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

Optimal. Leaf size=180 \[ -\frac{1}{2} \log \left (-\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-x+3}{x^2}\right )+\frac{1}{14} \left (7+\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{7}+\sqrt{3}+1\right )+\frac{1}{14} \left (7-\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}+\sqrt{7}+\sqrt{3}+1\right )+\tan ^{-1}\left (\frac{\sqrt{3}-\sqrt{-x^2-2 x+3}}{x}\right ) \]

[Out]

ArcTan[(Sqrt[3] - Sqrt[3 - 2*x - x^2])/x] - Log[-((3 - x - Sqrt[3]*Sqrt[3 - 2*x
- x^2])/x^2)]/2 + ((7 + Sqrt[7])*Log[1 + Sqrt[3] - Sqrt[7] - (Sqrt[3]*(Sqrt[3] -
Sqrt[3 - 2*x - x^2]))/x])/14 + ((7 - Sqrt[7])*Log[1 + Sqrt[3] + Sqrt[7] - (Sqrt
[3]*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x])/14

_______________________________________________________________________________________

Rubi [A]  time = 0.41385, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{2} \log \left (-\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-x+3}{x^2}\right )+\frac{1}{14} \left (7+\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{7}+\sqrt{3}+1\right )+\frac{1}{14} \left (7-\sqrt{7}\right ) \log \left (-\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}+\sqrt{7}+\sqrt{3}+1\right )+\tan ^{-1}\left (\frac{\sqrt{3}-\sqrt{-x^2-2 x+3}}{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(Sqrt[3] - Sqrt[3 - 2*x - x^2])/x] - Log[-((3 - x - Sqrt[3]*Sqrt[3 - 2*x
- x^2])/x^2)]/2 + ((7 + Sqrt[7])*Log[1 + Sqrt[3] - Sqrt[7] - (Sqrt[3]*(Sqrt[3] -
Sqrt[3 - 2*x - x^2]))/x])/14 + ((7 - Sqrt[7])*Log[1 + Sqrt[3] + Sqrt[7] - (Sqrt
[3]*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x])/14

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 84.3252, size = 180, normalized size = 1. \[ - \frac{\log{\left (1 + \frac{\left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )^{2}}{x^{2}} \right )}}{2} + \frac{\log{\left (-6 + 4 \sqrt{3} + \frac{\left (4 \sqrt{3} + 12\right ) \left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )}{x} + \frac{\left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )^{2}}{x^{2}} \right )}}{2} - \operatorname{atan}{\left (\frac{\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}}{x} \right )} + \frac{\sqrt{2} \left (- 2 \sqrt{3} + 5\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt{3} + 3 + \frac{\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}}{2 x}\right )}{\sqrt{10 \sqrt{3} + 27}} \right )}}{2 \sqrt{10 \sqrt{3} + 27}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(1 + (sqrt(-x**2 - 2*x + 3) - sqrt(3))**2/x**2)/2 + log(-6 + 4*sqrt(3) + (4*
sqrt(3) + 12)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))/x + (sqrt(-x**2 - 2*x + 3) - sqr
t(3))**2/x**2)/2 - atan((sqrt(-x**2 - 2*x + 3) - sqrt(3))/x) + sqrt(2)*(-2*sqrt(
3) + 5)*atanh(sqrt(2)*(sqrt(3) + 3 + (sqrt(-x**2 - 2*x + 3) - sqrt(3))/(2*x))/sq
rt(10*sqrt(3) + 27))/(2*sqrt(10*sqrt(3) + 27))

