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