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

Optimal. Leaf size=172 \[ \frac{2 \left (\frac{3 \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+4\right )}{7 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )}+\frac{8 \tanh ^{-1}\left (\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right )}{7 \sqrt{7}} \]

[Out]

(2*(4 - Sqrt[3] + (3*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(7*(2 - Sqrt[3] - (2*(
1 + Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(Sqrt[3] - Sqrt[3 - 2
*x - x^2])^2)/x^2)) + (8*ArcTanh[(3 - x - Sqrt[3]*x - Sqrt[3]*Sqrt[3 - 2*x - x^2
])/(Sqrt[7]*x)])/(7*Sqrt[7])

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Rubi [A]  time = 0.253153, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 \left (\frac{3 \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+4\right )}{7 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )}+\frac{8 \tanh ^{-1}\left (\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(4 - Sqrt[3] + (3*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(7*(2 - Sqrt[3] - (2*(
1 + Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(Sqrt[3] - Sqrt[3 - 2
*x - x^2])^2)/x^2)) + (8*ArcTanh[(3 - x - Sqrt[3]*x - Sqrt[3]*Sqrt[3 - 2*x - x^2
])/(Sqrt[7]*x)])/(7*Sqrt[7])

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Rubi in Sympy [A]  time = 30.4101, size = 264, normalized size = 1.53 \[ \frac{2 \sqrt{3} \left (- 16 \sqrt{3} + 12 + \frac{\left (\sqrt{3} \left (- \left (2 + 2 \sqrt{3}\right )^{2} + \sqrt{3} \left (- 2 \sqrt{3} + 4\right )\right ) + \sqrt{3} \left (4 \sqrt{3} + 10\right )\right ) \left (- \sqrt{- x^{2} - 2 x + 3} + \sqrt{3}\right )}{x}\right )}{3 \left (- \left (2 + 2 \sqrt{3}\right )^{2} - \sqrt{3} \left (-8 + 4 \sqrt{3}\right )\right ) \left (- \sqrt{3} + 2 + \frac{\left (2 + 2 \sqrt{3}\right ) \left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )}{x} + \frac{\sqrt{3} \left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )^{2}}{x^{2}}\right )} - \frac{2 \sqrt{21} \left (- \sqrt{3} \left (4 \sqrt{3} + 10\right ) - 6 \sqrt{3} + 12\right ) \operatorname{atanh}{\left (\sqrt{7} \left (\frac{1}{7} + \frac{\sqrt{3}}{7} + \frac{\sqrt{3} \left (\sqrt{- x^{2} - 2 x + 3} - \sqrt{3}\right )}{7 x}\right ) \right )}}{21 \left (- \left (2 + 2 \sqrt{3}\right )^{2} - \sqrt{3} \left (-8 + 4 \sqrt{3}\right )\right )} \]

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]

2*sqrt(3)*(-16*sqrt(3) + 12 + (sqrt(3)*(-(2 + 2*sqrt(3))**2 + sqrt(3)*(-2*sqrt(3
) + 4)) + sqrt(3)*(4*sqrt(3) + 10))*(-sqrt(-x**2 - 2*x + 3) + sqrt(3))/x)/(3*(-(
2 + 2*sqrt(3))**2 - sqrt(3)*(-8 + 4*sqrt(3)))*(-sqrt(3) + 2 + (2 + 2*sqrt(3))*(s
qrt(-x**2 - 2*x + 3) - sqrt(3))/x + sqrt(3)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))**2
/x**2)) - 2*sqrt(21)*(-sqrt(3)*(4*sqrt(3) + 10) - 6*sqrt(3) + 12)*atanh(sqrt(7)*
(1/7 + sqrt(3)/7 + sqrt(3)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))/(7*x)))/(21*(-(2 +
2*sqrt(3))**2 - sqrt(3)*(-8 + 4*sqrt(3))))

