\(\int \frac {e^{e^x+x} (1+e^x)}{e^x+x} \, dx\) [4]

Optimal result

Integrand size = 20, antiderivative size = 6 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {ExpIntegralEi}\left (e^x+x\right ) \]

[Out]

Ei(exp(x)+x)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6839, 2209} \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {ExpIntegralEi}\left (x+e^x\right ) \]

[In]

Int[(E^(E^x + x)*(1 + E^x))/(E^x + x),x]

[Out]

ExpIntegralEi[E^x + x]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 6839

Int[(F_)^(v_)*(u_)*(w_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Dist[q, Subst[Int[x^m*F^x,
x], x, v], x] /;  !FalseQ[q]] /; FreeQ[{F, m}, x] && EqQ[w, v]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x+x\right ) \\ & = \operatorname {ExpIntegralEi}\left (e^x+x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {ExpIntegralEi}\left (e^x+x\right ) \]

[In]

Integrate[(E^(E^x + x)*(1 + E^x))/(E^x + x),x]

[Out]

ExpIntegralEi[E^x + x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(12\) vs. \(2(5)=10\).

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 2.17

method	result	size
derivativedivides	\(-\operatorname {Ei}_{1}\left (-{\mathrm e}^{x}-x \right )\)	\(13\)
default	\(-\operatorname {Ei}_{1}\left (-{\mathrm e}^{x}-x \right )\)	\(13\)
risch	\(-\operatorname {Ei}_{1}\left (-{\mathrm e}^{x}-x \right )\)	\(13\)

[In]

int((1+exp(x))*exp(exp(x)+x)/(exp(x)+x),x,method=_RETURNVERBOSE)

[Out]

-Ei(1,-exp(x)-x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx={\rm Ei}\left (x + e^{x}\right ) \]

[In]

integrate((1+exp(x))*exp(exp(x)+x)/(exp(x)+x),x, algorithm="fricas")

[Out]

Ei(x + e^x)

Sympy [A] (verification not implemented)

Time = 0.97 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {Ei}{\left (x + e^{x} \right )} \]

[In]

integrate((1+exp(x))*exp(exp(x)+x)/(exp(x)+x),x)

[Out]

Ei(x + exp(x))

Maxima [F]

\[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\int { \frac {{\left (e^{x} + 1\right )} e^{\left (x + e^{x}\right )}}{x + e^{x}} \,d x } \]

[In]

integrate((1+exp(x))*exp(exp(x)+x)/(exp(x)+x),x, algorithm="maxima")

[Out]

(e^x + 1)*e^(e^x)/(x + e^x) - integrate(((x - 2)*e^x - 1)*e^(e^x)/(x^2 + 2*x*e^x + e^(2*x)), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx={\rm Ei}\left (x + e^{x}\right ) \]

[In]

integrate((1+exp(x))*exp(exp(x)+x)/(exp(x)+x),x, algorithm="giac")

[Out]

Ei(x + e^x)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\mathrm {ei}\left (x+{\mathrm {e}}^x\right ) \]

[In]

int((exp(x + exp(x))*(exp(x) + 1))/(x + exp(x)),x)

[Out]

ei(x + exp(x))