\(\int \frac {1}{(x+\sqrt {3-2 x-x^2})^3} \, dx\) [24]

Optimal result

Integrand size = 18, antiderivative size = 311 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=-\frac {4 \left (9-5 \sqrt {3}+\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{21 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )^2}-\frac {2 \left (43-6 \sqrt {3}+\frac {\left (49+6 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{49 \sqrt {3} \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )}+\frac {12 \text {arctanh}\left (\frac {3-x-\sqrt {3} x-\sqrt {3} \sqrt {3-2 x-x^2}}{\sqrt {7} x}\right )}{49 \sqrt {7}} \] Output:

1/21*(-36+20*3^(1/2)-4*(21+5*3^(1/2))*(3^(1/2)-(-x^2-2*x+3)^(1/2))/x)/(2-3 
^(1/2)-2*(1+3^(1/2))*(3^(1/2)-(-x^2-2*x+3)^(1/2))/x+3^(1/2)*(3^(1/2)-(-x^2 
-2*x+3)^(1/2))^2/x^2)^2-2/147*(43-6*3^(1/2)+(49+6*3^(1/2))*(3^(1/2)-(-x^2- 
2*x+3)^(1/2))/x)*3^(1/2)/(2-3^(1/2)-2*(1+3^(1/2))*(3^(1/2)-(-x^2-2*x+3)^(1 
/2))/x+3^(1/2)*(3^(1/2)-(-x^2-2*x+3)^(1/2))^2/x^2)+12/343*arctanh(1/7*(3-x 
-x*3^(1/2)-3^(1/2)*(-x^2-2*x+3)^(1/2))*7^(1/2)/x)*7^(1/2)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\frac {\frac {7 \left (-279+300 x+26 x^2-48 x^3\right )}{\left (-3+2 x+2 x^2\right )^2}+\frac {14 \sqrt {3-2 x-x^2} \left (15+83 x-58 x^2-34 x^3\right )}{\left (-3+2 x+2 x^2\right )^2}+48 \sqrt {7} \text {arctanh}\left (\frac {2-2 x+\sqrt {3-2 x-x^2}}{\sqrt {7} (-1+x)}\right )}{1372} \] Input:

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

Output:

((7*(-279 + 300*x + 26*x^2 - 48*x^3))/(-3 + 2*x + 2*x^2)^2 + (14*Sqrt[3 - 
2*x - x^2]*(15 + 83*x - 58*x^2 - 34*x^3))/(-3 + 2*x + 2*x^2)^2 + 48*Sqrt[7 
]*ArcTanh[(2 - 2*x + Sqrt[3 - 2*x - x^2])/(Sqrt[7]*(-1 + x))])/1372
 

Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {7285, 2191, 27, 2191, 27, 1083, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7285

\(\displaystyle 2 \int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^4}{x^4}+\frac {2 \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^3}{x^3}+\frac {2 \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}}{\left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^3}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\)

\(\Big \downarrow \) 2191

\(\displaystyle 2 \left (-\frac {1}{56} \int -\frac {8 \left (-\frac {21 \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {42 \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+16 \sqrt {3}+21\right )}{3 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {2 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {1}{21} \int \frac {-\frac {21 \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {42 \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+16 \sqrt {3}+21}{\left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {2 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}\right )\)

\(\Big \downarrow \) 2191

\(\displaystyle 2 \left (\frac {1}{21} \left (\frac {-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-43 \sqrt {3}+18}{7 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}-\frac {1}{28} \int -\frac {72}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )-\frac {2 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {1}{21} \left (\frac {18}{7} \int \frac {1}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )+\frac {-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-43 \sqrt {3}+18}{7 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}\right )-\frac {2 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle 2 \left (\frac {1}{21} \left (\frac {-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-43 \sqrt {3}+18}{7 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}-\frac {36}{7} \int \frac {1}{28-\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}}d\left (2 \left (1+\sqrt {3}\right )-\frac {2 \sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}\right )\right )-\frac {2 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (\frac {1}{21} \left (\frac {18 \text {arctanh}\left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{2 \sqrt {7} x}\right )}{7 \sqrt {7}}+\frac {-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-43 \sqrt {3}+18}{7 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}\right )-\frac {2 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}\right )\)

Input:

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

Output:

2*((-2*(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]*(Sqrt[3] - Sqrt[3 - 2*x - x^2])^2)/x^2)^2) + ((18 - 43*Sqrt[3 
] - ((18 + 49*Sqrt[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)) + (18*ArcTanh[(Sqrt[3] - Sqrt[3 - 2*x - x 
^2])/(2*Sqrt[7]*x)])/(7*Sqrt[7]))/21)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[p, -1]
 

rule 7285

Int[u_, x_Symbol] :> With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, Si 
mp[2   Subst[Int[lst[[1]], x], x, lst[[2]]], x] /;  !FalseQ[lst] && EqQ[lst 
[[3]], 1]] /; EulerIntegrandQ[u, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.41

method	result	size
trager	\(\frac {\left (62 x^{3}+100 x^{2}-111 x -36\right ) x}{98 \left (2 x^{2}+2 x -3\right )^{2}}-\frac {\left (34 x^{3}+58 x^{2}-83 x -15\right ) \sqrt {-x^{2}-2 x +3}}{98 \left (2 x^{2}+2 x -3\right )^{2}}-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )-7 \sqrt {-x^{2}-2 x +3}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x -x +3}\right )}{343}\)	\(129\)
default	\(\text {Expression too large to display}\)	\(5984\)

Input:

int(1/(x+(-x^2-2*x+3)^(1/2))^3,x,method=_RETURNVERBOSE)
 

