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

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

[Out]

Ei(x+exp(x))

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Rubi [A]
time = 0.06, 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 = {6839, 2209} \begin {gather*} \text {Ei}\left (x+e^x\right ) \end {gather*}

Antiderivative was successfully verified.

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

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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Mathics [A]
time = 2.60, size = 6, normalized size = 1.00 \begin {gather*} \text {ExpIntegralEi}\left [x+E^x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(Exp[x] + 1)*(Exp[Exp[x] + x]/(Exp[x] + x)),x]')

[Out]

ExpIntegralEi[x + E ^ x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(12\) vs. \(2(5)=10\).
time = 0.02, size = 13, normalized size = 2.17

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.31, size = 5, normalized size = 0.83 \begin {gather*} {\rm Ei}\left (x + e^{x}\right ) \end {gather*}

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 = 1.03, size = 5, normalized size = 0.83 \begin {gather*} \operatorname {Ei}{\left (x + e^{x} \right )} \end {gather*}

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 = 0.00, size = 5, normalized size = 0.83 \begin {gather*} \mathrm {Ei}\left (x+\mathrm {e}^{x}\right ) \end {gather*}

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(e^x + x)

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

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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