3.75.76 \(\int \frac {1}{4} e^{9+e^x} (1+e^x x) \, dx\)

Optimal. Leaf size=14 \[ 4+\frac {1}{4} e^{9+e^x} x \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 12, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2288} \begin {gather*} \frac {1}{4} e^{e^x+9} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(9 + E^x)*(1 + E^x*x))/4,x]

[Out]

(E^(9 + E^x)*x)/4

Rule 12

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int e^{9+e^x} \left (1+e^x x\right ) \, dx\\ &=\frac {1}{4} e^{9+e^x} x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{4} e^{9+e^x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(9 + E^x)*(1 + E^x*x))/4,x]

[Out]

(E^(9 + E^x)*x)/4

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 8, normalized size = 0.57 \begin {gather*} \frac {1}{4} \, x e^{\left (e^{x} + 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(x)*x+1)*exp(9+exp(x)),x, algorithm="fricas")

[Out]

1/4*x*e^(e^x + 9)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 8, normalized size = 0.57 \begin {gather*} \frac {1}{4} \, x e^{\left (e^{x} + 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(x)*x+1)*exp(9+exp(x)),x, algorithm="giac")

[Out]

1/4*x*e^(e^x + 9)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 9, normalized size = 0.64

method	result	size
norman	\(\frac {{\mathrm e}^{9+{\mathrm e}^{x}} x}{4}\)	\(9\)
risch	\(\frac {{\mathrm e}^{9+{\mathrm e}^{x}} x}{4}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(exp(x)*x+1)*exp(9+exp(x)),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(9+exp(x))*x

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (e^{x}\right ) e^{9} + \frac {1}{4} \, x e^{\left (e^{x} + 9\right )} - \frac {1}{4} \, \int e^{\left (e^{x} + 9\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(x)*x+1)*exp(9+exp(x)),x, algorithm="maxima")

[Out]

1/4*Ei(e^x)*e^9 + 1/4*x*e^(e^x + 9) - 1/4*integrate(e^(e^x + 9), x)

________________________________________________________________________________________

mupad [B]  time = 4.39, size = 8, normalized size = 0.57 \begin {gather*} \frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^9}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x) + 9)*(x*exp(x) + 1))/4,x)

[Out]

(x*exp(exp(x))*exp(9))/4

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 8, normalized size = 0.57 \begin {gather*} \frac {x e^{e^{x} + 9}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(x)*x+1)*exp(9+exp(x)),x)

[Out]

x*exp(exp(x) + 9)/4

________________________________________________________________________________________