Output:

1/98*(62*x^3+100*x^2-111*x-36)*x/(2*x^2+2*x-3)^2-1/98*(34*x^3+58*x^2-83*x- 
15)/(2*x^2+2*x-3)^2*(-x^2-2*x+3)^(1/2)-6/343*RootOf(_Z^2-7)*ln((RootOf(_Z^ 
2-7)*x-3*RootOf(_Z^2-7)-7*(-x^2-2*x+3)^(1/2))/(RootOf(_Z^2-7)*x-x+3))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=-\frac {336 \, x^{3} - 6 \, \sqrt {7} {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac {x^{4} + 44 \, x^{3} - \sqrt {7} {\left (3 \, x^{3} + x^{2} - 45 \, x + 45\right )} \sqrt {-x^{2} - 2 \, x + 3} + 26 \, x^{2} - 276 \, x + 207}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) - 12 \, \sqrt {7} {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac {2 \, x^{2} + \sqrt {7} {\left (2 \, x + 1\right )} + 2 \, x + 4}{2 \, x^{2} + 2 \, x - 3}\right ) - 182 \, x^{2} + 14 \, {\left (34 \, x^{3} + 58 \, x^{2} - 83 \, x - 15\right )} \sqrt {-x^{2} - 2 \, x + 3} - 2100 \, x + 1953}{1372 \, {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )}} \] Input:

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

Output:

-1/1372*(336*x^3 - 6*sqrt(7)*(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)*log((x^4 + 
 44*x^3 - sqrt(7)*(3*x^3 + x^2 - 45*x + 45)*sqrt(-x^2 - 2*x + 3) + 26*x^2 
- 276*x + 207)/(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)) - 12*sqrt(7)*(4*x^4 + 8 
*x^3 - 8*x^2 - 12*x + 9)*log((2*x^2 + sqrt(7)*(2*x + 1) + 2*x + 4)/(2*x^2 
+ 2*x - 3)) - 182*x^2 + 14*(34*x^3 + 58*x^2 - 83*x - 15)*sqrt(-x^2 - 2*x + 
 3) - 2100*x + 1953)/(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)
 

Sympy [F]

\[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\int \frac {1}{\left (x + \sqrt {- x^{2} - 2 x + 3}\right )^{3}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\int { \frac {1}{{\left (x + \sqrt {-x^{2} - 2 \, x + 3}\right )}^{3}} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=-\frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt {7} + 2 \right |}}\right ) + \frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) - \frac {48 \, x^{3} - 26 \, x^{2} - 300 \, x + 279}{196 \, {\left (2 \, x^{2} + 2 \, x - 3\right )}^{2}} + \frac {4 \, {\left (\frac {231 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {3286 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - \frac {4441 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac {18906 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - \frac {12487 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{5}}{{\left (x + 1\right )}^{5}} + \frac {946 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{6}}{{\left (x + 1\right )}^{6}} + \frac {1977 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{7}}{{\left (x + 1\right )}^{7}} - 414\right )}}{441 \, {\left (\frac {8 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {26 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {8 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac {3 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - 3\right )}^{2}} \] Input:

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

Output:

-3/343*sqrt(7)*log(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 3/ 
343*sqrt(7)*log(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 
)*log(abs(-2*sqrt(7) + 2*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) - 4)/abs(2*sqr 
t(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 + 946*(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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\int \frac {1}{{\left (x+\sqrt {-x^2-2\,x+3}\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*( - 81*sqrt(7)*log( - sqrt(7) + 3*tan(asin((x + 1)/2)/2) - 2)*tan(asin( 
(x + 1)/2)/2)**4 + 216*sqrt(7)*log( - sqrt(7) + 3*tan(asin((x + 1)/2)/2) - 
 2)*tan(asin((x + 1)/2)/2)**3 - 90*sqrt(7)*log( - sqrt(7) + 3*tan(asin((x 
+ 1)/2)/2) - 2)*tan(asin((x + 1)/2)/2)**2 - 72*sqrt(7)*log( - sqrt(7) + 3* 
tan(asin((x + 1)/2)/2) - 2)*tan(asin((x + 1)/2)/2) - 9*sqrt(7)*log( - sqrt 
(7) + 3*tan(asin((x + 1)/2)/2) - 2) + 81*sqrt(7)*log(sqrt(7) + 3*tan(asin( 
(x + 1)/2)/2) - 2)*tan(asin((x + 1)/2)/2)**4 - 216*sqrt(7)*log(sqrt(7) + 3 
*tan(asin((x + 1)/2)/2) - 2)*tan(asin((x + 1)/2)/2)**3 + 90*sqrt(7)*log(sq 
rt(7) + 3*tan(asin((x + 1)/2)/2) - 2)*tan(asin((x + 1)/2)/2)**2 + 72*sqrt( 
7)*log(sqrt(7) + 3*tan(asin((x + 1)/2)/2) - 2)*tan(asin((x + 1)/2)/2) + 9* 
sqrt(7)*log(sqrt(7) + 3*tan(asin((x + 1)/2)/2) - 2) - 399*tan(asin((x + 1) 
/2)/2)**4 + 770*tan(asin((x + 1)/2)/2)**2 - 140*tan(asin((x + 1)/2)/2) - 2 
59))/(1029*(9*tan(asin((x + 1)/2)/2)**4 - 24*tan(asin((x + 1)/2)/2)**3 + 1 
0*tan(asin((x + 1)/2)/2)**2 + 8*tan(asin((x + 1)/2)/2) + 1))