Optimal. Leaf size=29 \[ -\sqrt{1-x} \left (2-\sqrt{x}\right )-\sin ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0258106, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1398, 785, 780, 216} \[ -\sqrt{1-x} \left (2-\sqrt{x}\right )-\sin ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 1398
Rule 785
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x}}{1+\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x \sqrt{1-x^2}}{1+x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{(1-x) x}{\sqrt{1-x^2}} \, dx,x,\sqrt{x}\right )\\ &=-\left (2-\sqrt{x}\right ) \sqrt{1-x}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\sqrt{x}\right )\\ &=-\left (2-\sqrt{x}\right ) \sqrt{1-x}-\sin ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0269454, size = 26, normalized size = 0.9 \[ \left (\sqrt{x}-2\right ) \sqrt{1-x}-\sin ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 48, normalized size = 1.7 \begin{align*} -{\frac{1}{2}\sqrt{1-x}\sqrt{x} \left ( -2\,\sqrt{-x \left ( x-1 \right ) }+\arcsin \left ( 2\,x-1 \right ) \right ){\frac{1}{\sqrt{-x \left ( x-1 \right ) }}}}-2\,\sqrt{1-x} \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{-x + 1}}{\sqrt{x} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44996, size = 95, normalized size = 3.28 \begin{align*} \sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} + \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.71087, size = 32, normalized size = 1.1 \begin{align*} i \sqrt{x} \sqrt{x - 1} - 2 i \sqrt{x - 1} + i \operatorname{asinh}{\left (\sqrt{x - 1} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15073, size = 39, normalized size = 1.34 \begin{align*} \sqrt{x} \sqrt{-x + 1} - 2 \, \sqrt{-x + 1} + \arcsin \left (\sqrt{-x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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