3.64 \(\int \frac{1-2 x^2}{1+5 x^2+4 x^4} \, dx\)

Optimal. Leaf size=9 \[ \tan ^{-1}(2 x)-\tan ^{-1}(x) \]

[Out]

-ArcTan[x] + ArcTan[2*x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0225844, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \tan ^{-1}(2 x)-\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x^2)/(1 + 5*x^2 + 4*x^4),x]

[Out]

-ArcTan[x] + ArcTan[2*x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.9074, size = 7, normalized size = 0.78 \[ - \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (2 x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*x**2+1)/(4*x**4+5*x**2+1),x)

[Out]

-atan(x) + atan(2*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0114487, size = 12, normalized size = 1.33 \[ \tan ^{-1}\left (\frac{x}{2 x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x^2)/(1 + 5*x^2 + 4*x^4),x]

[Out]

ArcTan[x/(1 + 2*x^2)]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 10, normalized size = 1.1 \[ -\arctan \left ( x \right ) +\arctan \left ( 2\,x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*x^2+1)/(4*x^4+5*x^2+1),x)

[Out]

-arctan(x)+arctan(2*x)

_______________________________________________________________________________________

Maxima [A]  time = 0.840871, size = 12, normalized size = 1.33 \[ \arctan \left (2 \, x\right ) - \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 5*x^2 + 1),x, algorithm="maxima")

[Out]

arctan(2*x) - arctan(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.31455, size = 23, normalized size = 2.56 \[ \arctan \left (4 \, x^{3} + 3 \, x\right ) - \arctan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 5*x^2 + 1),x, algorithm="fricas")

[Out]

arctan(4*x^3 + 3*x) - arctan(2*x)

_______________________________________________________________________________________

Sympy [A]  time = 0.214699, size = 14, normalized size = 1.56 \[ - \operatorname{atan}{\left (2 x \right )} + \operatorname{atan}{\left (4 x^{3} + 3 x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x**2+1)/(4*x**4+5*x**2+1),x)

[Out]

-atan(2*x) + atan(4*x**3 + 3*x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268504, size = 12, normalized size = 1.33 \[ \arctan \left (2 \, x\right ) - \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 + 5*x^2 + 1),x, algorithm="giac")

[Out]

arctan(2*x) - arctan(x)