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

Optimal. Leaf size=11 \[ 2 \log (x)-\log (x+1) \]

[Out]

2*Log[x] - Log[1 + x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0200133, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ 2 \log (x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/(x + x^2),x]

[Out]

2*Log[x] - Log[1 + x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.57101, size = 8, normalized size = 0.73 \[ 2 \log{\left (x \right )} - \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x) - log(x + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00316559, size = 11, normalized size = 1. \[ 2 \log (x)-\log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x)/(x + x^2),x]

[Out]

2*Log[x] - Log[1 + x]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 12, normalized size = 1.1 \[ 2\,\ln \left ( x \right ) -\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(x^2+x),x)

[Out]

2*ln(x)-ln(1+x)

_______________________________________________________________________________________

Maxima [A]  time = 1.33706, size = 15, normalized size = 1.36 \[ -\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 1) + 2*log(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.231548, size = 15, normalized size = 1.36 \[ -\log \left (x + 1\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 1) + 2*log(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.089173, size = 8, normalized size = 0.73 \[ 2 \log{\left (x \right )} - \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(x**2+x),x)

[Out]

2*log(x) - log(x + 1)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211365, size = 18, normalized size = 1.64 \[ -{\rm ln}\left ({\left | x + 1 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(x + 1)) + 2*ln(abs(x))