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Mathematica [A]  time = 1.02062, size = 306, normalized size = 1.78 \[ \frac{1}{98} \left (\frac{7 (3-8 x)}{2 x^2+2 x-3}-\frac{14 (x-3) \sqrt{-x^2-2 x+3}}{2 x^2+2 x-3}-2 \left (1+\sqrt{7}\right ) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left (\sqrt{14 \left (4+\sqrt{7}\right )} \sqrt{-x^2-2 x+3}-\sqrt{7} x+7 x+7 \sqrt{7}+7\right )-\frac{2}{3} \left (\sqrt{7}-1\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (-\sqrt{14} \sqrt{\left (\sqrt{7}-4\right ) \left (x^2+2 x-3\right )}+\left (7+\sqrt{7}\right ) x-7 \sqrt{7}+7\right )-4 \sqrt{7} \log \left (-2 x+\sqrt{7}-1\right )+\frac{2}{3} \left (\sqrt{7}-1\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (2 x-\sqrt{7}+1\right )+2 \left (1+\sqrt{7}\right ) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left (2 x+\sqrt{7}+1\right )+4 \sqrt{7} \log \left (2 x+\sqrt{7}+1\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((7*(3 - 8*x))/(-3 + 2*x + 2*x^2) - (14*(-3 + x)*Sqrt[3 - 2*x - x^2])/(-3 + 2*x
+ 2*x^2) - 4*Sqrt[7]*Log[-1 + Sqrt[7] - 2*x] + (2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sq
rt[7])]*Log[1 - Sqrt[7] + 2*x])/3 + 4*Sqrt[7]*Log[1 + Sqrt[7] + 2*x] + 2*(1 + Sq
rt[7])*Sqrt[14/(4 + Sqrt[7])]*Log[1 + Sqrt[7] + 2*x] - 2*(1 + Sqrt[7])*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]] - (2*(-1 + 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)/98

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Maple [B]  time = 0.042, size = 1066, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/28*(4*x+2)/(2*x^2+2*x-3)+4/49*7^(1/2)*arctanh(1/14*(4*x+2)*7^(1/2))+1/14*(-2*
x+6)/(2*x^2+2*x-3)-1/49*7^(1/2)*(1/4*(-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*(-1-7^(1/2))*arcsin(1/(2-1/2*7^(1/2)+1/
4*(-1-7^(1/2))^2)^(1/2)*(1+x))-1/2*(2-1/2*7^(1/2))/(-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)))+1/49*7^(1/2)*(1/
4*(-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*(-1+7^(1/2))*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(1/2))^2)^(1/2)*(1+x))-1/2*
(2+1/2*7^(1/2))/(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)))-2*(-1/14-1/14*7^(1/2))*(-1/4/(2+1/2*7^(1/2))/(x+1/2+1
/2*7^(1/2))*(-(x+1/2+1/2*7^(1/2))^2+(-1+7^(1/2))*(x+1/2+1/2*7^(1/2))+2+1/2*7^(1/
2))^(3/2)+1/8*(-1+7^(1/2))/(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)+1/2*(-1+7^(1/2))*arcsin(1/(2+1/2
*7^(1/2)+1/4*(-1+7^(1/2))^2)^(1/2)*(1+x))-(2+1/2*7^(1/2))/(1/2*7^(1/2)+1/2)*arct
anh((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)))-1/2/(2+1/2*
7^(1/2))*(-1/4*(-2*x-2)*(-(x+1/2+1/2*7^(1/2))^2+(-1+7^(1/2))*(x+1/2+1/2*7^(1/2))
+2+1/2*7^(1/2))^(1/2)-1/8*(-8-2*7^(1/2)-(-1+7^(1/2))^2)*arcsin(1/(2+1/2*7^(1/2)+
1/4*(-1+7^(1/2))^2)^(1/2)*(1+x))))-2*(-1/14+1/14*7^(1/2))*(-1/4/(2-1/2*7^(1/2))/
(x+1/2-1/2*7^(1/2))*(-(x+1/2-1/2*7^(1/2))^2+(-1-7^(1/2))*(x+1/2-1/2*7^(1/2))+2-1
/2*7^(1/2))^(3/2)+1/8*(-1-7^(1/2))/(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)+1/2*(-1-7^(1/2))*arcsin(
1/(2-1/2*7^(1/2)+1/4*(-1-7^(1/2))^2)^(1/2)*(1+x))-(2-1/2*7^(1/2))/(-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
/2/(2-1/2*7^(1/2))*(-1/4*(-2*x-2)*(-(x+1/2-1/2*7^(1/2))^2+(-1-7^(1/2))*(x+1/2-1/
2*7^(1/2))+2-1/2*7^(1/2))^(1/2)-1/8*(-8+2*7^(1/2)-(-1-7^(1/2))^2)*arcsin(1/(2-1/
2*7^(1/2)+1/4*(-1-7^(1/2))^2)^(1/2)*(1+x))))

