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

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

[Out]

(1 + x)^(-1) + Log[1 + x]

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Rubi [A]  time = 0.0111296, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{x+1}+\log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x)^2,x]

[Out]

(1 + x)^(-1) + Log[1 + x]

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Rubi in Sympy [A]  time = 1.06328, size = 8, normalized size = 0.8 \[ \log{\left (x + 1 \right )} + \frac{1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(1+x)**2,x)

[Out]

log(x + 1) + 1/(x + 1)

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Mathematica [A]  time = 0.00397579, size = 10, normalized size = 1. \[ \frac{1}{x+1}+\log (x+1) \]

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 = 0.004, size = 11, normalized size = 1.1 \[ \left ( 1+x \right ) ^{-1}+\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+x)^2,x)

[Out]

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

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Maxima [A]  time = 1.35402, size = 14, normalized size = 1.4 \[ \frac{1}{x + 1} + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x + 1) + log(x + 1)

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Fricas [A]  time = 0.190112, size = 22, normalized size = 2.2 \[ \frac{{\left (x + 1\right )} \log \left (x + 1\right ) + 1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((x + 1)*log(x + 1) + 1)/(x + 1)

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Sympy [A]  time = 0.063238, size = 8, normalized size = 0.8 \[ \log{\left (x + 1 \right )} + \frac{1}{x + 1} \]

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/XCAS [A]  time = 0.210926, size = 15, normalized size = 1.5 \[ \frac{1}{x + 1} +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x + 1) + ln(abs(x + 1))