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

Optimal. Leaf size=82 \[ \frac{4 x}{3 \left (1-3 x^2\right )}-\frac{2 \sqrt{x^2+1}}{3 \left (1-3 x^2\right )}+\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{3} \sqrt{x^2+1}\right )}{3 \sqrt{3}}-\frac{\tanh ^{-1}\left (\sqrt{3} x\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.143501, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{4 x}{3 \left (1-3 x^2\right )}-\frac{2 \sqrt{x^2+1}}{3 \left (1-3 x^2\right )}+\frac{\tanh ^{-1}\left (\frac{1}{2} \sqrt{3} \sqrt{x^2+1}\right )}{3 \sqrt{3}}-\frac{\tanh ^{-1}\left (\sqrt{3} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2*x + Sqrt[1 + x^2])^(-2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((2*x + sqrt(x**2 + 1))**(-2), x)

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Mathematica [A]  time = 0.158967, size = 119, normalized size = 1.45 \[ \frac{1}{18} \left (\sqrt{3} \log \left (2 \sqrt{3} \sqrt{x^2+1}-\sqrt{3} x+3\right )+\frac{-12 \sqrt{x^2+1}+\sqrt{3} \left (1-3 x^2\right ) \log \left (2 \sqrt{3} \sqrt{x^2+1}+\sqrt{3} x+3\right )+24 x}{1-3 x^2}-2 \sqrt{3} \log \left (3 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2*x + Sqrt[1 + x^2])^(-2),x]

[Out]

(-2*Sqrt[3]*Log[Sqrt[3] + 3*x] + Sqrt[3]*Log[3 - Sqrt[3]*x + 2*Sqrt[3]*Sqrt[1 +
x^2]] + (24*x - 12*Sqrt[1 + x^2] + Sqrt[3]*(1 - 3*x^2)*Log[3 + Sqrt[3]*x + 2*Sqr
t[3]*Sqrt[1 + x^2]])/(1 - 3*x^2))/18

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Maple [B]  time = 0.062, size = 370, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x+(x^2+1)^(1/2))^2,x)

[Out]

-1/2*x/(3*x^2-1)-1/9*arctanh(x*3^(1/2))*3^(1/2)-5/18*x/(x^2-1/3)-3^(1/2)*(-1/12/
(x-1/3*3^(1/2))*((x-1/3*3^(1/2))^2+2/3*3^(1/2)*(x-1/3*3^(1/2))+4/3)^(3/2)+1/36*3
^(1/2)*(1/3*(9*(x-1/3*3^(1/2))^2+6*3^(1/2)*(x-1/3*3^(1/2))+12)^(1/2)+1/3*3^(1/2)
*arcsinh(x)-2/3*3^(1/2)*arctanh(3/4*(8/3+2/3*3^(1/2)*(x-1/3*3^(1/2)))*3^(1/2)/(9
*(x-1/3*3^(1/2))^2+6*3^(1/2)*(x-1/3*3^(1/2))+12)^(1/2)))+1/12*x*((x-1/3*3^(1/2))
^2+2/3*3^(1/2)*(x-1/3*3^(1/2))+4/3)^(1/2)+1/12*arcsinh(x))+3^(1/2)*(-1/12/(x+1/3
*3^(1/2))*((x+1/3*3^(1/2))^2-2/3*3^(1/2)*(x+1/3*3^(1/2))+4/3)^(3/2)-1/36*3^(1/2)
*(1/3*(9*(x+1/3*3^(1/2))^2-6*3^(1/2)*(x+1/3*3^(1/2))+12)^(1/2)-1/3*3^(1/2)*arcsi
nh(x)-2/3*3^(1/2)*arctanh(3/4*(8/3-2/3*3^(1/2)*(x+1/3*3^(1/2)))*3^(1/2)/(9*(x+1/
3*3^(1/2))^2-6*3^(1/2)*(x+1/3*3^(1/2))+12)^(1/2)))+1/12*x*((x+1/3*3^(1/2))^2-2/3
*3^(1/2)*(x+1/3*3^(1/2))+4/3)^(1/2)+1/12*arcsinh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*x + sqrt(x^2 + 1))^(-2), x)

