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

Optimal. Leaf size=307 \[ -\frac{4 \left (\frac{\left (21+5 \sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-5 \sqrt{3}+9\right )}{21 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )^2}+\frac{2 \left (-\frac{\left (18+49 \sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-43 \sqrt{3}+18\right )}{147 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )}+\frac{12 \tanh ^{-1}\left (\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right )}{49 \sqrt{7}} \]

[Out]

(-4*(9 - 5*Sqrt[3] + ((21 + 5*Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(21*
(2 - Sqrt[3] - (2*(1 + Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(S
qrt[3] - Sqrt[3 - 2*x - x^2])^2)/x^2)^2) + (2*(18 - 43*Sqrt[3] - ((18 + 49*Sqrt[
3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(147*(2 - Sqrt[3] - (2*(1 + Sqrt[3])*(S
qrt[3] - Sqrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(Sqrt[3] - Sqrt[3 - 2*x - x^2])^2)/x
^2)) + (12*ArcTanh[(3 - x - Sqrt[3]*x - Sqrt[3]*Sqrt[3 - 2*x - x^2])/(Sqrt[7]*x)
])/(49*Sqrt[7])

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Rubi [A]  time = 0.445297, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{4 \left (\frac{\left (21+5 \sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-5 \sqrt{3}+9\right )}{21 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )^2}+\frac{2 \left (-\frac{\left (18+49 \sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-43 \sqrt{3}+18\right )}{147 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )}+\frac{12 \tanh ^{-1}\left (\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(9 - 5*Sqrt[3] + ((21 + 5*Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(21*
(2 - Sqrt[3] - (2*(1 + Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(S
qrt[3] - Sqrt[3 - 2*x - x^2])^2)/x^2)^2) + (2*(18 - 43*Sqrt[3] - ((18 + 49*Sqrt[
3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(147*(2 - Sqrt[3] - (2*(1 + Sqrt[3])*(S
qrt[3] - Sqrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(Sqrt[3] - Sqrt[3 - 2*x - x^2])^2)/x
^2)) + (12*ArcTanh[(3 - x - Sqrt[3]*x - Sqrt[3]*Sqrt[3 - 2*x - x^2])/(Sqrt[7]*x)
])/(49*Sqrt[7])

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Rubi in Sympy [A]  time = 92.8832, size = 733, normalized size = 2.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*(-(-3*sqrt(3) + 6)*(sqrt(3)*(-2*sqrt(3) + 4) + (2 + 2*sqrt(3))**2) + 3*s
qrt(3)*(6 + (2 + 2*sqrt(3))*(2*sqrt(3)/3 + 2)))*(2 + 2*sqrt(3) - 2*sqrt(3)*(-sqr
t(-x**2 - 2*x + 3) + sqrt(3))/x)/(3*(-(2 + 2*sqrt(3))**2 - sqrt(3)*(-8 + 4*sqrt(
3)))**2*(-sqrt(3) + 2 + (2 + 2*sqrt(3))*(sqrt(-x**2 - 2*x + 3) - sqrt(3))/x + sq
rt(3)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))**2/x**2)) + sqrt(3)*(-8*sqrt(3) + 32 - (
(-3*sqrt(3) + 6)*(-(2 + 2*sqrt(3))**2 + sqrt(3)*(-2*sqrt(3) + 4)) + sqrt(3)*(6 +
(2 + 2*sqrt(3))*(2*sqrt(3)/3 + 2)))*(-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))*(
sqrt(-x**2 - 2*x + 3) - sqrt(3))/x + sqrt(3)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))**
2/x**2)**2) - 2*sqrt(7)*(-(-3*sqrt(3) + 6)*(sqrt(3)*(-2*sqrt(3) + 4) + (2 + 2*sq
rt(3))**2) + 3*sqrt(3)*(6 + (2 + 2*sqrt(3))*(2*sqrt(3)/3 + 2)))*atanh(sqrt(7)*(1
/7 + sqrt(3)/7 + sqrt(3)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))/(7*x)))/(7*(-(2 + 2*s
qrt(3))**2 - sqrt(3)*(-8 + 4*sqrt(3)))**2) + sqrt(3)*(2*sqrt(3) + 4)*(-sqrt(-x**
2 - 2*x + 3) + sqrt(3))**2/(3*x**2*(-sqrt(3) + 2 + 2*(sqrt(-x**2 - 2*x + 3) - sq
rt(3))/x + 2*sqrt(3)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))/x + sqrt(3)*(sqrt(-x**2 -
2*x + 3) - sqrt(3))**2/x**2)**2) - 2*(-sqrt(-x**2 - 2*x + 3) + sqrt(3))**3/(x**
3*(-sqrt(3) + 2 + 2*(sqrt(-x**2 - 2*x + 3) - sqrt(3))/x + 2*sqrt(3)*(sqrt(-x**2
- 2*x + 3) - sqrt(3))/x + sqrt(3)*(sqrt(-x**2 - 2*x + 3) - sqrt(3))**2/x**2)**2)

