3.601 \(\int \frac{1}{\left (x+\sqrt{-3-2 x+x^2}\right )^2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{3}{2 \left (\sqrt{x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0764641, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+\frac{3}{2 \left (\sqrt{x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x + Sqrt[-3 - 2*x + x^2])^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.63109, size = 70, normalized size = 0.84 \[ - 4 \log{\left (x + \sqrt{x^{2} - 2 x - 3} \right )} + 4 \log{\left (- x - \sqrt{x^{2} - 2 x - 3} + 1 \right )} - \frac{2}{- x - \sqrt{x^{2} - 2 x - 3} + 1} + \frac{3}{2 \left (x + \sqrt{x^{2} - 2 x - 3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x+(x**2-2*x-3)**(1/2))**2,x)

[Out]

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

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Mathematica [A]  time = 0.078783, size = 91, normalized size = 1.1 \[ -\frac{(x+3) \sqrt{x^2-2 x-3}}{2 x+3}+2 \log \left (-3 \sqrt{x^2-2 x-3}+5 x+3\right )+2 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )+\frac{x}{2}-\frac{9}{8 x+12}-4 \log (2 x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-2),x]

[Out]

x/2 - 9/(12 + 8*x) - ((3 + x)*Sqrt[-3 - 2*x + x^2])/(3 + 2*x) - 4*Log[3 + 2*x] +
2*Log[3 + 5*x - 3*Sqrt[-3 - 2*x + x^2]] + 2*Log[1 - x - Sqrt[-3 - 2*x + x^2]]

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Maple [A]  time = 0.026, size = 118, normalized size = 1.4 \[ -2\,\ln \left ( 3+2\,x \right ) +{\frac{x}{2}}-{\frac{9}{12+8\,x}}-{\frac{1}{3} \left ( \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{2}{3}\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}+2\,\ln \left ( -1+x+\sqrt{ \left ( x+3/2 \right ) ^{2}-5\,x-{\frac{21}{4}}} \right ) +2\,{\it Artanh} \left ( 2/3\,{\frac{-3-5\,x}{\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}} \right ) +{\frac{2\,x-2}{6}\sqrt{ \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x+(x^2-2*x-3)^(1/2))^2,x)

[Out]

-2*ln(3+2*x)+1/2*x-9/4/(3+2*x)-1/3/(x+3/2)*((x+3/2)^2-5*x-21/4)^(3/2)-2/3*(4*(x+
3/2)^2-20*x-21)^(1/2)+2*ln(-1+x+((x+3/2)^2-5*x-21/4)^(1/2))+2*arctanh(2/3*(-3-5*
x)/(4*(x+3/2)^2-20*x-21)^(1/2))+1/6*(2*x-2)*((x+3/2)^2-5*x-21/4)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x + \sqrt{x^{2} - 2 \, x - 3}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + sqrt(x^2 - 2*x - 3))^(-2),x, algorithm="maxima")

[Out]

integrate((x + sqrt(x^2 - 2*x - 3))^(-2), x)

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Fricas [A]  time = 0.267667, size = 317, normalized size = 3.82 \[ \frac{8 \, x^{4} - 6 \, x^{3} - 63 \, x^{2} - 8 \,{\left (2 \, x^{3} - x^{2} -{\left (2 \, x^{2} + x - 3\right )} \sqrt{x^{2} - 2 \, x - 3} - 8 \, x - 3\right )} \log \left (x^{2} - \sqrt{x^{2} - 2 \, x - 3}{\left (x + 1\right )} - 3\right ) - 8 \,{\left (2 \, x^{3} - x^{2} - 8 \, x - 3\right )} \log \left (2 \, x + 3\right ) + 8 \,{\left (2 \, x^{3} - x^{2} -{\left (2 \, x^{2} + x - 3\right )} \sqrt{x^{2} - 2 \, x - 3} - 8 \, x - 3\right )} \log \left (-x + \sqrt{x^{2} - 2 \, x - 3}\right ) -{\left (8 \, x^{3} + 2 \, x^{2} - 8 \,{\left (2 \, x^{2} + x - 3\right )} \log \left (2 \, x + 3\right ) - 45 \, x + 3\right )} \sqrt{x^{2} - 2 \, x - 3} + 28 \, x + 51}{4 \,{\left (2 \, x^{3} - x^{2} -{\left (2 \, x^{2} + x - 3\right )} \sqrt{x^{2} - 2 \, x - 3} - 8 \, x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + sqrt(x^2 - 2*x - 3))^(-2),x, algorithm="fricas")

[Out]

1/4*(8*x^4 - 6*x^3 - 63*x^2 - 8*(2*x^3 - x^2 - (2*x^2 + x - 3)*sqrt(x^2 - 2*x -
3) - 8*x - 3)*log(x^2 - sqrt(x^2 - 2*x - 3)*(x + 1) - 3) - 8*(2*x^3 - x^2 - 8*x
- 3)*log(2*x + 3) + 8*(2*x^3 - x^2 - (2*x^2 + x - 3)*sqrt(x^2 - 2*x - 3) - 8*x -
3)*log(-x + sqrt(x^2 - 2*x - 3)) - (8*x^3 + 2*x^2 - 8*(2*x^2 + x - 3)*log(2*x +
3) - 45*x + 3)*sqrt(x^2 - 2*x - 3) + 28*x + 51)/(2*x^3 - x^2 - (2*x^2 + x - 3)*
sqrt(x^2 - 2*x - 3) - 8*x - 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + \sqrt{x^{2} - 2 x - 3}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x+(x**2-2*x-3)**(1/2))**2,x)

[Out]

Integral((x + sqrt(x**2 - 2*x - 3))**(-2), x)

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GIAC/XCAS [A]  time = 0.276419, size = 193, normalized size = 2.33 \[ \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} - 2 \, x - 3} - \frac{3 \,{\left (5 \, x - 5 \, \sqrt{x^{2} - 2 \, x - 3} + 3\right )}}{4 \,{\left ({\left (x - \sqrt{x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt{x^{2} - 2 \, x - 3}\right )}} - \frac{9}{4 \,{\left (2 \, x + 3\right )}} - 2 \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - 2 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 2 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} \right |}\right ) - 2 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + sqrt(x^2 - 2*x - 3))^(-2),x, algorithm="giac")

[Out]

1/2*x - 1/2*sqrt(x^2 - 2*x - 3) - 3/4*(5*x - 5*sqrt(x^2 - 2*x - 3) + 3)/((x - sq
rt(x^2 - 2*x - 3))^2 + 3*x - 3*sqrt(x^2 - 2*x - 3)) - 9/4/(2*x + 3) - 2*ln(abs(2
*x + 3)) - 2*ln(abs(-x + sqrt(x^2 - 2*x - 3) + 1)) + 2*ln(abs(-x + sqrt(x^2 - 2*
x - 3))) - 2*ln(abs(-x + sqrt(x^2 - 2*x - 3) - 3))