3.3.62 \(\int \frac {x}{(1+x)^2} \, dx\) [262]

Optimal. Leaf size=10 \[ \frac {1}{1+x}+\log (1+x) \]

[Out]

1/(1+x)+ln(1+x)

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {45} \begin {gather*} \frac {1}{x+1}+\log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(1 + x)^2,x]

[Out]

(1 + x)^(-1) + Log[1 + x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{1+x}+\log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x)^2,x]

[Out]

(1 + x)^(-1) + Log[1 + x]

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Maple [A]
time = 0.03, size = 11, normalized size = 1.10

method	result	size
default	\(\frac {1}{1+x}+\ln \left (1+x \right )\)	\(11\)
norman	\(\frac {1}{1+x}+\ln \left (1+x \right )\)	\(11\)
risch	\(\frac {1}{1+x}+\ln \left (1+x \right )\)	\(11\)
meijerg	\(-\frac {x}{1+x}+\ln \left (1+x \right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(1+x)+ln(1+x)

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Maxima [A]
time = 1.82, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{x + 1} + \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + 1) + log(x + 1)

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Fricas [A]
time = 0.58, size = 16, normalized size = 1.60 \begin {gather*} \frac {{\left (x + 1\right )} \log \left (x + 1\right ) + 1}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x + 1)*log(x + 1) + 1)/(x + 1)

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Sympy [A]
time = 0.02, size = 8, normalized size = 0.80 \begin {gather*} \log {\left (x + 1 \right )} + \frac {1}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 1) + 1/(x + 1)

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Giac [A]
time = 0.45, size = 11, normalized size = 1.10 \begin {gather*} \frac {1}{x + 1} + \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x + 1) + log(abs(x + 1))

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Mupad [B]
time = 0.03, size = 10, normalized size = 1.00 \begin {gather*} \ln \left (x+1\right )+\frac {1}{x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x + 1)^2,x)

[Out]

log(x + 1) + 1/(x + 1)

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