Optimal. Leaf size=20 \[ -\sinh ^{-1}\left (\frac {1-2 \sqrt {-1+x}}{\sqrt {3}}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 633, 221}
\begin {gather*} -\sinh ^{-1}\left (\frac {1-2 \sqrt {x-1}}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{2 \sqrt {-1+x} \sqrt {-\sqrt {-1+x}+x}} \, dx &=\frac {1}{2} \int \frac {1}{\sqrt {-1+x} \sqrt {-\sqrt {-1+x}+x}} \, dx\\ &=\text {Subst}\left (\int \frac {1}{\sqrt {1-x+x^2}} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,-1+2 \sqrt {-1+x}\right )}{\sqrt {3}}\\ &=-\sinh ^{-1}\left (\frac {1-2 \sqrt {-1+x}}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 31, normalized size = 1.55 \begin {gather*} -\log \left (1-2 \sqrt {-1+x}+2 \sqrt {-\sqrt {-1+x}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 14, normalized size = 0.70
method | result | size |
derivativedivides | \(\arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}-\frac {1}{2}\right )}{3}\right )\) | \(14\) |
default | \(\arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}-\frac {1}{2}\right )}{3}\right )\) | \(14\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (17) = 34\).
time = 0.55, size = 37, normalized size = 1.85 \begin {gather*} \frac {1}{2} \, \log \left (4 \, \sqrt {x - \sqrt {x - 1}} {\left (2 \, \sqrt {x - 1} - 1\right )} + 8 \, x - 8 \, \sqrt {x - 1} - 3\right ) \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*} \frac {\int \frac {1}{\sqrt {x - 1} \sqrt {x - \sqrt {x - 1}}}\, dx}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.54, size = 25, normalized size = 1.25 \begin {gather*} -\log \left (2 \, \sqrt {x - \sqrt {x - 1}} - 2 \, \sqrt {x - 1} + 1\right ) \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 = -0.05 \begin {gather*} \int \frac {1}{2\,\sqrt {x-\sqrt {x-1}}\,\sqrt {x-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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