3.13 \(\int \frac{1+x}{1+x^3} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0288778, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{2 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/(1 + x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.58611, size = 22, normalized size = 1.16 \[ \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(x**3+1),x)

[Out]

2*sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/3

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Mathematica [A]  time = 0.00913423, size = 19, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)/(1 + x^3),x]

[Out]

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

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Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \[{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.55926, size = 22, normalized size = 1.16 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^3 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.210897, size = 22, normalized size = 1.16 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^3 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.101568, size = 26, normalized size = 1.37 \[ \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209195, size = 22, normalized size = 1.16 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^3 + 1),x, algorithm="giac")

[Out]

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