3.675 \(\int \frac{\sqrt{-1+4 x^2}}{x+\sqrt{-1+4 x^2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{1}{3} \sqrt{4 x^2-1}+\frac{\tanh ^{-1}\left (\sqrt{3} \sqrt{4 x^2-1}\right )}{3 \sqrt{3}}+\frac{4 x}{3}-\frac{\tanh ^{-1}\left (\sqrt{3} x\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.256962, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{3} \sqrt{4 x^2-1}+\frac{\tanh ^{-1}\left (\sqrt{3} \sqrt{4 x^2-1}\right )}{3 \sqrt{3}}+\frac{4 x}{3}-\frac{\tanh ^{-1}\left (\sqrt{3} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 x^{2} - 1}}{x + \sqrt{4 x^{2} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0742754, size = 98, normalized size = 1.51 \[ \frac{1}{18} \left (-6 \sqrt{4 x^2-1}+\sqrt{3} \log \left (-\sqrt{12 x^2-3}-4 \sqrt{3} x+3\right )+\sqrt{3} \log \left (-\sqrt{12 x^2-3}+4 \sqrt{3} x+3\right )+24 x-2 \sqrt{3} \log \left (3 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(24*x - 6*Sqrt[-1 + 4*x^2] - 2*Sqrt[3]*Log[Sqrt[3] + 3*x] + Sqrt[3]*Log[3 - 4*Sq
rt[3]*x - Sqrt[-3 + 12*x^2]] + Sqrt[3]*Log[3 + 4*Sqrt[3]*x - Sqrt[-3 + 12*x^2]])
/18

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Maple [B]  time = 0.049, size = 262, normalized size = 4. \[{\frac{4\,x}{3}}-{\frac{{\it Artanh} \left ( x\sqrt{3} \right ) \sqrt{3}}{9}}-{\frac{1}{18}\sqrt{36\, \left ( x-1/3\,\sqrt{3} \right ) ^{2}+24\, \left ( x-1/3\,\sqrt{3} \right ) \sqrt{3}+3}}-{\frac{\sqrt{3}\sqrt{4}}{18}\ln \left ( x\sqrt{4}+\sqrt{4\, \left ( x-1/3\,\sqrt{3} \right ) ^{2}+{\frac{8\,\sqrt{3}}{3} \left ( x-{\frac{\sqrt{3}}{3}} \right ) }+{\frac{1}{3}}} \right ) }+{\frac{\sqrt{3}}{18}{\it Artanh} \left ({\frac{3\,\sqrt{3}}{2} \left ({\frac{2}{3}}+{\frac{8\,\sqrt{3}}{3} \left ( x-{\frac{\sqrt{3}}{3}} \right ) } \right ){\frac{1}{\sqrt{36\, \left ( x-1/3\,\sqrt{3} \right ) ^{2}+24\, \left ( x-1/3\,\sqrt{3} \right ) \sqrt{3}+3}}}} \right ) }-{\frac{1}{18}\sqrt{36\, \left ( x+1/3\,\sqrt{3} \right ) ^{2}-24\, \left ( x+1/3\,\sqrt{3} \right ) \sqrt{3}+3}}+{\frac{\sqrt{3}\sqrt{4}}{18}\ln \left ( x\sqrt{4}+\sqrt{4\, \left ( x+1/3\,\sqrt{3} \right ) ^{2}-{\frac{8\,\sqrt{3}}{3} \left ( x+{\frac{\sqrt{3}}{3}} \right ) }+{\frac{1}{3}}} \right ) }+{\frac{\sqrt{3}}{18}{\it Artanh} \left ({\frac{3\,\sqrt{3}}{2} \left ({\frac{2}{3}}-{\frac{8\,\sqrt{3}}{3} \left ( x+{\frac{\sqrt{3}}{3}} \right ) } \right ){\frac{1}{\sqrt{36\, \left ( x+1/3\,\sqrt{3} \right ) ^{2}-24\, \left ( x+1/3\,\sqrt{3} \right ) \sqrt{3}+3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*x-1/9*arctanh(x*3^(1/2))*3^(1/2)-1/18*(36*(x-1/3*3^(1/2))^2+24*(x-1/3*3^(1/2
))*3^(1/2)+3)^(1/2)-1/18*3^(1/2)*ln(x*4^(1/2)+(4*(x-1/3*3^(1/2))^2+8/3*(x-1/3*3^
