3.787 \(\int \frac{\sqrt{x-x^2}}{1+x} \, dx\)

Optimal. Leaf size=54 \[ \sqrt{x-x^2}+\sqrt{2} \tan ^{-1}\left (\frac{1-3 x}{2 \sqrt{2} \sqrt{x-x^2}}\right )-\frac{3}{2} \sin ^{-1}(1-2 x) \]

[Out]

Sqrt[x - x^2] - (3*ArcSin[1 - 2*x])/2 + Sqrt[2]*ArcTan[(1 - 3*x)/(2*Sqrt[2]*Sqrt
[x - x^2])]

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Rubi [A]  time = 0.114361, 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 \[ \sqrt{x-x^2}+\sqrt{2} \tan ^{-1}\left (\frac{1-3 x}{2 \sqrt{2} \sqrt{x-x^2}}\right )-\frac{3}{2} \sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x - x^2]/(1 + x),x]

[Out]

Sqrt[x - x^2] - (3*ArcSin[1 - 2*x])/2 + Sqrt[2]*ArcTan[(1 - 3*x)/(2*Sqrt[2]*Sqrt
[x - x^2])]

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Rubi in Sympy [A]  time = 6.65732, size = 44, normalized size = 0.81 \[ \sqrt{- x^{2} + x} + \frac{3 \operatorname{asin}{\left (2 x - 1 \right )}}{2} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (3 x - 1\right )}{4 \sqrt{- x^{2} + x}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+x)**(1/2)/(1+x),x)

[Out]

sqrt(-x**2 + x) + 3*asin(2*x - 1)/2 - sqrt(2)*atan(sqrt(2)*(3*x - 1)/(4*sqrt(-x*
*2 + x)))

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Mathematica [B]  time = 0.0817704, size = 120, normalized size = 2.22 \[ \frac{\sqrt{-(x-1) x} \left (2 \sqrt{x-1} \sqrt{x}-6 \log \left (\sqrt{x-1}+\sqrt{x}\right )+\sqrt{2} \log \left (-3 x-2 \sqrt{2} \sqrt{x-1} \sqrt{x}+1\right )-\sqrt{2} \log \left (-3 x+2 \sqrt{2} \sqrt{x-1} \sqrt{x}+1\right )\right )}{2 \sqrt{x-1} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x - x^2]/(1 + x),x]

[Out]

(Sqrt[-((-1 + x)*x)]*(2*Sqrt[-1 + x]*Sqrt[x] - 6*Log[Sqrt[-1 + x] + Sqrt[x]] + S
qrt[2]*Log[1 - 2*Sqrt[2]*Sqrt[-1 + x]*Sqrt[x] - 3*x] - Sqrt[2]*Log[1 + 2*Sqrt[2]
*Sqrt[-1 + x]*Sqrt[x] - 3*x]))/(2*Sqrt[-1 + x]*Sqrt[x])

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Maple [A]  time = 0.01, size = 54, normalized size = 1. \[ \sqrt{- \left ( 1+x \right ) ^{2}+1+3\,x}+{\frac{3\,\arcsin \left ( 2\,x-1 \right ) }{2}}-\sqrt{2}\arctan \left ({\frac{ \left ( 3\,x-1 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( 1+x \right ) ^{2}+1+3\,x}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+x)^(1/2)/(1+x),x)

[Out]

(-(1+x)^2+1+3*x)^(1/2)+3/2*arcsin(2*x-1)-2^(1/2)*arctan(1/4*(3*x-1)*2^(1/2)/(-(1
+x)^2+1+3*x)^(1/2))

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Maxima [A]  time = 0.76602, size = 57, normalized size = 1.06 \[ -\sqrt{2} \arcsin \left (\frac{3 \, x}{{\left | x + 1 \right |}} - \frac{1}{{\left | x + 1 \right |}}\right ) + \sqrt{-x^{2} + x} + \frac{3}{2} \, \arcsin \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + x)/(x + 1),x, algorithm="maxima")

[Out]

-sqrt(2)*arcsin(3*x/abs(x + 1) - 1/abs(x + 1)) + sqrt(-x^2 + x) + 3/2*arcsin(2*x
- 1)

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Fricas [A]  time = 0.274778, size = 66, normalized size = 1.22 \[ 2 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + x}}{2 \, x}\right ) + \sqrt{-x^{2} + x} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} + x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + x)/(x + 1),x, algorithm="fricas")

[Out]

2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-x^2 + x)/x) + sqrt(-x^2 + x) - 3*arctan(sqrt(
-x^2 + x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- x \left (x - 1\right )}}{x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+x)**(1/2)/(1+x),x)

[Out]

Integral(sqrt(-x*(x - 1))/(x + 1), x)

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GIAC/XCAS [A]  time = 0.266554, size = 72, normalized size = 1.33 \[ 2 \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (\frac{3 \,{\left (2 \, \sqrt{-x^{2} + x} - 1\right )}}{2 \, x - 1} - 1\right )}\right ) + \sqrt{-x^{2} + x} + \frac{3}{2} \, \arcsin \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + x)/(x + 1),x, algorithm="giac")

[Out]

2*sqrt(2)*arctan(1/4*sqrt(2)*(3*(2*sqrt(-x^2 + x) - 1)/(2*x - 1) - 1)) + sqrt(-x
^2 + x) + 3/2*arcsin(2*x - 1)