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 = 0.0203589, antiderivative size = 36, normalized size of antiderivative = 1., 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 519
Rule 41
Rule 216
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*}
Mathematica [A] time = 0.0163369, size = 36, normalized size = 1. \[ \frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.601, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{-1-\sqrt{x}}}}{\frac{1}{\sqrt{-1+\sqrt{x}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.70318, size = 194, normalized size = 5.39 \begin{align*} -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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \sqrt{x} - 1} \sqrt{\sqrt{x} - 1} \sqrt{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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