3.11.43 \(\int e^{x+e^{90+10 e^2} x+2 e^{45+5 e^2} x \log (e^{-x} x)+x \log ^2(e^{-x} x)} (1+e^{90+10 e^2}+e^{45+5 e^2} (2-2 x)+(2+2 e^{45+5 e^2}-2 x) \log (e^{-x} x)+\log ^2(e^{-x} x)) \, dx\)

Optimal. Leaf size=26 \[ e^{x+x \left (e^{5 \left (9+e^2\right )}+\log \left (e^{-x} x\right )\right )^2} \]

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Rubi [A]  time = 2.56, antiderivative size = 48, normalized size of antiderivative = 1.85, number of steps used = 1, number of rules used = 1, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {6706} \begin {gather*} \left (e^{-x} x\right )^{2 e^{45+5 e^2} x} e^{e^{10 \left (9+e^2\right )} x+x+x \log ^2\left (e^{-x} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(x + E^(90 + 10*E^2)*x + 2*E^(45 + 5*E^2)*x*Log[x/E^x] + x*Log[x/E^x]^2)*(1 + E^(90 + 10*E^2) + E^(45 +
5*E^2)*(2 - 2*x) + (2 + 2*E^(45 + 5*E^2) - 2*x)*Log[x/E^x] + Log[x/E^x]^2),x]

[Out]

E^(x + E^(10*(9 + E^2))*x + x*Log[x/E^x]^2)*(x/E^x)^(2*E^(45 + 5*E^2)*x)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{x+e^{10 \left (9+e^2\right )} x+x \log ^2\left (e^{-x} x\right )} \left (e^{-x} x\right )^{2 e^{45+5 e^2} x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 44, normalized size = 1.69 \begin {gather*} e^{x \left (1+e^{10 \left (9+e^2\right )}+2 e^{5 \left (9+e^2\right )} \log \left (e^{-x} x\right )+\log ^2\left (e^{-x} x\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(x + E^(90 + 10*E^2)*x + 2*E^(45 + 5*E^2)*x*Log[x/E^x] + x*Log[x/E^x]^2)*(1 + E^(90 + 10*E^2) + E^
(45 + 5*E^2)*(2 - 2*x) + (2 + 2*E^(45 + 5*E^2) - 2*x)*Log[x/E^x] + Log[x/E^x]^2),x]

[Out]

E^(x*(1 + E^(10*(9 + E^2)) + 2*E^(5*(9 + E^2))*Log[x/E^x] + Log[x/E^x]^2))

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fricas [A]  time = 0.93, size = 40, normalized size = 1.54 \begin {gather*} e^{\left (2 \, x e^{\left (5 \, e^{2} + 45\right )} \log \left (x e^{\left (-x\right )}\right ) + x \log \left (x e^{\left (-x\right )}\right )^{2} + x e^{\left (10 \, e^{2} + 90\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x/exp(x))^2+(2*exp(5*exp(1)^2+45)-2*x+2)*log(x/exp(x))+exp(5*exp(1)^2+45)^2+(-2*x+2)*exp(5*exp(
1)^2+45)+1)*exp(x*log(x/exp(x))^2+2*x*exp(5*exp(1)^2+45)*log(x/exp(x))+x*exp(5*exp(1)^2+45)^2+x),x, algorithm=
"fricas")

[Out]

e^(2*x*e^(5*e^2 + 45)*log(x*e^(-x)) + x*log(x*e^(-x))^2 + x*e^(10*e^2 + 90) + x)

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giac [A]  time = 1.06, size = 40, normalized size = 1.54 \begin {gather*} e^{\left (2 \, x e^{\left (5 \, e^{2} + 45\right )} \log \left (x e^{\left (-x\right )}\right ) + x \log \left (x e^{\left (-x\right )}\right )^{2} + x e^{\left (10 \, e^{2} + 90\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x/exp(x))^2+(2*exp(5*exp(1)^2+45)-2*x+2)*log(x/exp(x))+exp(5*exp(1)^2+45)^2+(-2*x+2)*exp(5*exp(
1)^2+45)+1)*exp(x*log(x/exp(x))^2+2*x*exp(5*exp(1)^2+45)*log(x/exp(x))+x*exp(5*exp(1)^2+45)^2+x),x, algorithm=
"giac")