_______________________________________________________________________________________

Mathematica [A]  time = 0.89127, size = 250, normalized size = 1.39 \[ \frac{1}{28} \left (-\sqrt{14 \left (4+\sqrt{7}\right )} \log \left (\sqrt{14 \left (4+\sqrt{7}\right )} \sqrt{-x^2-2 x+3}-\sqrt{7} x+7 x+7 \sqrt{7}+7\right )-\frac{1}{3} \left (\sqrt{7}-4\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (-\sqrt{14} \sqrt{\left (\sqrt{7}-4\right ) \left (x^2+2 x-3\right )}+\left (7+\sqrt{7}\right ) x-7 \sqrt{7}+7\right )-\left (\sqrt{7}-7\right ) \log \left (-2 x+\sqrt{7}-1\right )+\frac{1}{3} \left (\sqrt{7}-4\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (2 x-\sqrt{7}+1\right )+\left (7+\sqrt{7}\right ) \log \left (2 x+\sqrt{7}+1\right )+\sqrt{14 \left (4+\sqrt{7}\right )} \log \left (2 x+\sqrt{7}+1\right )+14 \sin ^{-1}\left (\frac{x+1}{2}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(14*ArcSin[(1 + x)/2] - (-7 + Sqrt[7])*Log[-1 + Sqrt[7] - 2*x] + ((-4 + Sqrt[7])
*Sqrt[14*(4 + Sqrt[7])]*Log[1 - Sqrt[7] + 2*x])/3 + Sqrt[14*(4 + Sqrt[7])]*Log[1
+ Sqrt[7] + 2*x] + (7 + Sqrt[7])*Log[1 + Sqrt[7] + 2*x] - Sqrt[14*(4 + Sqrt[7])
]*Log[7 + 7*Sqrt[7] + 7*x - Sqrt[7]*x + Sqrt[14*(4 + Sqrt[7])]*Sqrt[3 - 2*x - x^
2]] - ((-4 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[7 - 7*Sqrt[7] + (7 + Sqrt[7])*x
- Sqrt[14]*Sqrt[(-4 + Sqrt[7])*(-3 + 2*x + x^2)]])/3)/28

_______________________________________________________________________________________

Maple [B]  time = 0.084, size = 551, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x+(-x^2-2*x+3)^(1/2)),x)

[Out]

-1/28*7^(1/2)*(-4*(x+1/2-1/2*7^(1/2))^2+4*(-1-7^(1/2))*(x+1/2-1/2*7^(1/2))+8-2*7
^(1/2))^(1/2)+1/28*arcsin(1/(2-1/2*7^(1/2)+1/4*(-1-7^(1/2))^2)^(1/2)*(1+x))*7^(1
/2)+1/4*arcsin(1/(2-1/2*7^(1/2)+1/4*(-1-7^(1/2))^2)^(1/2)*(1+x))+1/7/(-1/2+1/2*7
^(1/2))*arctanh((4-7^(1/2)+(-1-7^(1/2))*(x+1/2-1/2*7^(1/2)))/(-1/2+1/2*7^(1/2))/
(-4*(x+1/2-1/2*7^(1/2))^2+4*(-1-7^(1/2))*(x+1/2-1/2*7^(1/2))+8-2*7^(1/2))^(1/2))
*7^(1/2)-1/4/(-1/2+1/2*7^(1/2))*arctanh((4-7^(1/2)+(-1-7^(1/2))*(x+1/2-1/2*7^(1/
2)))/(-1/2+1/2*7^(1/2))/(-4*(x+1/2-1/2*7^(1/2))^2+4*(-1-7^(1/2))*(x+1/2-1/2*7^(1
/2))+8-2*7^(1/2))^(1/2))+1/28*7^(1/2)*(-4*(x+1/2+1/2*7^(1/2))^2+4*(-1+7^(1/2))*(
x+1/2+1/2*7^(1/2))+8+2*7^(1/2))^(1/2)-1/28*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(1/
2))^2)^(1/2)*(1+x))*7^(1/2)+1/4*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(1/2))^2)^(1/2
)*(1+x))-1/7/(1/2*7^(1/2)+1/2)*arctanh((4+7^(1/2)+(-1+7^(1/2))*(x+1/2+1/2*7^(1/2
)))/(1/2*7^(1/2)+1/2)/(-4*(x+1/2+1/2*7^(1/2))^2+4*(-1+7^(1/2))*(x+1/2+1/2*7^(1/2
))+8+2*7^(1/2))^(1/2))*7^(1/2)-1/4/(1/2*7^(1/2)+1/2)*arctanh((4+7^(1/2)+(-1+7^(1
/2))*(x+1/2+1/2*7^(1/2)))/(1/2*7^(1/2)+1/2)/(-4*(x+1/2+1/2*7^(1/2))^2+4*(-1+7^(1
/2))*(x+1/2+1/2*7^(1/2))+8+2*7^(1/2))^(1/2))+1/4*ln(2*x^2+2*x-3)+1/14*7^(1/2)*ar
ctanh(1/14*(4*x+2)*7^(1/2))

