3.45 \(\int \frac {e^x x}{(1+x)^2} \, dx\)

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

[Out]

exp(x)/(1+x)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2197} \[ \frac {e^x}{x+1} \]

Antiderivative was successfully verified.

[In]

Int[(E^x*x)/(1 + x)^2,x]

[Out]

E^x/(1 + x)

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {align*} \int \frac {e^x x}{(1+x)^2} \, dx &=\frac {e^x}{1+x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 9, normalized size = 1.00 \[ \frac {e^x}{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x)/(1 + x)^2,x]

[Out]

E^x/(1 + x)

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 8, normalized size = 0.89 \[ \frac {e^{x}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x/(1+x)^2,x, algorithm="fricas")

[Out]

e^x/(x + 1)

________________________________________________________________________________________

giac [B]  time = 0.92, size = 30, normalized size = 3.33 \[ -\frac {e^{\left (-{\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )}\right )}}{{\left (x + 1\right )} {\left (\frac {1}{x + 1} - 1\right )} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x/(1+x)^2,x, algorithm="giac")

[Out]

-e^(-(x + 1)*(1/(x + 1) - 1))/((x + 1)*(1/(x + 1) - 1) - 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 9, normalized size = 1.00 \[ \frac {{\mathrm e}^{x}}{x +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*x/(x+1)^2,x)

[Out]

exp(x)/(x+1)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 8, normalized size = 0.89 \[ \frac {e^{x}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x/(1+x)^2,x, algorithm="maxima")

[Out]

e^x/(x + 1)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 8, normalized size = 0.89 \[ \frac {{\mathrm {e}}^x}{x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x))/(x + 1)^2,x)

[Out]

exp(x)/(x + 1)

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 5, normalized size = 0.56 \[ \frac {e^{x}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x/(1+x)**2,x)

[Out]

exp(x)/(x + 1)

________________________________________________________________________________________