Optimal. Leaf size=16 \[ \frac{1}{2 (x+1)}+\frac{1}{2} \tan ^{-1}(x) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0374706, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2 (x+1)}+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x/((1 + x)^2*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.70513, size = 10, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{1}{2 \left (x + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(1+x)**2/(x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0105242, size = 12, normalized size = 0.75 \[ \frac{1}{2} \left (\frac{1}{x+1}+\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/((1 + x)^2*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 13, normalized size = 0.8 \[{\frac{1}{2+2\,x}}+{\frac{\arctan \left ( x \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(1+x)^2/(x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.876112, size = 16, normalized size = 1. \[ \frac{1}{2 \,{\left (x + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 1)*(x + 1)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.248934, size = 20, normalized size = 1.25 \[ \frac{{\left (x + 1\right )} \arctan \left (x\right ) + 1}{2 \,{\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 1)*(x + 1)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.242806, size = 10, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{1}{2 x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(1+x)**2/(x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.264845, size = 43, normalized size = 2.69 \[ -\frac{1}{8} \, \pi - \frac{1}{2} \, \pi \left \lfloor -\frac{\pi - 4 \, \arctan \left (x\right )}{4 \, \pi } + \frac{1}{2} \right \rfloor + \frac{1}{2 \,{\left (x + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 1)*(x + 1)^2),x, algorithm="giac")
[Out]