Optimal. Leaf size=39 \[ \sqrt {x-\sqrt {x^2-1}}+\frac {1}{3 \left (x-\sqrt {x^2-1}\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 39, 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} \begin {gather*} \sqrt {x-\sqrt {x^2-1}}+\frac {1}{3 \left (x-\sqrt {x^2-1}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x-\sqrt {-1+x^2}}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x^2}{x^{5/2}} \, dx,x,x-\sqrt {-1+x^2}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,x-\sqrt {-1+x^2}\right )\\ &=\frac {1}{3 \left (x-\sqrt {-1+x^2}\right )^{3/2}}+\sqrt {x-\sqrt {-1+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.00 \begin {gather*} \sqrt {x-\sqrt {x^2-1}}+\frac {1}{3 \left (x-\sqrt {x^2-1}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{3 \left (x-\sqrt {-1+x^2}\right )^{3/2}}+\sqrt {x-\sqrt {-1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, {\left (x^{2} + \sqrt {x^{2} - 1} x + 1\right )} \sqrt {x - \sqrt {x^{2} - 1}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x - \sqrt {x^{2} - 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {x -\sqrt {x^{2}-1}}}\, dx\]
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 \begin {gather*} \int \frac {1}{\sqrt {x - \sqrt {x^{2} - 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {x-\sqrt {x^2-1}}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x - \sqrt {x^{2} - 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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