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Mathematica [A]  time = 1.09902, size = 333, normalized size = 1.08 \[ \frac{\frac{7 (37-24 x)}{2 x^2+2 x-3}+\frac{98 (11 x-12)}{\left (2 x^2+2 x-3\right )^2}-6 \left (1+\sqrt{7}\right ) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left (\sqrt{14 \left (4+\sqrt{7}\right )} \sqrt{-x^2-2 x+3}-\sqrt{7} x+7 x+7 \sqrt{7}+7\right )-2 \left (\sqrt{7}-1\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (-\sqrt{14} \sqrt{\left (\sqrt{7}-4\right ) \left (x^2+2 x-3\right )}+\left (7+\sqrt{7}\right ) x-7 \sqrt{7}+7\right )-\frac{14 \sqrt{-x^2-2 x+3} \left (34 x^3+58 x^2-83 x-15\right )}{\left (2 x^2+2 x-3\right )^2}-12 \sqrt{7} \log \left (-2 x+\sqrt{7}-1\right )+2 \left (\sqrt{7}-1\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (2 x-\sqrt{7}+1\right )+6 \left (1+\sqrt{7}\right ) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left (2 x+\sqrt{7}+1\right )+12 \sqrt{7} \log \left (2 x+\sqrt{7}+1\right )}{1372} \]

Warning: Unable to verify antiderivative.

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

[Out]

((98*(-12 + 11*x))/(-3 + 2*x + 2*x^2)^2 + (7*(37 - 24*x))/(-3 + 2*x + 2*x^2) - (
14*Sqrt[3 - 2*x - x^2]*(-15 - 83*x + 58*x^2 + 34*x^3))/(-3 + 2*x + 2*x^2)^2 - 12
*Sqrt[7]*Log[-1 + Sqrt[7] - 2*x] + 2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[1
- Sqrt[7] + 2*x] + 12*Sqrt[7]*Log[1 + Sqrt[7] + 2*x] + 6*(1 + Sqrt[7])*Sqrt[14/
(4 + Sqrt[7])]*Log[1 + Sqrt[7] + 2*x] - 6*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7])]*L
og[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)]])/1372

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Maple [B]  time = 0.062, size = 6000, normalized size = 19.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27693, size = 306, normalized size = 1. \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (34 \, x^{3} + 58 \, x^{2} - 83 \, x - 15\right )} \sqrt{-x^{2} - 2 \, x + 3} - 6 \,{\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac{\sqrt{7}{\left (x^{4} + 44 \, x^{3} + 26 \, x^{2} - 276 \, x + 207\right )} - 7 \,{\left (3 \, x^{3} + x^{2} - 45 \, x + 45\right )} \sqrt{-x^{2} - 2 \, x + 3}}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) - 12 \,{\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac{2 \, \sqrt{7}{\left (x^{2} + x + 2\right )} + 14 \, x + 7}{2 \, x^{2} + 2 \, x - 3}\right ) + \sqrt{7}{\left (48 \, x^{3} - 26 \, x^{2} - 300 \, x + 279\right )}\right )}}{1372 \,{\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1372*sqrt(7)*(2*sqrt(7)*(34*x^3 + 58*x^2 - 83*x - 15)*sqrt(-x^2 - 2*x + 3) -
6*(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)*log((sqrt(7)*(x^4 + 44*x^3 + 26*x^2 - 276*x
+ 207) - 7*(3*x^3 + x^2 - 45*x + 45)*sqrt(-x^2 - 2*x + 3))/(4*x^4 + 8*x^3 - 8*x
^2 - 12*x + 9)) - 12*(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)*log((2*sqrt(7)*(x^2 + x
+ 2) + 14*x + 7)/(2*x^2 + 2*x - 3)) + sqrt(7)*(48*x^3 - 26*x^2 - 300*x + 279))/(
4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.311365, size = 610, normalized size = 1.99 \[ -\frac{3}{343} \, \sqrt{7}{\rm ln}\left (\frac{{\left | 4 \, x - 2 \, \sqrt{7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt{7} + 2 \right |}}\right ) + \frac{3}{343} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{6 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac{3}{343} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt{7} + \frac{2 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) - \frac{48 \, x^{3} - 26 \, x^{2} - 300 \, x + 279}{196 \,{\left (2 \, x^{2} + 2 \, x - 3\right )}^{2}} + \frac{4 \,{\left (\frac{231 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{3286 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - \frac{4441 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac{18906 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - \frac{12487 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{5}}{{\left (x + 1\right )}^{5}} + \frac{946 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{6}}{{\left (x + 1\right )}^{6}} + \frac{1977 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{7}}{{\left (x + 1\right )}^{7}} - 414\right )}}{441 \,{\left (\frac{8 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{26 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{8 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac{3 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/343*sqrt(7)*ln(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 3/343*sqr
t(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)) - 3/343*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)) - 1/196*(48*x^3 - 26*x^2 - 300*x + 279)/(2*x^2 + 2*x - 3)
^2 + 4/441*(231*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 3286*(sqrt(-x^2 - 2*x + 3)
- 2)^2/(x + 1)^2 - 4441*(sqrt(-x^2 - 2*x + 3) - 2)^3/(x + 1)^3 - 18906*(sqrt(-x^
2 - 2*x + 3) - 2)^4/(x + 1)^4 - 12487*(sqrt(-x^2 - 2*x + 3) - 2)^5/(x + 1)^5 + 9
46*(sqrt(-x^2 - 2*x + 3) - 2)^6/(x + 1)^6 + 1977*(sqrt(-x^2 - 2*x + 3) - 2)^7/(x
+ 1)^7 - 414)/(8*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 26*(sqrt(-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)^2