3.274 \(\int \frac{-1+x+x^3}{\left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=29 \[ -\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{2} \tan ^{-1}(x) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.032797, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x + x^3)/(1 + x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.91506, size = 20, normalized size = 0.69 \[ - \frac{x}{2 \left (x^{2} + 1\right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} - \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+x-1)/(x**2+1)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0153806, size = 25, normalized size = 0.86 \[ \frac{1}{2} \left (-\frac{x}{x^2+1}+\log \left (x^2+1\right )-\tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x + x^3)/(1 + x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 24, normalized size = 0.8 \[ -{\frac{x}{2\,{x}^{2}+2}}-{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x-1)/(x^2+1)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.898812, size = 31, normalized size = 1.07 \[ -\frac{x}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x - 1)/(x^2 + 1)^2,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.245212, size = 43, normalized size = 1.48 \[ -\frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + x}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x - 1)/(x^2 + 1)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.263982, size = 20, normalized size = 0.69 \[ - \frac{x}{2 x^{2} + 2} + \frac{\log{\left (x^{2} + 1 \right )}}{2} - \frac{\operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x-1)/(x**2+1)**2,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260965, size = 31, normalized size = 1.07 \[ -\frac{x}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x - 1)/(x^2 + 1)^2,x, algorithm="giac")

[Out]

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