Optimal. Leaf size=54 \[ \sqrt{x-x^2}+\sqrt{2} \tan ^{-1}\left (\frac{1-3 x}{2 \sqrt{2} \sqrt{x-x^2}}\right )-\frac{3}{2} \sin ^{-1}(1-2 x) \]
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Rubi [A] time = 0.039589, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {734, 843, 619, 216, 724, 204} \[ \sqrt{x-x^2}+\sqrt{2} \tan ^{-1}\left (\frac{1-3 x}{2 \sqrt{2} \sqrt{x-x^2}}\right )-\frac{3}{2} \sin ^{-1}(1-2 x) \]
Antiderivative was successfully verified.
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Rule 734
Rule 843
Rule 619
Rule 216
Rule 724
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{x-x^2}}{1+x} \, dx &=\sqrt{x-x^2}-\frac{1}{2} \int \frac{1-3 x}{(1+x) \sqrt{x-x^2}} \, dx\\ &=\sqrt{x-x^2}+\frac{3}{2} \int \frac{1}{\sqrt{x-x^2}} \, dx-2 \int \frac{1}{(1+x) \sqrt{x-x^2}} \, dx\\ &=\sqrt{x-x^2}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,1-2 x\right )+4 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,\frac{-1+3 x}{\sqrt{x-x^2}}\right )\\ &=\sqrt{x-x^2}-\frac{3}{2} \sin ^{-1}(1-2 x)+\sqrt{2} \tan ^{-1}\left (\frac{1-3 x}{2 \sqrt{2} \sqrt{x-x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0576623, size = 95, normalized size = 1.76 \[ \sqrt{-(x-1) x}-\frac{3 \sqrt{-(x-1) x} \sin ^{-1}\left (\sqrt{1-x}\right )}{\sqrt{1-x} \sqrt{x}}+\frac{2 \sqrt{2} \sqrt{-(x-1) x} \tanh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2} \sqrt{x}}\right )}{\sqrt{x-1} \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 54, normalized size = 1. \begin{align*} \sqrt{- \left ( 1+x \right ) ^{2}+3\,x+1}+{\frac{3\,\arcsin \left ( 2\,x-1 \right ) }{2}}-\sqrt{2}\arctan \left ({\frac{ \left ( 3\,x-1 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( 1+x \right ) ^{2}+3\,x+1}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60678, size = 57, normalized size = 1.06 \begin{align*} -\sqrt{2} \arcsin \left (\frac{3 \, x}{{\left | x + 1 \right |}} - \frac{1}{{\left | x + 1 \right |}}\right ) + \sqrt{-x^{2} + x} + \frac{3}{2} \, \arcsin \left (2 \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50108, size = 127, normalized size = 2.35 \begin{align*} 2 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + x}}{2 \, x}\right ) + \sqrt{-x^{2} + x} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} + x}}{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{\sqrt{- x \left (x - 1\right )}}{x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13924, size = 72, normalized size = 1.33 \begin{align*} 2 \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (\frac{3 \,{\left (2 \, \sqrt{-x^{2} + x} - 1\right )}}{2 \, x - 1} - 1\right )}\right ) + \sqrt{-x^{2} + x} + \frac{3}{2} \, \arcsin \left (2 \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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