Optimal. Leaf size=36 \[ \frac {\sqrt {1-x} \sin ^{-1}(x)}{\sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}}} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {533, 41, 222}
\begin {gather*} \frac {\sqrt {1-x} \text {ArcSin}(x)}{\sqrt {-\sqrt {x}-1} \sqrt {\sqrt {x}-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 222
Rule 533
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}} \sqrt {1+x}} \, dx &=\frac {\sqrt {1-x} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx}{\sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}}}\\ &=\frac {\sqrt {1-x} \int \frac {1}{\sqrt {1-x^2}} \, dx}{\sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}}}\\ &=\frac {\sqrt {1-x} \sin ^{-1}(x)}{\sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.18, size = 44, normalized size = 1.22 \begin {gather*} -i \log \left (-x+i \sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}} \sqrt {1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {x +1}\, \sqrt {-1-\sqrt {x}}\, \sqrt {-1+\sqrt {x}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.90, size = 69, normalized size = 1.92 \begin {gather*} -i \, \log \left (\frac {\sqrt {x + 1} \sqrt {\sqrt {x} - 1} \sqrt {-\sqrt {x} - 1} + i \, x - 1}{x}\right ) + i \, \log \left (\frac {\sqrt {x + 1} \sqrt {\sqrt {x} - 1} \sqrt {-\sqrt {x} - 1} - i \, x - 1}{x}\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 {- \sqrt {x} - 1} \sqrt {\sqrt {x} - 1} \sqrt {x + 1}}\, dx \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*} \text {could not integrate} \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 {\sqrt {x}-1}\,\sqrt {-\sqrt {x}-1}\,\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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