Integrand size = 18, antiderivative size = 108 \[ \int \frac {1}{x+\sqrt {-3-4 x-x^2}} \, dx=-\arctan \left (\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )-\sqrt {2} \arctan \left (\frac {1-\frac {3 \sqrt {-1-x}}{\sqrt {3+x}}}{\sqrt {2}}\right )+\frac {1}{2} \log (3+x)+\frac {1}{2} \log \left (\frac {3 \sqrt {-1-x}+\sqrt {-1-x} x+x \sqrt {3+x}}{(3+x)^{3/2}}\right ) \]
-arctan((-1-x)^(1/2)/(3+x)^(1/2))+1/2*ln(3+x)+1/2*ln((3*(-1-x)^(1/2)+x*(-1 -x)^(1/2)+x*(3+x)^(1/2))/(3+x)^(3/2))-arctan(1/2*(1-3*(-1-x)^(1/2)/(3+x)^( 1/2))*2^(1/2))*2^(1/2)
Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x+\sqrt {-3-4 x-x^2}} \, dx=\frac {1}{2} \left (-2 \arctan \left (\frac {\sqrt {-3-4 x-x^2}}{3+x}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} (1+x)}{1+x+\sqrt {-3-4 x-x^2}}\right )+\log \left (x+\sqrt {-3-4 x-x^2}\right )\right ) \]
(-2*ArcTan[Sqrt[-3 - 4*x - x^2]/(3 + x)] - 2*Sqrt[2]*ArcTan[(Sqrt[2]*(1 + x))/(1 + x + Sqrt[-3 - 4*x - x^2])] + Log[x + Sqrt[-3 - 4*x - x^2]])/2
Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {7287, 27, 1356, 27, 452, 216, 240, 1142, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\sqrt {-x^2-4 x-3}+x} \, dx\) |
| \(\Big \downarrow \) 7287 |
| \(\displaystyle 2 \int \frac {2 \sqrt {-x-1}}{\sqrt {x+3} \left (\frac {-x-1}{x+3}+1\right ) \left (\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {\sqrt {-x-1}}{\sqrt {x+3} \left (\frac {-x-1}{x+3}+1\right ) \left (\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\) |
| \(\Big \downarrow \) 1356 |
| \(\displaystyle 4 \left (\frac {1}{8} \int -\frac {2 \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}+1\right )}{\frac {-x-1}{x+3}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}+\frac {1}{8} \int \frac {2 \left (\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {1}{4} \int \frac {\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}+1}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}-\frac {1}{4} \int \frac {\frac {\sqrt {-x-1}}{\sqrt {x+3}}+1}{\frac {-x-1}{x+3}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle 4 \left (\frac {1}{4} \left (-\int \frac {1}{\frac {-x-1}{x+3}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}-\int \frac {\sqrt {-x-1}}{\sqrt {x+3} \left (\frac {-x-1}{x+3}+1\right )}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )+\frac {1}{4} \int \frac {\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}+1}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 4 \left (\frac {1}{4} \left (-\int \frac {\sqrt {-x-1}}{\sqrt {x+3} \left (\frac {-x-1}{x+3}+1\right )}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}-\arctan \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )\right )+\frac {1}{4} \int \frac {\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}+1}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle 4 \left (\frac {1}{4} \int \frac {\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}+1}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}+\frac {1}{4} \left (-\arctan \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )-\frac {1}{2} \log \left (\frac {-x-1}{x+3}+1\right )\right )\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 4 \left (\frac {1}{4} \left (2 \int \frac {1}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}+\frac {1}{2} \int -\frac {2 \left (1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}\right )}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )+\frac {1}{4} \left (-\arctan \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )-\frac {1}{2} \log \left (\frac {-x-1}{x+3}+1\right )\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {1}{4} \left (2 \int \frac {1}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}-\int \frac {1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )+\frac {1}{4} \left (-\arctan \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )-\frac {1}{2} \log \left (\frac {-x-1}{x+3}+1\right )\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 4 \left (\frac {1}{4} \left (-4 \int \frac {1}{-\frac {-x-1}{x+3}-8}d\left (\frac {6 \sqrt {-x-1}}{\sqrt {x+3}}-2\right )-\int \frac {1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )+\frac {1}{4} \left (-\arctan \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )-\frac {1}{2} \log \left (\frac {-x-1}{x+3}+1\right )\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 4 \left (\frac {1}{4} \left (\sqrt {2} \arctan \left (\frac {\frac {6 \sqrt {-x-1}}{\sqrt {x+3}}-2}{2 \sqrt {2}}\right )-\int \frac {1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}}{\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}d\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )+\frac {1}{4} \left (-\arctan \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )-\frac {1}{2} \log \left (\frac {-x-1}{x+3}+1\right )\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 4 \left (\frac {1}{4} \left (-\arctan \left (\frac {\sqrt {-x-1}}{\sqrt {x+3}}\right )-\frac {1}{2} \log \left (\frac {-x-1}{x+3}+1\right )\right )+\frac {1}{4} \left (\sqrt {2} \arctan \left (\frac {\frac {6 \sqrt {-x-1}}{\sqrt {x+3}}-2}{2 \sqrt {2}}\right )+\frac {1}{2} \log \left (\frac {3 (-x-1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1\right )\right )\right )\) |
4*((-ArcTan[Sqrt[-1 - x]/Sqrt[3 + x]] - Log[1 + (-1 - x)/(3 + x)]/2)/4 + ( Sqrt[2]*ArcTan[(-2 + (6*Sqrt[-1 - x])/Sqrt[3 + x])/(2*Sqrt[2])] + Log[1 + (3*(-1 - x))/(3 + x) - (2*Sqrt[-1 - x])/Sqrt[3 + x]]/2)/4)
3.8.59.3.1 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]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 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[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)* (x_)^2)), x_Symbol] :> With[{q = Simplify[c^2*d^2 + b^2*d*f - 2*a*c*d*f + a ^2*f^2]}, Simp[1/q Int[Simp[g*c^2*d + g*b^2*f - a*b*h*f - a*g*c*f + c*(h* c*d + g*b*f - a*h*f)*x, x]/(a + b*x + c*x^2), x], x] + Simp[1/q Int[Simp[ b*h*d*f - g*c*d*f + a*g*f^2 - f*(h*c*d + g*b*f - a*h*f)*x, x]/(d + f*x^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0]
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]], 3]] /; EulerIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(85)=170\).
Time = 1.04 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.43
| method | result | size |
| default | \(\frac {\arcsin \left (x +2\right )}{2}-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{12 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (\frac {x}{-\frac {3}{2}-x}+1\right )^{2}}}\, \left (\frac {x}{-\frac {3}{2}-x}+1\right )}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )}{3 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (\frac {x}{-\frac {3}{2}-x}+1\right )^{2}}}\, \left (\frac {x}{-\frac {3}{2}-x}+1\right )}-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{6 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (\frac {x}{-\frac {3}{2}-x}+1\right )^{2}}}\, \left (\frac {x}{-\frac {3}{2}-x}+1\right )}+\frac {\ln \left (2 x^{2}+4 x +3\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (4+4 x \right ) \sqrt {2}}{4}\right )}{2}\) | \(370\) |
| trager | \(\operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) \ln \left (4 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )^{2} x +4 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x +\operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x -4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) \sqrt {-x^{2}-4 x -3}-6 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )+\sqrt {-x^{2}-4 x -3}-3 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )\right )-\ln \left (4 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )^{2} x -12 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x +9 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) \sqrt {-x^{2}-4 x -3}+6 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )-3 \sqrt {-x^{2}-4 x -3}-9 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )-\ln \left (2 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +\sqrt {-x^{2}-4 x -3}+4 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+x +2\right ) \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+\ln \left (4 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )^{2} x -12 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) x +9 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x +4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right ) \sqrt {-x^{2}-4 x -3}+6 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +3\right )-3 \sqrt {-x^{2}-4 x -3}-9 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )\right )-\ln \left (2 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +\sqrt {-x^{2}-4 x -3}+4 \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+x +2\right )\) | \(575\) |
1/2*arcsin(x+2)-1/12*3^(1/2)*4^(1/2)*(3*x^2/(-3/2-x)^2-12)^(1/2)*(2^(1/2)* arctan(1/6*(3*x^2/(-3/2-x)^2-12)^(1/2)*2^(1/2))-arctanh(3*x/(-3/2-x)/(3*x^ 2/(-3/2-x)^2-12)^(1/2)))/((x^2/(-3/2-x)^2-4)/(x/(-3/2-x)+1)^2)^(1/2)/(x/(- 3/2-x)+1)+1/3*3^(1/2)*4^(1/2)/((x^2/(-3/2-x)^2-4)/(x/(-3/2-x)+1)^2)^(1/2)/ (x/(-3/2-x)+1)*(3*x^2/(-3/2-x)^2-12)^(1/2)*2^(1/2)*arctan(1/6*(3*x^2/(-3/2 -x)^2-12)^(1/2)*2^(1/2))-1/6*3^(1/2)*4^(1/2)*(3*x^2/(-3/2-x)^2-12)^(1/2)*( 2^(1/2)*arctan(1/6*(3*x^2/(-3/2-x)^2-12)^(1/2)*2^(1/2))+arctanh(3*x/(-3/2- x)/(3*x^2/(-3/2-x)^2-12)^(1/2)))/((x^2/(-3/2-x)^2-4)/(x/(-3/2-x)+1)^2)^(1/ 2)/(x/(-3/2-x)+1)+1/4*ln(2*x^2+4*x+3)-1/2*2^(1/2)*arctan(1/4*(4+4*x)*2^(1/ 2))
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (85) = 170\).
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.73 \[ \int \frac {1}{x+\sqrt {-3-4 x-x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (x + 1\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x - 3} {\left (x + 2\right )}}{x^{2} + 4 \, x + 3}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 4 \, x + 3\right ) - \frac {1}{8} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) + \frac {1}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]
-1/2*sqrt(2)*arctan(sqrt(2)*(x + 1)) + 1/4*sqrt(2)*arctan(1/2*(sqrt(2)*x + 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x + 3)) + 1/4*sqrt(2)*arctan(-1/2*(sqr t(2)*x - 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x + 3)) - 1/2*arctan(sqrt(-x^2 - 4*x - 3)*(x + 2)/(x^2 + 4*x + 3)) + 1/4*log(2*x^2 + 4*x + 3) - 1/8*log( -(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) + 1/8*log((2*sqrt(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2)
\[ \int \frac {1}{x+\sqrt {-3-4 x-x^2}} \, dx=\int \frac {1}{x + \sqrt {- x^{2} - 4 x - 3}}\, dx \]
\[ \int \frac {1}{x+\sqrt {-3-4 x-x^2}} \, dx=\int { \frac {1}{x + \sqrt {-x^{2} - 4 \, x - 3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (85) = 170\).
Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x+\sqrt {-3-4 x-x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (x + 1\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {1}{2} \, \arcsin \left (x + 2\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 4 \, x + 3\right ) + \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
-1/2*sqrt(2)*arctan(sqrt(2)*(x + 1)) + 1/2*sqrt(2)*arctan(1/2*sqrt(2)*(3*( sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 1/2*sqrt(2)*arctan(1/2*sqrt(2)*( (sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 1/2*arcsin(x + 2) + 1/4*log(2*x ^2 + 4*x + 3) + 1/4*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3*(sqrt(-x^ 2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) - 1/4*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/ (x + 2) + (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 3)
Timed out. \[ \int \frac {1}{x+\sqrt {-3-4 x-x^2}} \, dx=\int \frac {1}{x+\sqrt {-x^2-4\,x-3}} \,d x \]