Integrand size = 18, antiderivative size = 307 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=-\frac {4 \left (9-5 \sqrt {3}+\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{21 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )^2}+\frac {2 \left (18-43 \sqrt {3}-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{147 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )}+\frac {12 \text {arctanh}\left (\frac {3-x-\sqrt {3} x-\sqrt {3} \sqrt {3-2 x-x^2}}{\sqrt {7} x}\right )}{49 \sqrt {7}} \]
12/343*arctanh(1/7*(3-x-x*3^(1/2)-3^(1/2)*(-x^2-2*x+3)^(1/2))/x*7^(1/2))*7 ^(1/2)-4/21*(9-5*3^(1/2)+(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*(18-43*3^(1/2)-(18+49*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)
Time = 0.44 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\frac {\frac {7 \left (-279+300 x+26 x^2-48 x^3\right )}{\left (-3+2 x+2 x^2\right )^2}+\frac {14 \sqrt {3-2 x-x^2} \left (15+83 x-58 x^2-34 x^3\right )}{\left (-3+2 x+2 x^2\right )^2}+48 \sqrt {7} \text {arctanh}\left (\frac {2-2 x+\sqrt {3-2 x-x^2}}{\sqrt {7} (-1+x)}\right )}{1372} \]
((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
Time = 0.54 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.99, 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 |
\(\displaystyle 2 \left (\frac {1}{21} \left (\frac {18 \text {arctanh}\left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{2 \sqrt {7} x}\right )}{7 \sqrt {7}}+\frac {-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-43 \sqrt {3}+18}{7 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}\right )-\frac {2 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}\right )\) |
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)
3.8.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.42
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 | \(\text {Expression too large to display}\) | \(5984\) |
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))
Time = 0.27 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=-\frac {336 \, x^{3} - 6 \, \sqrt {7} {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac {x^{4} + 44 \, x^{3} - \sqrt {7} {\left (3 \, x^{3} + x^{2} - 45 \, x + 45\right )} \sqrt {-x^{2} - 2 \, x + 3} + 26 \, x^{2} - 276 \, x + 207}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) - 12 \, \sqrt {7} {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac {2 \, x^{2} + \sqrt {7} {\left (2 \, x + 1\right )} + 2 \, x + 4}{2 \, x^{2} + 2 \, x - 3}\right ) - 182 \, x^{2} + 14 \, {\left (34 \, x^{3} + 58 \, x^{2} - 83 \, x - 15\right )} \sqrt {-x^{2} - 2 \, x + 3} - 2100 \, x + 1953}{1372 \, {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )}} \]
-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)
Timed out. \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\int { \frac {1}{{\left (x + \sqrt {-x^{2} - 2 \, x + 3}\right )}^{3}} \,d x } \]
Time = 0.38 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=-\frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt {7} + 2 \right |}}\right ) + \frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) - \frac {48 \, x^{3} - 26 \, x^{2} - 300 \, x + 279}{196 \, {\left (2 \, x^{2} + 2 \, x - 3\right )}^{2}} + \frac {4 \, {\left (\frac {231 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {3286 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - \frac {4441 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac {18906 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - \frac {12487 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{5}}{{\left (x + 1\right )}^{5}} + \frac {946 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{6}}{{\left (x + 1\right )}^{6}} + \frac {1977 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{7}}{{\left (x + 1\right )}^{7}} - 414\right )}}{441 \, {\left (\frac {8 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {26 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {8 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac {3 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - 3\right )}^{2}} \]
-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
Timed out. \[ \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx=\int \frac {1}{{\left (x+\sqrt {-x^2-2\,x+3}\right )}^3} \,d x \]