Optimal. Leaf size=25 \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0526596, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x/((1 + x)*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.40097, size = 19, normalized size = 0.76 \[ - \frac{\log{\left (x + 1 \right )}}{2} + \frac{\log{\left (x^{2} + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(1+x)/(x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.00928431, size = 25, normalized size = 1. \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[x/((1 + x)*(1 + x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 20, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(1+x)/(x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.881122, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 1)*(x + 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.281666, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 1)*(x + 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.264509, size = 19, normalized size = 0.76 \[ - \frac{\log{\left (x + 1 \right )}}{2} + \frac{\log{\left (x^{2} + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(1+x)/(x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.260959, size = 27, normalized size = 1.08 \[ \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 1)*(x + 1)),x, algorithm="giac")
[Out]