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 = 0.13, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6742, 444, 50, 63, 207, 388} \[ -\frac {1}{3} \sqrt {4 x^2-1}+\frac {\tanh ^{-1}\left (\sqrt {3} \sqrt {4 x^2-1}\right )}{3 \sqrt {3}}+\frac {4 x}{3}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 207
Rule 388
Rule 444
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+4 x^2}}{x+\sqrt {-1+4 x^2}} \, dx &=\int \left (-\frac {x \sqrt {-1+4 x^2}}{-1+3 x^2}+\frac {-1+4 x^2}{-1+3 x^2}\right ) \, dx\\ &=-\int \frac {x \sqrt {-1+4 x^2}}{-1+3 x^2} \, dx+\int \frac {-1+4 x^2}{-1+3 x^2} \, dx\\ &=\frac {4 x}{3}+\frac {1}{3} \int \frac {1}{-1+3 x^2} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-1+4 x}}{-1+3 x} \, dx,x,x^2\right )\\ &=\frac {4 x}{3}-\frac {1}{3} \sqrt {-1+4 x^2}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{(-1+3 x) \sqrt {-1+4 x}} \, dx,x,x^2\right )\\ &=\frac {4 x}{3}-\frac {1}{3} \sqrt {-1+4 x^2}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{4}+\frac {3 x^2}{4}} \, dx,x,\sqrt {-1+4 x^2}\right )\\ &=\frac {4 x}{3}-\frac {1}{3} \sqrt {-1+4 x^2}-\frac {\tanh ^{-1}\left (\sqrt {3} x\right )}{3 \sqrt {3}}+\frac {\tanh ^{-1}\left (\sqrt {3} \sqrt {-1+4 x^2}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 0.83 \[ \frac {1}{9} \left (-3 \sqrt {4 x^2-1}+\sqrt {3} \tanh ^{-1}\left (\sqrt {12 x^2-3}\right )+12 x-\sqrt {3} \tanh ^{-1}\left (\sqrt {3} x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 80, normalized size = 1.23 \[ \frac {1}{18} \, \sqrt {3} \log \left (\frac {6 \, x^{2} + \sqrt {3} \sqrt {4 \, x^{2} - 1} - 1}{3 \, x^{2} - 1}\right ) + \frac {1}{18} \, \sqrt {3} \log \left (\frac {3 \, x^{2} - 2 \, \sqrt {3} x + 1}{3 \, x^{2} - 1}\right ) + \frac {4}{3} \, x - \frac {1}{3} \, \sqrt {4 \, x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 133, normalized size = 2.05 \[ \frac {1}{18} \, \sqrt {3} \log \left (\frac {{\left | 6 \, x - 2 \, \sqrt {3} \right |}}{{\left | 6 \, x + 2 \, \sqrt {3} \right |}}\right ) - \frac {1}{18} \, \sqrt {3} \log \left (-\frac {{\left | -12 \, x - 4 \, \sqrt {3} + 6 \, \sqrt {4 \, x^{2} - 1} + \frac {6}{2 \, x - \sqrt {4 \, x^{2} - 1}} \right |}}{2 \, {\left (6 \, x - 2 \, \sqrt {3} - 3 \, \sqrt {4 \, x^{2} - 1} - \frac {3}{2 \, x - \sqrt {4 \, x^{2} - 1}}\right )}}\right ) + \frac {4}{3} \, x - \frac {1}{3} \, \sqrt {4 \, x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 262, normalized size = 4.03 \[ \frac {4 x}{3}-\frac {\sqrt {3}\, \arctanh \left (\sqrt {3}\, x \right )}{9}+\frac {\sqrt {3}\, \arctanh \left (\frac {3 \left (\frac {2}{3}+\frac {8 \left (x -\frac {\sqrt {3}}{3}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 \sqrt {36 \left (x -\frac {\sqrt {3}}{3}\right )^{2}+24 \left (x -\frac {\sqrt {3}}{3}\right ) \sqrt {3}+3}}\right )}{18}+\frac {\sqrt {3}\, \arctanh \left (\frac {3 \left (\frac {2}{3}-\frac {8 \left (x +\frac {\sqrt {3}}{3}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 \sqrt {36 \left (x +\frac {\sqrt {3}}{3}\right )^{2}-24 \left (x +\frac {\sqrt {3}}{3}\right ) \sqrt {3}+3}}\right )}{18}-\frac {\sqrt {3}\, \sqrt {4}\, \ln \left (\sqrt {4}\, x +\sqrt {4 \left (x -\frac {\sqrt {3}}{3}\right )^{2}+\frac {8 \left (x -\frac {\sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {1}{3}}\right )}{18}+\frac {\sqrt {3}\, \sqrt {4}\, \ln \left (\sqrt {4}\, x +\sqrt {4 \left (x +\frac {\sqrt {3}}{3}\right )^{2}-\frac {8 \left (x +\frac {\sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {1}{3}}\right )}{18}-\frac {\sqrt {36 \left (x -\frac {\sqrt {3}}{3}\right )^{2}+24 \left (x -\frac {\sqrt {3}}{3}\right ) \sqrt {3}+3}}{18}-\frac {\sqrt {36 \left (x +\frac {\sqrt {3}}{3}\right )^{2}-24 \left (x +\frac {\sqrt {3}}{3}\right ) \sqrt {3}+3}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x - \int \frac {x}{\sqrt {2 \, x + 1} \sqrt {2 \, x - 1} + x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.44, size = 60, normalized size = 0.92 \[ \frac {4\,x}{3}+\frac {\sqrt {3}\,\ln \left (x-\frac {\sqrt {3}}{3}\right )}{18}-\frac {\sqrt {3}\,\ln \left (x+\frac {\sqrt {3}}{3}\right )}{18}+\frac {\sqrt {3}\,\mathrm {atanh}\left (\sqrt {3}\,\sqrt {4\,x^2-1}\right )}{9}-\frac {\sqrt {4\,x^2-1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (2 x - 1\right ) \left (2 x + 1\right )}}{x + \sqrt {4 x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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