(1/2))*3^(1/2)+1/3)^(1/2))*4^(1/2)+1/18*3^(1/2)*arctanh(3/2*(2/3+8/3*(x-1/3*3^(1
/2))*3^(1/2))*3^(1/2)/(36*(x-1/3*3^(1/2))^2+24*(x-1/3*3^(1/2))*3^(1/2)+3)^(1/2))
-1/18*(36*(x+1/3*3^(1/2))^2-24*(x+1/3*3^(1/2))*3^(1/2)+3)^(1/2)+1/18*3^(1/2)*ln(
x*4^(1/2)+(4*(x+1/3*3^(1/2))^2-8/3*(x+1/3*3^(1/2))*3^(1/2)+1/3)^(1/2))*4^(1/2)+1
/18*3^(1/2)*arctanh(3/2*(2/3-8/3*(x+1/3*3^(1/2))*3^(1/2))*3^(1/2)/(36*(x+1/3*3^(
1/2))^2-24*(x+1/3*3^(1/2))*3^(1/2)+3)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ x - \int \frac{x}{\sqrt{2 \, x + 1} \sqrt{2 \, x - 1} + x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.277461, size = 300, normalized size = 4.62 \[ \frac{{\left (2 \, x - \sqrt{4 \, x^{2} - 1}\right )} \log \left (-\frac{48 \, x^{3} - \sqrt{3}{\left (48 \, x^{4} - 14 \, x^{2} + 1\right )} -{\left (24 \, x^{2} - 4 \, \sqrt{3}{\left (6 \, x^{3} - x\right )} - 3\right )} \sqrt{4 \, x^{2} - 1} - 12 \, x}{24 \, x^{4} - 11 \, x^{2} - 4 \,{\left (3 \, x^{3} - x\right )} \sqrt{4 \, x^{2} - 1} + 1}\right ) + 2 \, x \log \left (\frac{\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 6 \, x}{3 \, x^{2} - 1}\right ) + 2 \, \sqrt{3}{\left (12 \, x^{2} - 1\right )} - \sqrt{4 \, x^{2} - 1}{\left (12 \, \sqrt{3} x + \log \left (\frac{\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 6 \, x}{3 \, x^{2} - 1}\right )\right )}}{6 \,{\left (2 \, \sqrt{3} x - \sqrt{3} \sqrt{4 \, x^{2} - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*((2*x - sqrt(4*x^2 - 1))*log(-(48*x^3 - sqrt(3)*(48*x^4 - 14*x^2 + 1) - (24*
x^2 - 4*sqrt(3)*(6*x^3 - x) - 3)*sqrt(4*x^2 - 1) - 12*x)/(24*x^4 - 11*x^2 - 4*(3
*x^3 - x)*sqrt(4*x^2 - 1) + 1)) + 2*x*log((sqrt(3)*(3*x^2 + 1) - 6*x)/(3*x^2 - 1
)) + 2*sqrt(3)*(12*x^2 - 1) - sqrt(4*x^2 - 1)*(12*sqrt(3)*x + log((sqrt(3)*(3*x^
2 + 1) - 6*x)/(3*x^2 - 1))))/(2*sqrt(3)*x - sqrt(3)*sqrt(4*x^2 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277628, size = 180, normalized size = 2.77 \[ \frac{1}{18} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 6 \, x - 2 \, \sqrt{3} \right |}}{{\left | 6 \, x + 2 \, \sqrt{3} \right |}}\right ) - \frac{1}{18} \, \sqrt{3}{\rm ln}\left (-\frac{{\left | -12 \, x - 4 \, \sqrt{3} + 6 \, \sqrt{4 \, x^{2} - 1} + \frac{6}{2 \, x - \sqrt{4 \, x^{2} - 1}} \right |}}{2 \,{\left (6 \, x - 2 \, \sqrt{3} - 3 \, \sqrt{4 \, x^{2} - 1} - \frac{3}{2 \, x - \sqrt{4 \, x^{2} - 1}}\right )}}\right ) + \frac{4}{3} \, x - \frac{1}{3} \, \sqrt{4 \, x^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 1)/(x + sqrt(4*x^2 - 1)),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(-12*x - 4*sqrt(3) + 6*sqrt(4*x^2 - 1) + 6/(2*x - sqrt(4*x^2 - 1)))/(6*x -
2*sqrt(3) - 3*sqrt(4*x^2 - 1) - 3/(2*x - sqrt(4*x^2 - 1)))) + 4/3*x - 1/3*sqrt(4
*x^2 - 1)