3.183 \(\int \frac{x^2}{2+x^3+x^6} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3+1}{\sqrt{7}}\right )}{3 \sqrt{7}} \]

[Out]

(2*ArcTan[(1 + 2*x^3)/Sqrt[7]])/(3*Sqrt[7])

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Rubi [A]  time = 0.0488604, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3+1}{\sqrt{7}}\right )}{3 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + x^3 + x^6),x]

[Out]

(2*ArcTan[(1 + 2*x^3)/Sqrt[7]])/(3*Sqrt[7])

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Rubi in Sympy [A]  time = 5.27161, size = 24, normalized size = 1.04 \[ \frac{2 \sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{2 x^{3}}{7} + \frac{1}{7}\right ) \right )}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**6+x**3+2),x)

[Out]

2*sqrt(7)*atan(sqrt(7)*(2*x**3/7 + 1/7))/21

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Mathematica [A]  time = 0.013718, size = 23, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3+1}{\sqrt{7}}\right )}{3 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(2 + x^3 + x^6),x]

[Out]

(2*ArcTan[(1 + 2*x^3)/Sqrt[7]])/(3*Sqrt[7])

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Maple [A]  time = 0.004, size = 19, normalized size = 0.8 \[{\frac{2\,\sqrt{7}}{21}\arctan \left ({\frac{ \left ( 2\,{x}^{3}+1 \right ) \sqrt{7}}{7}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^6+x^3+2),x)

[Out]

2/21*arctan(1/7*(2*x^3+1)*7^(1/2))*7^(1/2)

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Maxima [A]  time = 0.84682, size = 24, normalized size = 1.04 \[ \frac{2}{21} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{3} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + x^3 + 2),x, algorithm="maxima")

[Out]

2/21*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^3 + 1))

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Fricas [A]  time = 0.255343, size = 24, normalized size = 1.04 \[ \frac{2}{21} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{3} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + x^3 + 2),x, algorithm="fricas")

[Out]

2/21*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^3 + 1))

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Sympy [A]  time = 0.238163, size = 27, normalized size = 1.17 \[ \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{2 \sqrt{7} x^{3}}{7} + \frac{\sqrt{7}}{7} \right )}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**6+x**3+2),x)

[Out]

2*sqrt(7)*atan(2*sqrt(7)*x**3/7 + sqrt(7)/7)/21

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GIAC/XCAS [A]  time = 0.28272, size = 24, normalized size = 1.04 \[ \frac{2}{21} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{3} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + x^3 + 2),x, algorithm="giac")

[Out]

2/21*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^3 + 1))