3.4 \(\int \frac{e^{e^x+x} (1+e^x)}{e^x+x} \, dx\)

Optimal. Leaf size=6 \[ \text{ExpIntegralEi}\left (x+e^x\right ) \]

[Out]

ExpIntegralEi[E^x + x]

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Rubi [A]  time = 0.105787, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6707, 2178} \[ \text{ExpIntegralEi}\left (x+e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[E^x + x]

Rule 6707

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]

Rule 2178

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

Rubi steps

\begin{align*} \int \frac{e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx &=\operatorname{Subst}\left (\int \frac{e^x}{x} \, dx,x,e^x+x\right )\\ &=\text{Ei}\left (e^x+x\right )\\ \end{align*}

Mathematica [A]  time = 0.0357706, size = 6, normalized size = 1. \[ \text{ExpIntegralEi}\left (x+e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[E^x + x]

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Maple [B]  time = 0.004, size = 13, normalized size = 2.2 \begin{align*} -{\it Ei} \left ( 1,-{{\rm e}^{x}}-x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (e^{x} + 1\right )} e^{\left (e^{x}\right )}}{x + e^{x}} - \int \frac{{\left ({\left (x - 2\right )} e^{x} - 1\right )} e^{\left (e^{x}\right )}}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.4728, size = 18, normalized size = 3. \begin{align*}{\rm Ei}\left (x + e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(x + e^x)

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Sympy [A]  time = 2.00931, size = 5, normalized size = 0.83 \begin{align*} \operatorname{Ei}{\left (x + e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(x + exp(x))

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Giac [A]  time = 1.07683, size = 7, normalized size = 1.17 \begin{align*}{\rm Ei}\left (x + e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(x + e^x)