3.2244 \(\int \frac{x}{5+2 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]

[Out]

-ArcTan[(1 + x)/2]/2 + Log[5 + 2*x + x^2]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0329877, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(5 + 2*x + x^2),x]

[Out]

-ArcTan[(1 + x)/2]/2 + Log[5 + 2*x + x^2]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.10193, size = 20, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 2 x + 5 \right )}}{2} - \frac{\operatorname{atan}{\left (\frac{x}{2} + \frac{1}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**2+2*x+5),x)

[Out]

log(x**2 + 2*x + 5)/2 - atan(x/2 + 1/2)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0064435, size = 26, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(5 + 2*x + x^2),x]

[Out]

-ArcTan[(1 + x)/2]/2 + Log[5 + 2*x + x^2]/2

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 21, normalized size = 0.8 \[ -{\frac{1}{2}\arctan \left ({\frac{1}{2}}+{\frac{x}{2}} \right ) }+{\frac{\ln \left ({x}^{2}+2\,x+5 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+2*x+5),x)

[Out]

-1/2*arctan(1/2+1/2*x)+1/2*ln(x^2+2*x+5)

_______________________________________________________________________________________

Maxima [A]  time = 0.755293, size = 27, normalized size = 1.04 \[ -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^2 + 2*x + 5),x, algorithm="maxima")

[Out]

-1/2*arctan(1/2*x + 1/2) + 1/2*log(x^2 + 2*x + 5)

_______________________________________________________________________________________

Fricas [A]  time = 0.204423, size = 27, normalized size = 1.04 \[ -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^2 + 2*x + 5),x, algorithm="fricas")

[Out]

-1/2*arctan(1/2*x + 1/2) + 1/2*log(x^2 + 2*x + 5)

_______________________________________________________________________________________

Sympy [A]  time = 0.189723, size = 20, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 2 x + 5 \right )}}{2} - \frac{\operatorname{atan}{\left (\frac{x}{2} + \frac{1}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**2+2*x+5),x)

[Out]

log(x**2 + 2*x + 5)/2 - atan(x/2 + 1/2)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205571, size = 27, normalized size = 1.04 \[ -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 2 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^2 + 2*x + 5),x, algorithm="giac")

[Out]

-1/2*arctan(1/2*x + 1/2) + 1/2*ln(x^2 + 2*x + 5)