Integrand size = 27, antiderivative size = 95 \[ \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{3} \sqrt {2} \arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )+\frac {1}{3} \sqrt {2} \arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
1/3*arctanh(x/(-x^2-4*x-3)^(1/2))-1/3*arctan(1/2*(1+(-3-x)/(-x^2-4*x-3)^(1 /2))*2^(1/2))*2^(1/2)+1/3*arctan(1/2*(1+(3+x)/(-x^2-4*x-3)^(1/2))*2^(1/2)) *2^(1/2)
Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{3} \left (\sqrt {2} \arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )+\text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\right ) \]
(Sqrt[2]*ArcTan[(3 + 2*x)/(Sqrt[2]*Sqrt[-3 - 4*x - x^2])] + ArcTanh[x/Sqrt [-3 - 4*x - x^2]])/3
Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.61, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1317, 27, 1359, 27, 1360, 219, 1475, 1083, 217}
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.
| \(\displaystyle \int \frac {1}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )} \, dx\) |
| \(\Big \downarrow \) 1317 |
| \(\displaystyle \frac {1}{6} \int -\frac {4 x}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {1}{6} \int -\frac {2 (2 x+3)}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {2}{3} \int \frac {x}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx\) |
| \(\Big \downarrow \) 1359 |
| \(\displaystyle \frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx-\frac {16}{3} \int -\frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{4 \left (\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1\right )}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {2 x+3}{\sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}dx+\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\) |
| \(\Big \downarrow \) 1360 |
| \(\displaystyle \int \frac {1}{3-\frac {3 x^2}{-x^2-4 x-3}}d\frac {x}{\sqrt {-x^2-4 x-3}}+\frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {4}{3} \int \frac {\frac {(x+3)^2}{3 \left (-x^2-4 x-3\right )}+1}{\frac {(x+3)^4}{9 \left (-x^2-4 x-3\right )^2}+\frac {2 (x+3)^2}{9 \left (-x^2-4 x-3\right )}+1}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle \frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {4}{3} \left (-\frac {1}{6} \int \frac {1}{\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {2 (x+3)}{9 \sqrt {-x^2-4 x-3}}+\frac {1}{3}}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}-\frac {1}{6} \int \frac {1}{\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}+\frac {2 (x+3)}{9 \sqrt {-x^2-4 x-3}}+\frac {1}{3}}d\frac {x+3}{3 \sqrt {-x^2-4 x-3}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )-\frac {4}{3} \left (\frac {1}{3} \int \frac {1}{-\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {8}{9}}d\left (\frac {2 (x+3)}{3 \sqrt {-x^2-4 x-3}}-\frac {2}{3}\right )+\frac {1}{3} \int \frac {1}{-\frac {(x+3)^2}{9 \left (-x^2-4 x-3\right )}-\frac {8}{9}}d\left (\frac {2 (x+3)}{3 \sqrt {-x^2-4 x-3}}+\frac {2}{3}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{3} \sqrt {2} \arctan \left (\frac {x+3}{2 \sqrt {2} \sqrt {-x^2-4 x-3}}\right )+\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )\) |
(2*Sqrt[2]*ArcTan[(3 + x)/(2*Sqrt[2]*Sqrt[-3 - 4*x - x^2])])/3 + ArcTanh[x /Sqrt[-3 - 4*x - x^2]]/3
3.2.30.3.1 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[(-(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[((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[1/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)* (x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f) , 2]}, Simp[1/(2*q) Int[(c*d - a*f + q + (c*e - b*f)*x)/((a + b*x + c*x^2 )*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[(c*d - a*f - q + (c*e - b*f)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[c*e - b*f, 0] && NegQ[b^2 - 4*a*c]
Int[(x_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_.)*(x_) + (f_.) *(x_)^2]), x_Symbol] :> Simp[-2*e Subst[Int[(1 - d*x^2)/(c*e - b*f - e*(2 *c*d - b*e + 2*a*f)*x^2 + d^2*(c*e - b*f)*x^4), x], x, (1 + (e + Sqrt[e^2 - 4*d*f])*(x/(2*d)))/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0]
Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (e_ .)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[g Subst[Int[1/(a + (c*d - a*f )*x^2), x], x, x/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0] & & EqQ[2*h*d - g*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^ 2, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Time = 0.68 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{18 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(121\) |
| trager | \(\frac {\ln \left (\frac {36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x -36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x -36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-6 \sqrt {-x^{2}-4 x -3}+5 x +6}{6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +x +3}\right )}{3}-\ln \left (\frac {36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x -36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x -36 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-6 \sqrt {-x^{2}-4 x -3}+5 x +6}{6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +x +3}\right ) \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (\frac {12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x +4 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +12 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-2 \sqrt {-x^{2}-4 x -3}-x -2}{2 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x -x -1}\right )\) | \(280\) |
-1/18*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)/(1+x/(-3/2-x))^2)^(1/2)/(1+x/(-3/2-x))
Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - \frac {1}{12} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) + \frac {1}{12} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]
-1/6*sqrt(2)*arctan(1/2*(sqrt(2)*x + 3*sqrt(2)*sqrt(-x^2 - 4*x - 3))/(2*x + 3)) - 1/6*sqrt(2)*arctan(-1/2*(sqrt(2)*x - 3*sqrt(2)*sqrt(-x^2 - 4*x - 3 ))/(2*x + 3)) - 1/12*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) + 1/12 *log((2*sqrt(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2)
\[ \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {1}{\sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
\[ \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt {-x^{2} - 4 \, x - 3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (76) = 152\).
Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {1}{6} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
-1/3*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1) ) - 1/3*sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1 )) + 1/6*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3*(sqrt(-x^2 - 4*x - 3 ) - 1)^2/(x + 2)^2 + 1) - 1/6*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. \[ \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {1}{\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]