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Fricas [A]  time = 0.220191, size = 363, normalized size = 4.43 \[ -\frac{{\left (6 \, x^{4} + x^{2} - 2 \,{\left (3 \, x^{3} - x\right )} \sqrt{x^{2} + 1} - 1\right )} \log \left (-\frac{24 \, x^{3} - \sqrt{3}{\left (6 \, x^{4} + 17 \, x^{2} + 7\right )} - 2 \,{\left (12 \, x^{2} - \sqrt{3}{\left (3 \, x^{3} + 7 \, x\right )} + 6\right )} \sqrt{x^{2} + 1} + 24 \, x}{6 \, x^{4} + x^{2} - 2 \,{\left (3 \, x^{3} - x\right )} \sqrt{x^{2} + 1} - 1}\right ) +{\left (6 \, x^{4} + x^{2} - 1\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 6 \, x}{3 \, x^{2} - 1}\right ) - 8 \, \sqrt{3}{\left (3 \, x^{3} + 2 \, x\right )} - 2 \, \sqrt{x^{2} + 1}{\left ({\left (3 \, x^{3} - x\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 6 \, x}{3 \, x^{2} - 1}\right ) - 2 \, \sqrt{3}{\left (6 \, x^{2} + 1\right )}\right )}}{6 \,{\left (2 \, \sqrt{3}{\left (3 \, x^{3} - x\right )} \sqrt{x^{2} + 1} - \sqrt{3}{\left (6 \, x^{4} + x^{2} - 1\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*((6*x^4 + x^2 - 2*(3*x^3 - x)*sqrt(x^2 + 1) - 1)*log(-(24*x^3 - sqrt(3)*(6*
x^4 + 17*x^2 + 7) - 2*(12*x^2 - sqrt(3)*(3*x^3 + 7*x) + 6)*sqrt(x^2 + 1) + 24*x)
/(6*x^4 + x^2 - 2*(3*x^3 - x)*sqrt(x^2 + 1) - 1)) + (6*x^4 + x^2 - 1)*log((sqrt(
3)*(3*x^2 + 1) - 6*x)/(3*x^2 - 1)) - 8*sqrt(3)*(3*x^3 + 2*x) - 2*sqrt(x^2 + 1)*(
(3*x^3 - x)*log((sqrt(3)*(3*x^2 + 1) - 6*x)/(3*x^2 - 1)) - 2*sqrt(3)*(6*x^2 + 1)
))/(2*sqrt(3)*(3*x^3 - x)*sqrt(x^2 + 1) - sqrt(3)*(6*x^4 + x^2 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((2*x + sqrt(x**2 + 1))**(-2), x)

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GIAC/XCAS [A]  time = 0.206009, size = 239, normalized size = 2.91 \[ \frac{1}{18} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 6 \, x - 2 \, \sqrt{3} \right |}}{{\left | 6 \, x + 2 \, \sqrt{3} \right |}}\right ) - \frac{1}{18} \, \sqrt{3}{\rm ln}\left (-\frac{{\left | -6 \, x - 8 \, \sqrt{3} + 6 \, \sqrt{x^{2} + 1} - \frac{6}{x - \sqrt{x^{2} + 1}} \right |}}{2 \,{\left (3 \, x - 4 \, \sqrt{3} - 3 \, \sqrt{x^{2} + 1} + \frac{3}{x - \sqrt{x^{2} + 1}}\right )}}\right ) - \frac{4 \,{\left (x - \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}}\right )}}{3 \,{\left (3 \,{\left (x - \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}}\right )}^{2} - 16\right )}} - \frac{4 \, x}{3 \,{\left (3 \, x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*ln(abs(6*x - 2*sqrt(3))/abs(6*x + 2*sqrt(3))) - 1/18*sqrt(3)*ln(-1/
2*abs(-6*x - 8*sqrt(3) + 6*sqrt(x^2 + 1) - 6/(x - sqrt(x^2 + 1)))/(3*x - 4*sqrt(
3) - 3*sqrt(x^2 + 1) + 3/(x - sqrt(x^2 + 1)))) - 4/3*(x - sqrt(x^2 + 1) + 1/(x -
sqrt(x^2 + 1)))/(3*(x - sqrt(x^2 + 1) + 1/(x - sqrt(x^2 + 1)))^2 - 16) - 4/3*x/
(3*x^2 - 1)