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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 = 0.277265, size = 236, normalized size = 1.37 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7} \sqrt{-x^{2} - 2 \, x + 3}{\left (x - 3\right )} - 2 \,{\left (2 \, x^{2} + 2 \, x - 3\right )} \log \left (\frac{\sqrt{7}{\left (x^{4} + 44 \, x^{3} + 26 \, x^{2} - 276 \, x + 207\right )} - 7 \,{\left (3 \, x^{3} + x^{2} - 45 \, x + 45\right )} \sqrt{-x^{2} - 2 \, x + 3}}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) - 4 \,{\left (2 \, x^{2} + 2 \, x - 3\right )} \log \left (\frac{2 \, \sqrt{7}{\left (x^{2} + x + 2\right )} + 14 \, x + 7}{2 \, x^{2} + 2 \, x - 3}\right ) + \sqrt{7}{\left (8 \, x - 3\right )}\right )}}{98 \,{\left (2 \, x^{2} + 2 \, x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/98*sqrt(7)*(2*sqrt(7)*sqrt(-x^2 - 2*x + 3)*(x - 3) - 2*(2*x^2 + 2*x - 3)*log(
(sqrt(7)*(x^4 + 44*x^3 + 26*x^2 - 276*x + 207) - 7*(3*x^3 + x^2 - 45*x + 45)*sqr
t(-x^2 - 2*x + 3))/(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)) - 4*(2*x^2 + 2*x - 3)*log
((2*sqrt(7)*(x^2 + x + 2) + 14*x + 7)/(2*x^2 + 2*x - 3)) + sqrt(7)*(8*x - 3))/(2
*x^2 + 2*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 = 0.309997, size = 473, normalized size = 2.75 \[ -\frac{2}{49} \, \sqrt{7}{\rm ln}\left (\frac{{\left | 4 \, x - 2 \, \sqrt{7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt{7} + 2 \right |}}\right ) + \frac{2}{49} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac{2}{49} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) - \frac{8 \, x - 3}{14 \,{\left (2 \, x^{2} + 2 \, x - 3\right )}} - \frac{8 \,{\left (\frac{5 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{26 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{11 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - 6\right )}}{21 \,{\left (\frac{8 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{26 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{8 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac{3 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/49*sqrt(7)*ln(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 2/49*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)) - 2/49*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)) - 1/14*(8*x - 3)/(2*x^2 + 2*x - 3) - 8/21*(5*(sqrt(-x^2 - 2*
x + 3) - 2)/(x + 1) + 26*(sqrt(-x^2 - 2*x + 3) - 2)^2/(x + 1)^2 + 11*(sqrt(-x^2
- 2*x + 3) - 2)^3/(x + 1)^3 - 6)/(8*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 26*(sqr
t(-x^2 - 2*x + 3) - 2)^2/(x + 1)^2 + 8*(sqrt(-x^2 - 2*x + 3) - 2)^3/(x + 1)^3 -
3*(sqrt(-x^2 - 2*x + 3) - 2)^4/(x + 1)^4 - 3)