Optimal. Leaf size=29 \[ -\frac{\sqrt{\frac{1}{x}+1} \sqrt{x} \sin ^{-1}(1-2 x)}{\sqrt{x+1}} \]
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Rubi [A] time = 0.0122943, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1448, 26, 53, 619, 216} \[ -\frac{\sqrt{\frac{1}{x}+1} \sqrt{x} \sin ^{-1}(1-2 x)}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
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Rule 1448
Rule 26
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1+\frac{1}{x}}}{\sqrt{1-x^2}} \, dx &=\frac{\left (\sqrt{1+\frac{1}{x}} \sqrt{x}\right ) \int \frac{\sqrt{1+x}}{\sqrt{x} \sqrt{1-x^2}} \, dx}{\sqrt{1+x}}\\ &=\frac{\left (\sqrt{1+\frac{1}{x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{1-x} \sqrt{x}} \, dx}{\sqrt{1+x}}\\ &=\frac{\left (\sqrt{1+\frac{1}{x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x-x^2}} \, dx}{\sqrt{1+x}}\\ &=-\frac{\left (\sqrt{1+\frac{1}{x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,1-2 x\right )}{\sqrt{1+x}}\\ &=-\frac{\sqrt{1+\frac{1}{x}} \sqrt{x} \sin ^{-1}(1-2 x)}{\sqrt{1+x}}\\ \end{align*}
Mathematica [A] time = 0.228729, size = 41, normalized size = 1.41 \[ -\tan ^{-1}\left (\frac{\sqrt{\frac{x+1}{x}} (2 x-1) \sqrt{1-x^2}}{2 \left (x^2-1\right )}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 40, normalized size = 1.4 \begin{align*}{\frac{x\arcsin \left ( 2\,x-1 \right ) }{1+x}\sqrt{{\frac{1+x}{x}}}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{-x \left ( x-1 \right ) }}}} \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{\sqrt{\frac{1}{x} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84845, size = 82, normalized size = 2.83 \begin{align*} -\arctan \left (\frac{2 \, \sqrt{-x^{2} + 1} x \sqrt{\frac{x + 1}{x}}}{2 \, x^{2} + x - 1}\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{\sqrt{1 + \frac{1}{x}}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, 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{\sqrt{\frac{1}{x} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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