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

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

[Out]

E^x/(1 + x)

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Rubi [A]  time = 0.0256603, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 = 0.0229454, size = 9, normalized size = 1. \[ \frac{e^x}{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x)/(1 + x)^2,x]

[Out]

E^x/(1 + x)

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Maple [A]  time = 0.002, size = 9, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{x}}}{1+x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)/(1+x)

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Maxima [A]  time = 0.942497, size = 11, normalized size = 1.22 \begin{align*} \frac{e^{x}}{x + 1} \end{align*}

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)

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Fricas [A]  time = 1.74625, size = 18, normalized size = 2. \begin{align*} \frac{e^{x}}{x + 1} \end{align*}

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)

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Sympy [A]  time = 0.085104, size = 5, normalized size = 0.56 \begin{align*} \frac{e^{x}}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)/(x + 1)

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Giac [B]  time = 1.10392, size = 26, normalized size = 2.89 \begin{align*} \frac{e^{\left (-{\left (x + 1\right )}{\left (\frac{1}{x + 1} - 1\right )}\right )}}{x + 1} \end{align*}

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)