Optimal. Leaf size=36 \[ \frac {\sqrt {1-x} \sin ^{-1}(x)}{\sqrt {-\sqrt {x}-1} \sqrt {\sqrt {x}-1}} \]
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Rubi [A] time = 0.02, 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 = {519, 41, 216} \[ \frac {\sqrt {1-x} \sin ^{-1}(x)}{\sqrt {-\sqrt {x}-1} \sqrt {\sqrt {x}-1}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 216
Rule 519
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 [A] time = 0.02, size = 36, normalized size = 1.00 \[ \frac {\sqrt {1-x} \sin ^{-1}(x)}{\sqrt {-\sqrt {x}-1} \sqrt {\sqrt {x}-1}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.92, size = 69, normalized size = 1.92 \[ -i \, \log \left (\frac {\sqrt {x + 1} \sqrt {\sqrt {x} - 1} \sqrt {-\sqrt {x} - 1} + i \, x - 1}{x}\right ) + i \, \log \left (\frac {\sqrt {x + 1} \sqrt {\sqrt {x} - 1} \sqrt {-\sqrt {x} - 1} - i \, x - 1}{x}\right ) \]
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 \[ \int \frac {1}{\sqrt {x + 1} \sqrt {\sqrt {x} - 1} \sqrt {-\sqrt {x} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x +1}\, \sqrt {-\sqrt {x}-1}\, \sqrt {\sqrt {x}-1}}\, dx \]
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 \[ \int \frac {1}{\sqrt {x + 1} \sqrt {\sqrt {x} - 1} \sqrt {-\sqrt {x} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\sqrt {x}-1}\,\sqrt {-\sqrt {x}-1}\,\sqrt {x+1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \sqrt {x} - 1} \sqrt {\sqrt {x} - 1} \sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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