3.80 \(\int \frac{1+x^2}{1+2 x^2+x^4} \, dx\)

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

[Out]

ArcTan[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.00498854, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^2)/(1 + 2*x^2 + x^4),x]

[Out]

ArcTan[x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.76924, size = 2, normalized size = 1. \[ \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+1)/(x**4+2*x**2+1),x)

[Out]

atan(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00360941, size = 2, normalized size = 1. \[ \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^2)/(1 + 2*x^2 + x^4),x]

[Out]

ArcTan[x]

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 3, normalized size = 1.5 \[ \arctan \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^4+2*x^2+1),x)

[Out]

arctan(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.836428, size = 3, normalized size = 1.5 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^4 + 2*x^2 + 1),x, algorithm="maxima")

[Out]

arctan(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.283217, size = 3, normalized size = 1.5 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^4 + 2*x^2 + 1),x, algorithm="fricas")

[Out]

arctan(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.151909, size = 2, normalized size = 1. \[ \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)/(x**4+2*x**2+1),x)

[Out]

atan(x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268526, size = 3, normalized size = 1.5 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^4 + 2*x^2 + 1),x, algorithm="giac")

[Out]

arctan(x)