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

Fricas [A]  time = 0.291535, size = 379, normalized size = 2.11 \[ \frac{1}{56} \, \sqrt{7}{\left (4 \, \sqrt{7} \arctan \left (\frac{x + 1}{\sqrt{-x^{2} - 2 \, x + 3}}\right ) + 2 \, \sqrt{7} \log \left (2 \, x^{2} + 2 \, x - 3\right ) - \sqrt{7} \log \left (\frac{2 \, \sqrt{-x^{2} - 2 \, x + 3} x + 2 \, x - 3}{x^{2}}\right ) + \sqrt{7} \log \left (-\frac{2 \, \sqrt{-x^{2} - 2 \, x + 3} x - 2 \, x + 3}{x^{2}}\right ) + 2 \, \log \left (\frac{2 \, \sqrt{7}{\left (x^{2} + x + 2\right )} + 14 \, x + 7}{2 \, x^{2} + 2 \, x - 3}\right ) + \log \left (\frac{28 \, x^{2} + \sqrt{7}{\left (7 \, x^{2} - 30 \, x + 45\right )} + 3 \, \sqrt{-x^{2} - 2 \, x + 3}{\left (4 \, \sqrt{7} x + 7 \, x - 21\right )} - 84 \, x}{2 \, \sqrt{-x^{2} - 2 \, x + 3} x + 2 \, x - 3}\right ) + \log \left (\frac{28 \, x^{2} - \sqrt{7}{\left (7 \, x^{2} - 30 \, x + 45\right )} + 3 \, \sqrt{-x^{2} - 2 \, x + 3}{\left (4 \, \sqrt{7} x - 7 \, x + 21\right )} - 84 \, x}{2 \, \sqrt{-x^{2} - 2 \, x + 3} x - 2 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/56*sqrt(7)*(4*sqrt(7)*arctan((x + 1)/sqrt(-x^2 - 2*x + 3)) + 2*sqrt(7)*log(2*x
^2 + 2*x - 3) - sqrt(7)*log((2*sqrt(-x^2 - 2*x + 3)*x + 2*x - 3)/x^2) + sqrt(7)*
log(-(2*sqrt(-x^2 - 2*x + 3)*x - 2*x + 3)/x^2) + 2*log((2*sqrt(7)*(x^2 + x + 2)
+ 14*x + 7)/(2*x^2 + 2*x - 3)) + log((28*x^2 + sqrt(7)*(7*x^2 - 30*x + 45) + 3*s
qrt(-x^2 - 2*x + 3)*(4*sqrt(7)*x + 7*x - 21) - 84*x)/(2*sqrt(-x^2 - 2*x + 3)*x +
2*x - 3)) + log((28*x^2 - sqrt(7)*(7*x^2 - 30*x + 45) + 3*sqrt(-x^2 - 2*x + 3)*
(4*sqrt(7)*x - 7*x + 21) - 84*x)/(2*sqrt(-x^2 - 2*x + 3)*x - 2*x + 3)))

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.323625, size = 387, normalized size = 2.15 \[ -\frac{1}{28} \, \sqrt{7}{\rm ln}\left (\frac{{\left | 4 \, x - 2 \, \sqrt{7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt{7} + 2 \right |}}\right ) + \frac{1}{28} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac{1}{28} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) + \frac{1}{2} \, \arcsin \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | 2 \, x^{2} + 2 \, x - 3 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | \frac{4 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{3 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | -\frac{4 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/28*sqrt(7)*ln(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 1/28*sqrt(
7)*ln(abs(-2*sqrt(7) + 6*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)/abs(2*sqrt(7) +
6*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)) - 1/28*sqrt(7)*ln(abs(-2*sqrt(7) + 2
*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) - 4)/abs(2*sqrt(7) + 2*(sqrt(-x^2 - 2*x + 3)
- 2)/(x + 1) - 4)) + 1/2*arcsin(1/2*x + 1/2) + 1/4*ln(abs(2*x^2 + 2*x - 3)) + 1
/4*ln(abs(4*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 3*(sqrt(-x^2 - 2*x + 3) - 2)^2/
(x + 1)^2 - 1)) - 1/4*ln(abs(-4*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + (sqrt(-x^2
- 2*x + 3) - 2)^2/(x + 1)^2 - 3))