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

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

[Out]

Ei(x+exp(x))

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Rubi [A]  time = 0.11, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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

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]

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*}

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Mathematica [A]  time = 0.03, size = 6, normalized size = 1.00 \[ \operatorname {ExpIntegralEi}\left (x+e^x\right ) \]

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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fricas [A]  time = 0.41, size = 5, normalized size = 0.83 \[ {\rm Ei}\left (x + e^{x}\right ) \]

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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giac [A]  time = 1.06, size = 5, normalized size = 0.83 \[ {\rm Ei}\left (x + e^{x}\right ) \]

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)

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maple [B]  time = 0.01, size = 13, normalized size = 2.17 \[ -\Ei \left (1, -x -{\mathrm e}^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e^{x} + 1\right )} e^{\left (e^{x}\right )}}{x + e^{x}} - \int \frac {{\left ({\left (x - 2\right )} e^{x} - 1\right )} e^{\left (e^{x}\right )}}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}}\,{d x} \]

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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mupad [B]  time = 0.05, size = 5, normalized size = 0.83 \[ \mathrm {ei}\left (x+{\mathrm {e}}^x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ei(x + exp(x))

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sympy [A]  time = 2.12, size = 5, normalized size = 0.83 \[ \operatorname {Ei}{\left (x + e^{x} \right )} \]

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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