3.327 \(\int \frac{2+x}{2+x+x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{2} \log \left (x^2+x+2\right )+\frac{3 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{7}}\right )}{\sqrt{7}} \]

[Out]

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

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Rubi [A]  time = 0.0334638, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{2} \log \left (x^2+x+2\right )+\frac{3 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{7}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/(2 + x + x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.21122, size = 32, normalized size = 1.03 \[ \frac{\log{\left (x^{2} + x + 2 \right )}}{2} + \frac{3 \sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{2 x}{7} + \frac{1}{7}\right ) \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0167361, size = 31, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+x+2\right )+\frac{3 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{7}}\right )}{\sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 27, normalized size = 0.9 \[{\frac{\ln \left ({x}^{2}+x+2 \right ) }{2}}+{\frac{3\,\sqrt{7}}{7}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{7}}{7}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(x^2+x+2),x)

[Out]

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

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Maxima [A]  time = 1.48483, size = 35, normalized size = 1.13 \[ \frac{3}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} + x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.254523, size = 41, normalized size = 1.32 \[ \frac{1}{14} \, \sqrt{7}{\left (\sqrt{7} \log \left (x^{2} + x + 2\right ) + 6 \, \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*sqrt(7)*(sqrt(7)*log(x^2 + x + 2) + 6*arctan(1/7*sqrt(7)*(2*x + 1)))

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Sympy [A]  time = 0.116841, size = 36, normalized size = 1.16 \[ \frac{\log{\left (x^{2} + x + 2 \right )}}{2} + \frac{3 \sqrt{7} \operatorname{atan}{\left (\frac{2 \sqrt{7} x}{7} + \frac{\sqrt{7}}{7} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(x**2+x+2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214339, size = 35, normalized size = 1.13 \[ \frac{3}{7} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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