Optimal. Leaf size=31 \[ \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {2} \sqrt {-1+x}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {675, 209}
\begin {gather*} \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {1-x^2}}{\sqrt {2} \sqrt {x-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 675
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x} \sqrt {1-x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {-1+x}}\right )\\ &=\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {2} \sqrt {-1+x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 32, normalized size = 1.03 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+x}}{\sqrt {1-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 39, normalized size = 1.26
method | result | size |
default | \(\frac {\sqrt {-x^{2}+1}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )}{\sqrt {x -1}\, \sqrt {-1-x}}\) | \(39\) |
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 [A]
time = 1.86, size = 30, normalized size = 0.97 \begin {gather*} \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + 1} \sqrt {x - 1}}{x^{2} - 1}\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*} \int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {x - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.90, size = 36, normalized size = 1.16 \begin {gather*} \frac {1}{2} i \, {\left (\sqrt {2} \log \left (\sqrt {2} + \sqrt {x + 1}\right ) - \sqrt {2} \log \left (-\sqrt {2} + \sqrt {x + 1}\right )\right )} \mathrm {sgn}\left (x\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 = -0.03 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {x-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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