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

Optimal. Leaf size=65 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+2 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-\frac{3}{2} \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0678783, antiderivative size = 65, 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 \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+2 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-\frac{3}{2} \log \left (\sqrt{x^2-2 x-3}+x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x + Sqrt[-3 - 2*x + x^2])^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.97313, size = 53, normalized size = 0.82 \[ - \frac{3 \log{\left (x + \sqrt{x^{2} - 2 x - 3} \right )}}{2} + 2 \log{\left (- x - \sqrt{x^{2} - 2 x - 3} + 1 \right )} - \frac{2}{- x - \sqrt{x^{2} - 2 x - 3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0447807, size = 74, normalized size = 1.14 \[ \frac{1}{4} \left (-2 \sqrt{x^2-2 x-3}+3 \log \left (-3 \sqrt{x^2-2 x-3}+5 x+3\right )+5 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )+2 x-6 \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x + Sqrt[-3 - 2*x + x^2])^(-1),x]

[Out]

(2*x - 2*Sqrt[-3 - 2*x + x^2] - 6*Log[3 + 2*x] + 3*Log[3 + 5*x - 3*Sqrt[-3 - 2*x
+ x^2]] + 5*Log[1 - x - Sqrt[-3 - 2*x + x^2]])/4

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Maple [A]  time = 0.008, size = 71, normalized size = 1.1 \[ -{\frac{1}{4}\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}+{\frac{5}{4}\ln \left ( -1+x+\sqrt{ \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}}} \right ) }+{\frac{3}{4}{\it Artanh} \left ({\frac{-6-10\,x}{3}{\frac{1}{\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}}} \right ) }+{\frac{x}{2}}-{\frac{3\,\ln \left ( 3+2\,x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x+(x^2-2*x-3)^(1/2)),x)

[Out]

-1/4*(4*(x+3/2)^2-20*x-21)^(1/2)+5/4*ln(-1+x+((x+3/2)^2-5*x-21/4)^(1/2))+3/4*arc
tanh(2/3*(-3-5*x)/(4*(x+3/2)^2-20*x-21)^(1/2))+1/2*x-3/4*ln(3+2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x + sqrt(x^2 - 2*x - 3)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.272226, size = 220, normalized size = 3.38 \[ \frac{4 \, x^{2} - 3 \,{\left (x - 1\right )} \log \left (2 \, x + 3\right ) - 5 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3} - 1\right )} \log \left (-x + \sqrt{x^{2} - 2 \, x - 3} + 1\right ) + 3 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3} - 1\right )} \log \left (-x + \sqrt{x^{2} - 2 \, x - 3}\right ) - 3 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3} - 1\right )} \log \left (-x + \sqrt{x^{2} - 2 \, x - 3} - 3\right ) - \sqrt{x^{2} - 2 \, x - 3}{\left (4 \, x - 3 \, \log \left (2 \, x + 3\right ) - 1\right )} - 5 \, x - 7}{4 \,{\left (x - \sqrt{x^{2} - 2 \, x - 3} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x + sqrt(x^2 - 2*x - 3)),x, algorithm="fricas")

[Out]

1/4*(4*x^2 - 3*(x - 1)*log(2*x + 3) - 5*(x - sqrt(x^2 - 2*x - 3) - 1)*log(-x + s
qrt(x^2 - 2*x - 3) + 1) + 3*(x - sqrt(x^2 - 2*x - 3) - 1)*log(-x + sqrt(x^2 - 2*
x - 3)) - 3*(x - sqrt(x^2 - 2*x - 3) - 1)*log(-x + sqrt(x^2 - 2*x - 3) - 3) - sq
rt(x^2 - 2*x - 3)*(4*x - 3*log(2*x + 3) - 1) - 5*x - 7)/(x - sqrt(x^2 - 2*x - 3)
- 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x + sqrt(x**2 - 2*x - 3)), x)

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GIAC/XCAS [A]  time = 0.276838, size = 109, normalized size = 1.68 \[ \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} - 2 \, x - 3} - \frac{3}{4} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{5}{4} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} + 1 \right |}\right ) + \frac{3}{4} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} \right |}\right ) - \frac{3}{4} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 2 \, x - 3} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x + sqrt(x^2 - 2*x - 3)),x, algorithm="giac")

[Out]

1/2*x - 1/2*sqrt(x^2 - 2*x - 3) - 3/4*ln(abs(2*x + 3)) - 5/4*ln(abs(-x + sqrt(x^
2 - 2*x - 3) + 1)) + 3/4*ln(abs(-x + sqrt(x^2 - 2*x - 3))) - 3/4*ln(abs(-x + sqr
t(x^2 - 2*x - 3) - 3))