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

Optimal. Leaf size=44 \[ \sqrt{3} \tan ^{-1}\left (\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right )-\tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

[Out]

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

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Rubi [A]  time = 0.145462, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \sqrt{3} \tan ^{-1}\left (\frac{x+1}{\sqrt{3} \sqrt{x^2+2 x+5}}\right )-\tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 54.5304, size = 42, normalized size = 0.95 \[ \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (2 x + 2\right )}{6 \sqrt{x^{2} + 2 x + 5}} \right )} - \operatorname{atanh}{\left (\sqrt{x^{2} + 2 x + 5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4+x)/(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))) - atanh(sqrt(x**2 + 2*x
+ 5))

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Mathematica [B]  time = 0.152868, size = 109, normalized size = 2.48 \[ \frac{1}{2} \left (\log \left (\left (x^2+2 x+4\right )^2\right )-\log \left (\left (x^2+2 x+4\right ) \left (x^2+2 \sqrt{x^2+2 x+5}+2 x+6\right )\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (x^2+\left (\sqrt{x^2+2 x+5}+2\right ) x+\sqrt{x^2+2 x+5}+4\right )}{2 x^2+4 x+11}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3]*ArcTan[(Sqrt[3]*(4 + x^2 + Sqrt[5 + 2*x + x^2] + x*(2 + Sqrt[5 + 2*x
+ x^2])))/(11 + 4*x + 2*x^2)] + Log[(4 + 2*x + x^2)^2] - Log[(4 + 2*x + x^2)*(6
+ 2*x + x^2 + 2*Sqrt[5 + 2*x + x^2])])/2

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Maple [A]  time = 0.013, size = 40, normalized size = 0.9 \[ -{\it Artanh} \left ( \sqrt{{x}^{2}+2\,x+5} \right ) +\sqrt{3}\arctan \left ({\frac{\sqrt{3} \left ( 2\,x+2 \right ) }{6}{\frac{1}{\sqrt{{x}^{2}+2\,x+5}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctanh((x^2+2*x+5)^(1/2))+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. \[ \int \frac{x + 4}{\sqrt{x^{2} + 2 \, x + 5}{\left (x^{2} + 2 \, x + 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276216, size = 134, normalized size = 3.05 \[ -\sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5} + 2\right )}\right ) + \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5}{\left (x + 2\right )} + 3 \, x + 6\right ) - \frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5} x + x + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*arctan(-1/3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5) + 2)) + sqrt(3)*arctan(-1/
3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5))) + 1/2*log(x^2 - sqrt(x^2 + 2*x + 5)*(x + 2)
+ 3*x + 6) - 1/2*log(x^2 - sqrt(x^2 + 2*x + 5)*x + x + 4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.276905, size = 146, normalized size = 3.32 \[ -\sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5} + 2\right )}\right ) + \sqrt{3} \arctan \left (-\frac{1}{3} \, \sqrt{3}{\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 4 \, x - 4 \, \sqrt{x^{2} + 2 \, x + 5} + 7\right ) - \frac{1}{2} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*arctan(-1/3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5) + 2)) + sqrt(3)*arctan(-1/
3*sqrt(3)*(x - sqrt(x^2 + 2*x + 5))) + 1/2*ln((x - sqrt(x^2 + 2*x + 5))^2 + 4*x
- 4*sqrt(x^2 + 2*x + 5) + 7) - 1/2*ln((x - sqrt(x^2 + 2*x + 5))^2 + 3)