[Out]

e^(2*x*e^(5*e^2 + 45)*log(x*e^(-x)) + x*log(x*e^(-x))^2 + x*e^(10*e^2 + 90) + x)

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maple [C]  time = 0.20, size = 557, normalized size = 21.42

method	result	size
risch	\(x^{i \pi \,\mathrm {csgn}\left (i x \right ) x} x^{i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x} \left ({\mathrm e}^{x}\right )^{-i \pi \,\mathrm {csgn}\left (i x \right ) x} \left ({\mathrm e}^{x}\right )^{-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x} x^{2 x \,{\mathrm e}^{5 \,{\mathrm e}^{2}+45}} \left ({\mathrm e}^{x}\right )^{-2 x \ln \relax (x )} \left ({\mathrm e}^{x}\right )^{-2 x \,{\mathrm e}^{5 \,{\mathrm e}^{2}+45}} \left ({\mathrm e}^{x}\right )^{i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x} x^{-i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x} \left ({\mathrm e}^{x}\right )^{i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) x} x^{-i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) x} {\mathrm e}^{\frac {x \left (-\pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{6}+2 \pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{5} \mathrm {csgn}\left (i x \right )+2 \pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{5} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )-\pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{4} \mathrm {csgn}\left (i x \right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )-\pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{4} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i x \right )-\pi ^{2} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )^{2}-4 i {\mathrm e}^{5 \,{\mathrm e}^{2}+45} \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}+4 i {\mathrm e}^{5 \,{\mathrm e}^{2}+45} \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i x \right )+4 i {\mathrm e}^{5 \,{\mathrm e}^{2}+45} \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )-4 i {\mathrm e}^{5 \,{\mathrm e}^{2}+45} \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+4 \ln \relax (x )^{2}+4 \ln \left ({\mathrm e}^{x}\right )^{2}+4 \,{\mathrm e}^{10 \,{\mathrm e}^{2}+90}+4\right )}{4}}\)	\(557\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x/exp(x))^2+(2*exp(5*exp(1)^2+45)-2*x+2)*ln(x/exp(x))+exp(5*exp(1)^2+45)^2+(-2*x+2)*exp(5*exp(1)^2+45)
+1)*exp(x*ln(x/exp(x))^2+2*x*exp(5*exp(1)^2+45)*ln(x/exp(x))+x*exp(5*exp(1)^2+45)^2+x),x,method=_RETURNVERBOSE
)

[Out]

x^(I*Pi*csgn(I*x)*x)*x^(I*Pi*csgn(I*exp(-x))*x)*exp(x)^(-I*Pi*csgn(I*x)*x)*exp(x)^(-I*Pi*csgn(I*exp(-x))*x)*x^
(2*x*exp(5*exp(2)+45))*exp(x)^(-2*x*ln(x))*exp(x)^(-2*x*exp(5*exp(2)+45))*exp(x)^(I*Pi*csgn(I*x*exp(-x))*csgn(
I*x)*csgn(I*exp(-x))*x)*x^(-I*Pi*csgn(I*x*exp(-x))*csgn(I*x)*csgn(I*exp(-x))*x)*exp(x)^(I*Pi*csgn(I*x*exp(-x))
*x)*x^(-I*Pi*csgn(I*x*exp(-x))*x)*exp(1/4*x*(-Pi^2*csgn(I*x*exp(-x))^6+2*Pi^2*csgn(I*x*exp(-x))^5*csgn(I*x)+2*
Pi^2*csgn(I*x*exp(-x))^5*csgn(I*exp(-x))-Pi^2*csgn(I*x*exp(-x))^4*csgn(I*x)^2-4*Pi^2*csgn(I*x*exp(-x))^4*csgn(
I*x)*csgn(I*exp(-x))-Pi^2*csgn(I*x*exp(-x))^4*csgn(I*exp(-x))^2+2*Pi^2*csgn(I*x*exp(-x))^3*csgn(I*x)^2*csgn(I*
exp(-x))+2*Pi^2*csgn(I*x*exp(-x))^3*csgn(I*exp(-x))^2*csgn(I*x)-Pi^2*csgn(I*x*exp(-x))^2*csgn(I*x)^2*csgn(I*ex
p(-x))^2-4*I*exp(5*exp(2)+45)*Pi*csgn(I*x*exp(-x))^3+4*I*exp(5*exp(2)+45)*Pi*csgn(I*x*exp(-x))^2*csgn(I*x)+4*I
*exp(5*exp(2)+45)*Pi*csgn(I*x*exp(-x))^2*csgn(I*exp(-x))-4*I*exp(5*exp(2)+45)*Pi*csgn(I*x*exp(-x))*csgn(I*x)*c
sgn(I*exp(-x))+4*ln(x)^2+4*ln(exp(x))^2+4*exp(10*exp(2)+90)+4))

