Optimal. Leaf size=41 \[ \frac {1}{3} \left (x-\sqrt {x^2-4}\right )^{3/2}+\frac {4}{\sqrt {x-\sqrt {x^2-4}}} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2117, 14} \[ \frac {1}{3} \left (x-\sqrt {x^2-4}\right )^{3/2}+\frac {4}{\sqrt {x-\sqrt {x^2-4}}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2117
Rubi steps
\begin {align*} \int \sqrt {x-\sqrt {-4+x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-4+x^2}{x^{3/2}} \, dx,x,x-\sqrt {-4+x^2}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {4}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,x-\sqrt {-4+x^2}\right )\\ &=\frac {4}{\sqrt {x-\sqrt {-4+x^2}}}+\frac {1}{3} \left (x-\sqrt {-4+x^2}\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.98 \[ \frac {2 x^2-2 \sqrt {x^2-4} x+8}{3 \sqrt {x-\sqrt {x^2-4}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 26, normalized size = 0.63 \[ \frac {2}{3} \, {\left (2 \, x + \sqrt {x^{2} - 4}\right )} \sqrt {x - \sqrt {x^{2} - 4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x - \sqrt {x^{2} - 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \sqrt {x -\sqrt {x^{2}-4}}\, dx \]
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 \[ \int \sqrt {x - \sqrt {x^{2} - 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {x-\sqrt {x^2-4}} \,d x \]
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 \[ \int \sqrt {x - \sqrt {x^{2} - 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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