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

Optimal. Leaf size=8 \[ \frac{x}{\log (x)+1} \]

[Out]

x/(1 + Log[x])

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Rubi [A]  time = 0.0632274, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{x}{\log (x)+1} \]

Antiderivative was successfully verified.

[In]  Int[Log[x]/(1 + Log[x])^2,x]

[Out]

x/(1 + Log[x])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\log{\left (x \right )}}{\left (\log{\left (x \right )} + 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x)/(1+ln(x))**2,x)

[Out]

Integral(log(x)/(log(x) + 1)**2, x)

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Mathematica [A]  time = 0.00512997, size = 8, normalized size = 1. \[ \frac{x}{\log (x)+1} \]

Antiderivative was successfully verified.

[In]  Integrate[Log[x]/(1 + Log[x])^2,x]

[Out]

x/(1 + Log[x])

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Maple [A]  time = 0.051, size = 9, normalized size = 1.1 \[{\frac{x}{1+\ln \left ( x \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x)/(1+ln(x))^2,x)

[Out]

x/(1+ln(x))

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Maxima [A]  time = 1.45072, size = 11, normalized size = 1.38 \[ \frac{x}{\log \left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(log(x) + 1)

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Fricas [A]  time = 0.21709, size = 11, normalized size = 1.38 \[ \frac{x}{\log \left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(log(x) + 1)

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Sympy [A]  time = 0.077185, size = 5, normalized size = 0.62 \[ \frac{x}{\log{\left (x \right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x)/(1+ln(x))**2,x)

[Out]

x/(log(x) + 1)

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GIAC/XCAS [A]  time = 0.227057, size = 11, normalized size = 1.38 \[ \frac{x}{{\rm ln}\left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(ln(x) + 1)