3.80 \(\int \frac{x}{1+x^4} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{2} \tan ^{-1}\left (x^2\right ) \]

[Out]

ArcTan[x^2]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.00874866, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{2} \tan ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x^4),x]

[Out]

ArcTan[x^2]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.06419, size = 5, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**4+1),x)

[Out]

atan(x**2)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00453224, size = 8, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + x^4),x]

[Out]

ArcTan[x^2]/2

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 7, normalized size = 0.9 \[{\frac{\arctan \left ({x}^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+1),x)

[Out]

1/2*arctan(x^2)

_______________________________________________________________________________________

Maxima [A]  time = 1.54005, size = 8, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 + 1),x, algorithm="maxima")

[Out]

1/2*arctan(x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.203921, size = 8, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 + 1),x, algorithm="fricas")

[Out]

1/2*arctan(x^2)

_______________________________________________________________________________________

Sympy [A]  time = 0.087924, size = 5, normalized size = 0.62 \[ \frac{\operatorname{atan}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+1),x)

[Out]

atan(x**2)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205238, size = 8, normalized size = 1. \[ \frac{1}{2} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^4 + 1),x, algorithm="giac")

[Out]

1/2*arctan(x^2)