3.1.45 \(\int \frac {e^x x}{(1+x)^2} \, dx\) [45]

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

[Out]

exp(x)/(1+x)

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Rubi [A]
time = 0.01, 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 = {2228} \begin {gather*} \frac {e^x}{x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*x)/(1 + x)^2,x]

[Out]

E^x/(1 + x)

Rule 2228

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*}

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Mathematica [A]
time = 0.03, size = 9, normalized size = 1.00 \begin {gather*} \frac {e^x}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x)/(1 + x)^2,x]

[Out]

E^x/(1 + x)

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Mathics [A]
time = 1.69, size = 9, normalized size = 1.00 \begin {gather*} \frac {E^x}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(x/(x + 1)^2)*E^x,x]')

[Out]

E ^ x / (1 + x)

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Maple [A]
time = 0.03, size = 9, normalized size = 1.00

method	result	size
gosper	\(\frac {{\mathrm e}^{x}}{1+x}\)	\(9\)
default	\(\frac {{\mathrm e}^{x}}{1+x}\)	\(9\)
norman	\(\frac {{\mathrm e}^{x}}{1+x}\)	\(9\)
risch	\(\frac {{\mathrm e}^{x}}{1+x}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)/(1+x)

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Maxima [A]
time = 0.26, size = 8, normalized size = 0.89 \begin {gather*} \frac {e^{x}}{x + 1} \end {gather*}

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 = 0.31, size = 8, normalized size = 0.89 \begin {gather*} \frac {e^{x}}{x + 1} \end {gather*}

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 = 0.04, size = 5, normalized size = 0.56 \begin {gather*} \frac {e^{x}}{x + 1} \end {gather*}

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 [A]
time = 0.00, size = 7, normalized size = 0.78 \begin {gather*} \frac {\mathrm {e}^{x}}{x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x/(x + 1)

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Mupad [B]
time = 0.09, size = 8, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^x}{x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x))/(x + 1)^2,x)

[Out]

exp(x)/(x + 1)

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