3.278 \(\int \frac{1}{\left (1+x^2\right ) \left (4+x^2\right )} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (\frac{x}{2}\right ) \]

[Out]

-ArcTan[x/2]/6 + ArcTan[x]/3

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Rubi [A]  time = 0.022618, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x^2)*(4 + x^2)),x]

[Out]

-ArcTan[x/2]/6 + ArcTan[x]/3

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Rubi in Sympy [A]  time = 5.12088, size = 10, normalized size = 0.59 \[ - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{6} + \frac{\operatorname{atan}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(x/2)/6 + atan(x)/3

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Mathematica [A]  time = 0.00993099, size = 17, normalized size = 1. \[ \frac{1}{6} \tan ^{-1}\left (\frac{2}{x}\right )+\frac{1}{3} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[2/x]/6 + ArcTan[x]/3

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Maple [A]  time = 0.01, size = 12, normalized size = 0.7 \[ -{\frac{1}{6}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)/(x^2+4),x)

[Out]

-1/6*arctan(1/2*x)+1/3*arctan(x)

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Maxima [A]  time = 0.89781, size = 15, normalized size = 0.88 \[ -\frac{1}{6} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*arctan(1/2*x) + 1/3*arctan(x)

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Fricas [A]  time = 0.262774, size = 15, normalized size = 0.88 \[ -\frac{1}{6} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*arctan(1/2*x) + 1/3*arctan(x)

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Sympy [A]  time = 0.40096, size = 10, normalized size = 0.59 \[ - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{6} + \frac{\operatorname{atan}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(x/2)/6 + atan(x)/3

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GIAC/XCAS [A]  time = 0.262438, size = 15, normalized size = 0.88 \[ -\frac{1}{6} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*arctan(1/2*x) + 1/3*arctan(x)