) + log(x)**2 + 400)/x**40

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