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maxima [B]  time = 0.59, size = 52, normalized size = 2.00 \begin {gather*} e^{\left (x^{3} - 2 \, x^{2} e^{\left (5 \, e^{2} + 45\right )} - 2 \, x^{2} \log \relax (x) + 2 \, x e^{\left (5 \, e^{2} + 45\right )} \log \relax (x) + x \log \relax (x)^{2} + x e^{\left (10 \, e^{2} + 90\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x/exp(x))^2+(2*exp(5*exp(1)^2+45)-2*x+2)*log(x/exp(x))+exp(5*exp(1)^2+45)^2+(-2*x+2)*exp(5*exp(
1)^2+45)+1)*exp(x*log(x/exp(x))^2+2*x*exp(5*exp(1)^2+45)*log(x/exp(x))+x*exp(5*exp(1)^2+45)^2+x),x, algorithm=
"maxima")

[Out]

e^(x^3 - 2*x^2*e^(5*e^2 + 45) - 2*x^2*log(x) + 2*x*e^(5*e^2 + 45)*log(x) + x*log(x)^2 + x*e^(10*e^2 + 90) + x)

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mupad [B]  time = 1.28, size = 55, normalized size = 2.12 \begin {gather*} x^{2\,x\,{\mathrm {e}}^{5\,{\mathrm {e}}^2}\,{\mathrm {e}}^{45}-2\,x^2}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{5\,{\mathrm {e}}^2}\,{\mathrm {e}}^{45}}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{x\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{10\,{\mathrm {e}}^2}\,{\mathrm {e}}^{90}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + x*exp(10*exp(2) + 90) + x*log(x*exp(-x))^2 + 2*x*exp(5*exp(2) + 45)*log(x*exp(-x)))*(exp(10*exp(2)
+ 90) + log(x*exp(-x))*(2*exp(5*exp(2) + 45) - 2*x + 2) - exp(5*exp(2) + 45)*(2*x - 2) + log(x*exp(-x))^2 + 1
),x)

[Out]

x^(2*x*exp(5*exp(2))*exp(45) - 2*x^2)*exp(-2*x^2*exp(5*exp(2))*exp(45))*exp(x^3)*exp(x*log(x)^2)*exp(x*exp(10*
exp(2))*exp(90))*exp(x)

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sympy [B]  time = 97.46, size = 41, normalized size = 1.58 \begin {gather*} e^{x \log {\left (x e^{- x} \right )}^{2} + 2 x e^{5 e^{2} + 45} \log {\left (x e^{- x} \right )} + x + x e^{10 e^{2} + 90}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x/exp(x))**2+(2*exp(5*exp(1)**2+45)-2*x+2)*ln(x/exp(x))+exp(5*exp(1)**2+45)**2+(-2*x+2)*exp(5*ex
p(1)**2+45)+1)*exp(x*ln(x/exp(x))**2+2*x*exp(5*exp(1)**2+45)*ln(x/exp(x))+x*exp(5*exp(1)**2+45)**2+x),x)

[Out]

exp(x*log(x*exp(-x))**2 + 2*x*exp(5*exp(2) + 45)*log(x*exp(-x)) + x + x*exp(10*exp(2) + 90))

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