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

Optimal. Leaf size=29 \[ -\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x) \]

[Out]

(-13*x)/(24*(4 + x^2)) + (25*ArcTan[x/2])/144 + ArcTan[x]/9

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Rubi [A]  time = 0.193392, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-13*x)/(24*(4 + x^2)) + (25*ArcTan[x/2])/144 + ArcTan[x]/9

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Rubi in Sympy [A]  time = 30.8389, size = 22, normalized size = 0.76 \[ - \frac{13 x}{24 \left (x^{2} + 4\right )} + \frac{25 \operatorname{atan}{\left (\frac{x}{2} \right )}}{144} + \frac{\operatorname{atan}{\left (x \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13*x/(24*(x**2 + 4)) + 25*atan(x/2)/144 + atan(x)/9

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Mathematica [A]  time = 0.0272968, size = 29, normalized size = 1. \[ -\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-13*x)/(24*(4 + x^2)) + (25*ArcTan[x/2])/144 + ArcTan[x]/9

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Maple [A]  time = 0.013, size = 22, normalized size = 0.8 \[ -{\frac{13\,x}{24\,{x}^{2}+96}}+{\frac{25}{144}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/24*x/(x^2+4)+25/144*arctan(1/2*x)+1/9*arctan(x)

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Maxima [A]  time = 0.883595, size = 28, normalized size = 0.97 \[ -\frac{13 \, x}{24 \,{\left (x^{2} + 4\right )}} + \frac{25}{144} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{9} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/24*x/(x^2 + 4) + 25/144*arctan(1/2*x) + 1/9*arctan(x)

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Fricas [A]  time = 0.25219, size = 45, normalized size = 1.55 \[ \frac{25 \,{\left (x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, x\right ) + 16 \,{\left (x^{2} + 4\right )} \arctan \left (x\right ) - 78 \, x}{144 \,{\left (x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*(25*(x^2 + 4)*arctan(1/2*x) + 16*(x^2 + 4)*arctan(x) - 78*x)/(x^2 + 4)

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Sympy [A]  time = 0.510357, size = 22, normalized size = 0.76 \[ - \frac{13 x}{24 x^{2} + 96} + \frac{25 \operatorname{atan}{\left (\frac{x}{2} \right )}}{144} + \frac{\operatorname{atan}{\left (x \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13*x/(24*x**2 + 96) + 25*atan(x/2)/144 + atan(x)/9

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GIAC/XCAS [A]  time = 0.260678, size = 28, normalized size = 0.97 \[ -\frac{13 \, x}{24 \,{\left (x^{2} + 4\right )}} + \frac{25}{144} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{9} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/24*x/(x^2 + 4) + 25/144*arctan(1/2*x) + 1/9*arctan(x)