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 = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 204
Rule 216
Rule 619
Rule 724
Rule 734
Rule 843
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*}
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Mathematica [A] time = 0.08, 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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fricas [A] time = 0.61, size = 49, normalized size = 0.91 \[ 2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + x}}{2 \, x}\right ) + \sqrt {-x^{2} + x} - 3 \, \arctan \left (\frac {\sqrt {-x^{2} + x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 53, normalized size = 0.98 \[ 2 \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (\frac {3 \, {\left (2 \, \sqrt {-x^{2} + x} - 1\right )}}{2 \, x - 1} - 1\right )}\right ) + \sqrt {-x^{2} + x} + \frac {3}{2} \, \arcsin \left (2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 1.00 \[ \frac {3 \arcsin \left (2 x -1\right )}{2}-\sqrt {2}\, \arctan \left (\frac {\left (3 x -1\right ) \sqrt {2}}{4 \sqrt {3 x -\left (x +1\right )^{2}+1}}\right )+\sqrt {3 x -\left (x +1\right )^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.91, size = 42, normalized size = 0.78 \[ -\sqrt {2} \arcsin \left (\frac {3 \, x}{{\left | x + 1 \right |}} - \frac {1}{{\left | x + 1 \right |}}\right ) + \sqrt {-x^{2} + x} + \frac {3}{2} \, \arcsin \left (2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {x-x^2}}{x+1} \,d x \]
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 \[ \int \frac {\sqrt {- x \left (x - 1\right )}}{x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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