3.146 \(\int \frac{1+x^4}{1+x^6} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{2}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-ArcTan[Sqrt[3] - 2*x]/3 + (2*ArcTan[x])/3 + ArcTan[Sqrt[3] + 2*x]/3

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Rubi [A]  time = 0.835775, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{2}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)/(1 + x^6),x]

[Out]

-ArcTan[Sqrt[3] - 2*x]/3 + (2*ArcTan[x])/3 + ArcTan[Sqrt[3] + 2*x]/3

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Rubi in Sympy [A]  time = 77.2345, size = 29, normalized size = 0.83 \[ \frac{2 \operatorname{atan}{\left (x \right )}}{3} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{3} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+1)/(x**6+1),x)

[Out]

2*atan(x)/3 + atan(2*x - sqrt(3))/3 + atan(2*x + sqrt(3))/3

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Mathematica [A]  time = 0.0103454, size = 21, normalized size = 0.6 \[ \frac{2}{3} \tan ^{-1}(x)-\frac{1}{3} \tan ^{-1}\left (\frac{x}{x^2-1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^4)/(1 + x^6),x]

[Out]

(2*ArcTan[x])/3 - ArcTan[x/(-1 + x^2)]/3

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Maple [A]  time = 0.07, size = 28, normalized size = 0.8 \[{\frac{2\,\arctan \left ( x \right ) }{3}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{3}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/(x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.50584, size = 36, normalized size = 1.03 \[ \frac{1}{3} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{2}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)/(x^6 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201957, size = 12, normalized size = 0.34 \[ \frac{1}{3} \, \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)/(x^6 + 1),x, algorithm="fricas")

[Out]

1/3*arctan(x^3) + arctan(x)

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Sympy [A]  time = 0.131135, size = 8, normalized size = 0.23 \[ \operatorname{atan}{\left (x \right )} + \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+1)/(x**6+1),x)

[Out]

atan(x) + atan(x**3)/3

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GIAC/XCAS [A]  time = 0.200594, size = 36, normalized size = 1.03 \[ \frac{1}{3} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{2}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)/(x^6 + 1),x, algorithm="giac")

[Out]

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