3.8 \(\int \frac{1}{\left (4+2 x+x^2\right ) \sqrt{5+2 x+x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right )}{\sqrt{3}} \]

[Out]

ArcTan[(1 + x)/(Sqrt[3]*Sqrt[5 + 2*x + x^2])]/Sqrt[3]

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Rubi [A]  time = 0.0449474, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

ArcTan[(1 + x)/(Sqrt[3]*Sqrt[5 + 2*x + x^2])]/Sqrt[3]

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Rubi in Sympy [A]  time = 15.435, size = 31, normalized size = 1.11 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (2 x + 2\right )}{6 \sqrt{x^{2} + 2 x + 5}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+2*x+4)/(x**2+2*x+5)**(1/2),x)

[Out]

sqrt(3)*atan(sqrt(3)*(2*x + 2)/(6*sqrt(x**2 + 2*x + 5)))/3

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Mathematica [A]  time = 0.0626008, size = 28, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

ArcTan[(1 + x)/(Sqrt[3]*Sqrt[5 + 2*x + x^2])]/Sqrt[3]

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Maple [A]  time = 0.019, size = 27, normalized size = 1. \[{\frac{\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3} \left ( 2\,x+2 \right ) }{6}{\frac{1}{\sqrt{{x}^{2}+2\,x+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x)

[Out]

1/3*3^(1/2)*arctan(1/6*3^(1/2)/(x^2+2*x+5)^(1/2)*(2*x+2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 2 \, x + 5}{\left (x^{2} + 2 \, x + 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)), x)

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Fricas [A]  time = 0.274123, size = 51, normalized size = 1.82 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} \sqrt{x^{2} + 2 \, x + 5}{\left (x + 1\right )} - \frac{1}{3} \, \sqrt{3}{\left (x^{2} + 2 \, x + 4\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)),x, algorithm="fricas")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(x^2 + 2*x + 5)*(x + 1) - 1/3*sqrt(3)*(x^2 +
2*x + 4))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x^{2} + 2 x + 4\right ) \sqrt{x^{2} + 2 x + 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+2*x+4)/(x**2+2*x+5)**(1/2),x)

[Out]

Integral(1/((x**2 + 2*x + 4)*sqrt(x**2 + 2*x + 5)), x)

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GIAC/XCAS [A]  time = 0.265977, size = 70, normalized size = 2.5 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5} + 2\right )}\right ) + \frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5) + 2)) + 1/3*sqrt(3)*ar
ctan(-1/3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5)))