\(\int \frac {e^{39-e^3-x} (1+x)+(e^{39-e^3-x}+x) \log (\frac {x}{e^{39-e^3-x}+x})}{e^{39-e^3-x}+x} \, dx\) [7667]

Optimal result

Integrand size = 68, antiderivative size = 21 \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=x \log \left (\frac {x}{e^{39-e^3-x}+x}\right ) \]

[Out]

x*ln(x/(exp(-exp(3)-x+39)+x))

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6820, 6874, 2628, 12} \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=x \log \left (\frac {x}{x+e^{-x-e^3+39}}\right ) \]

[In]

Int[(E^(39 - E^3 - x)*(1 + x) + (E^(39 - E^3 - x) + x)*Log[x/(E^(39 - E^3 - x) + x)])/(E^(39 - E^3 - x) + x),x
]

[Out]

x*Log[x/(E^(39 - E^3 - x) + x)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2628

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{39} (1+x)}{e^{39}+e^{e^3+x} x}+\log \left (\frac {x}{e^{39-e^3-x}+x}\right )\right ) \, dx \\ & = e^{39} \int \frac {1+x}{e^{39}+e^{e^3+x} x} \, dx+\int \log \left (\frac {x}{e^{39-e^3-x}+x}\right ) \, dx \\ & = x \log \left (\frac {x}{e^{39-e^3-x}+x}\right )+e^{39} \int \left (\frac {1}{e^{39}+e^{e^3+x} x}+\frac {x}{e^{39}+e^{e^3+x} x}\right ) \, dx-\int \frac {e^{39} (1+x)}{e^{39}+e^{e^3+x} x} \, dx \\ & = x \log \left (\frac {x}{e^{39-e^3-x}+x}\right )+e^{39} \int \frac {1}{e^{39}+e^{e^3+x} x} \, dx+e^{39} \int \frac {x}{e^{39}+e^{e^3+x} x} \, dx-e^{39} \int \frac {1+x}{e^{39}+e^{e^3+x} x} \, dx \\ & = x \log \left (\frac {x}{e^{39-e^3-x}+x}\right )+e^{39} \int \frac {1}{e^{39}+e^{e^3+x} x} \, dx+e^{39} \int \frac {x}{e^{39}+e^{e^3+x} x} \, dx-e^{39} \int \left (\frac {1}{e^{39}+e^{e^3+x} x}+\frac {x}{e^{39}+e^{e^3+x} x}\right ) \, dx \\ & = x \log \left (\frac {x}{e^{39-e^3-x}+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=x \log \left (\frac {x}{e^{39-e^3-x}+x}\right ) \]

[In]

Integrate[(E^(39 - E^3 - x)*(1 + x) + (E^(39 - E^3 - x) + x)*Log[x/(E^(39 - E^3 - x) + x)])/(E^(39 - E^3 - x)
+ x),x]

[Out]

x*Log[x/(E^(39 - E^3 - x) + x)]

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
norman	\(x \ln \left (\frac {x}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right )\)	\(20\)
parallelrisch	\(x \ln \left (\frac {x}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right )\)	\(20\)
risch	\(-x \ln \left ({\mathrm e}^{-{\mathrm e}^{3}-x +39}+x \right )+x \ln \left (x \right )-\frac {i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right ) \operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right )}{2}+\frac {i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right )^{2}}{2}+\frac {i x \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right ) \operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right )^{2}}{2}-\frac {i x \pi \operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{-{\mathrm e}^{3}-x +39}+x}\right )^{3}}{2}\)	\(170\)

[In]

int(((exp(-exp(3)-x+39)+x)*ln(x/(exp(-exp(3)-x+39)+x))+(1+x)*exp(-exp(3)-x+39))/(exp(-exp(3)-x+39)+x),x,method
=_RETURNVERBOSE)

[Out]

x*ln(x/(exp(-exp(3)-x+39)+x))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=x \log \left (\frac {x}{x + e^{\left (-x - e^{3} + 39\right )}}\right ) \]

[In]

integrate(((exp(-exp(3)-x+39)+x)*log(x/(exp(-exp(3)-x+39)+x))+(1+x)*exp(-exp(3)-x+39))/(exp(-exp(3)-x+39)+x),x
, algorithm="fricas")

[Out]

x*log(x/(x + e^(-x - e^3 + 39)))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=x \log {\left (\frac {x}{x + e^{- x - e^{3} + 39}} \right )} \]

[In]

integrate(((exp(-exp(3)-x+39)+x)*ln(x/(exp(-exp(3)-x+39)+x))+(1+x)*exp(-exp(3)-x+39))/(exp(-exp(3)-x+39)+x),x)

[Out]

x*log(x/(x + exp(-x - exp(3) + 39)))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=x^{2} + x e^{3} - x \log \left (x e^{\left (x + e^{3}\right )} + e^{39}\right ) + x \log \left (x\right ) \]

[In]

integrate(((exp(-exp(3)-x+39)+x)*log(x/(exp(-exp(3)-x+39)+x))+(1+x)*exp(-exp(3)-x+39))/(exp(-exp(3)-x+39)+x),x
, algorithm="maxima")

[Out]

x^2 + x*e^3 - x*log(x*e^(x + e^3) + e^39) + x*log(x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=-x \log \left (x + e^{\left (-x - e^{3} + 39\right )}\right ) + x \log \left (x\right ) \]

[In]

integrate(((exp(-exp(3)-x+39)+x)*log(x/(exp(-exp(3)-x+39)+x))+(1+x)*exp(-exp(3)-x+39))/(exp(-exp(3)-x+39)+x),x
, algorithm="giac")

[Out]

-x*log(x + e^(-x - e^3 + 39)) + x*log(x)

Mupad [B] (verification not implemented)

Time = 13.67 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{39-e^3-x} (1+x)+\left (e^{39-e^3-x}+x\right ) \log \left (\frac {x}{e^{39-e^3-x}+x}\right )}{e^{39-e^3-x}+x} \, dx=x\,\ln \left (\frac {x}{x+{\mathrm {e}}^{-{\mathrm {e}}^3}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{39}}\right ) \]

[In]

int((log(x/(x + exp(39 - exp(3) - x)))*(x + exp(39 - exp(3) - x)) + exp(39 - exp(3) - x)*(x + 1))/(x + exp(39
- exp(3) - x)),x)

[Out]

x*log(x/(x + exp(-exp(3))*exp(-x)*exp(39)))