\(\int \frac {1}{4} e^{9+e^x} (1+e^x x) \, dx\) [7476]

Optimal result

Integrand size = 18, antiderivative size = 14 \[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=4+\frac {1}{4} e^{9+e^x} x \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2326} \[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=\frac {1}{4} e^{e^x+9} x \]

[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 2326

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=\frac {1}{4} e^{9+e^x} x \]

[In]

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

[Out]

(E^(9 + E^x)*x)/4

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=\frac {1}{4} \, x e^{\left (e^{x} + 9\right )} \]

[In]

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

[Out]

1/4*x*e^(e^x + 9)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=\frac {x e^{e^{x} + 9}}{4} \]

[In]

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

[Out]

x*exp(exp(x) + 9)/4

Maxima [F]

\[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=\int { \frac {1}{4} \, {\left (x e^{x} + 1\right )} e^{\left (e^{x} + 9\right )} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=\frac {1}{4} \, x e^{\left (e^{x} + 9\right )} \]

[In]

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

[Out]

1/4*x*e^(e^x + 9)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{4} e^{9+e^x} \left (1+e^x x\right ) \, dx=\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^9}{4} \]

[In]

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

[Out]

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