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

Optimal. Leaf size=31 \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

[Out]

3/(5*(4 + 5*x)) + Sqrt[1 - x^2]/(4 + 5*x)

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Rubi [A]  time = 1.24118, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 14, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.326 \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

Antiderivative was successfully verified.

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

[Out]

3/(5*(4 + 5*x)) + Sqrt[1 - x^2]/(4 + 5*x)

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Rubi in Sympy [A]  time = 22.256, size = 15, normalized size = 0.48 \[ \frac{1}{2 \left (1 - \frac{2 \left (\sqrt{- x^{2} + 1} - 1\right )}{x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*(1 - 2*(sqrt(-x**2 + 1) - 1)/x))

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Mathematica [A]  time = 0.0435279, size = 23, normalized size = 0.74 \[ \frac{5 \sqrt{1-x^2}+3}{25 x+20} \]

Antiderivative was successfully verified.

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

[Out]

(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)

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Maple [A]  time = 0.009, size = 32, normalized size = 1. \[{\frac{1}{5}\sqrt{- \left ( x+{\frac{4}{5}} \right ) ^{2}+{\frac{8\,x}{5}}+{\frac{41}{25}}} \left ( x+{\frac{4}{5}} \right ) ^{-1}}+{\frac{3}{20+25\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5/(x+4/5)*(-(x+4/5)^2+8/5*x+41/25)^(1/2)+3/5/(4+5*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{56} \, \sqrt{7} \log \left (\frac{3 \, x - 2 \, \sqrt{7} - 2}{3 \, x + 2 \, \sqrt{7} - 2}\right ) - \int -\frac{100 \, x^{7} + 285 \, x^{6} + 264 \, x^{5} + 80 \, x^{4}}{8 \,{\left (21 \, x^{9} + 278 \, x^{8} + 283 \, x^{7} - 2022 \, x^{6} - 3632 \, x^{5} + 2256 \, x^{4} + 7424 \, x^{3} + 1536 \, x^{2} - 8 \,{\left (9 \, x^{8} + 12 \, x^{7} - 101 \, x^{6} - 172 \, x^{5} + 284 \, x^{4} + 672 \, x^{3} + 64 \, x^{2} - 512 \, x - 256\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4096 \, x - 2048\right )}}\,{d x} - \frac{1}{24} \, \log \left (x + 2\right ) + \frac{1}{16} \, \log \left (x + 1\right ) - \frac{1}{48} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/56*sqrt(7)*log((3*x - 2*sqrt(7) - 2)/(3*x + 2*sqrt(7) - 2)) - integrate(-1/8*
(100*x^7 + 285*x^6 + 264*x^5 + 80*x^4)/(21*x^9 + 278*x^8 + 283*x^7 - 2022*x^6 -
3632*x^5 + 2256*x^4 + 7424*x^3 + 1536*x^2 - 8*(9*x^8 + 12*x^7 - 101*x^6 - 172*x^
5 + 284*x^4 + 672*x^3 + 64*x^2 - 512*x - 256)*sqrt(x + 1)*sqrt(-x + 1) - 4096*x
- 2048), x) - 1/24*log(x + 2) + 1/16*log(x + 1) - 1/48*log(x - 1)

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Fricas [A]  time = 0.26752, size = 68, normalized size = 2.19 \[ -\frac{20 \, x^{2} - \sqrt{-x^{2} + 1}{\left (25 \, x + 12\right )} + 25 \, x + 12}{20 \,{\left (\sqrt{-x^{2} + 1}{\left (5 \, x + 4\right )} - 5 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*x^2 - sqrt(-x^2 + 1)*(25*x + 12) + 25*x + 12)/(sqrt(-x^2 + 1)*(5*x + 4
) - 5*x - 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285564, size = 92, normalized size = 2.97 \[ \frac{\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - 4}{4 \,{\left (\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 2\right )}} + \frac{3}{5 \,{\left (5 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(5*(sqrt(-x^2 + 1) - 1)/x - 4)/(5*(sqrt(-x^2 + 1) - 1)/x - 2*(sqrt(-x^2 + 1)
- 1)^2/x^2 - 2) + 3/5/(5*x + 4)