Integrand size = 30, antiderivative size = 151 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \arctan \left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )+\frac {10}{27} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
10/27*arctanh(x/(-x^2-4*x-3)^(1/2))+2/27*arctan(1/2*(1+(-3-x)/(-x^2-4*x-3) ^(1/2))*2^(1/2))*2^(1/2)-2/27*arctan(1/2*(1+(3+x)/(-x^2-4*x-3)^(1/2))*2^(1 /2))*2^(1/2)+2/9*arctan(1/3*(3+2*x)*3^(1/2)/(-x^2-4*x-3)^(1/2))*3^(1/2)+1/ 9*(-x^2-4*x-3)^(1/2)/x
Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {-2 \sqrt {2} x \arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )+3 \left (\sqrt {-3-4 x-x^2}-4 \sqrt {3} x \arctan \left (\frac {\sqrt {3} \sqrt {-3-4 x-x^2}}{3+x}\right )\right )+10 x \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )}{27 x} \]
(-2*Sqrt[2]*x*ArcTan[(3 + 2*x)/(Sqrt[2]*Sqrt[-3 - 4*x - x^2])] + 3*(Sqrt[- 3 - 4*x - x^2] - 4*Sqrt[3]*x*ArcTan[(Sqrt[3]*Sqrt[-3 - 4*x - x^2])/(3 + x) ]) + 10*x*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/(27*x)
Time = 0.61 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {1}{x^2 \sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 (4 x+5)}{9 \sqrt {-x^2-4 x-3} \left (2 x^2+4 x+3\right )}-\frac {4}{9 x \sqrt {-x^2-4 x-3}}+\frac {1}{3 x^2 \sqrt {-x^2-4 x-3}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \arctan \left (\frac {2 x+3}{\sqrt {3} \sqrt {-x^2-4 x-3}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \arctan \left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \arctan \left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )+\frac {10}{27} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right )+\frac {\sqrt {-x^2-4 x-3}}{9 x}\) |
Sqrt[-3 - 4*x - x^2]/(9*x) + (2*ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])])/(3*Sqrt[3]) + (2*Sqrt[2]*ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2]) /Sqrt[2]])/27 - (2*Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[ 2]])/27 + (10*ArcTanh[x/Sqrt[-3 - 4*x - x^2]])/27
3.2.32.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.83 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (-6-4 x \right ) \sqrt {3}}{6 \sqrt {-x^{2}-4 x -3}}\right )}{9}+\frac {\sqrt {-x^{2}-4 x -3}}{9 x}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{81 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(169\) |
| risch | \(-\frac {x^{2}+4 x +3}{9 x \sqrt {-x^{2}-4 x -3}}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (-6-4 x \right ) \sqrt {3}}{6 \sqrt {-x^{2}-4 x -3}}\right )}{9}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )-5 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{81 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(177\) |
| trager | \(\frac {\sqrt {-x^{2}-4 x -3}}{9 x}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x +3 \sqrt {-x^{2}-4 x -3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x}\right )}{9}+\frac {10 \ln \left (-\frac {-96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2} x +4160 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2}+7232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x +9 \sqrt {-x^{2}-4 x -3}-41232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+468 x +3510}{16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-3 x}\right )}{27}-\frac {16 \ln \left (-\frac {-96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2} x +4160 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+96000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2}+7232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x +9 \sqrt {-x^{2}-4 x -3}-41232 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+468 x +3510}{16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-3 x}\right ) \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )}{9}+\frac {16 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) \ln \left (\frac {288000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2} x +12480 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-288000 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )^{2}-98304 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -2627 \sqrt {-x^{2}-4 x -3}-3696 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )+6576 x +2740}{48 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right ) x -48 \operatorname {RootOf}\left (768 \textit {\_Z}^{2}-160 \textit {\_Z} +9\right )-x +10}\right )}{9}\) | \(506\) |
-2/9*3^(1/2)*arctan(1/6*(-6-4*x)*3^(1/2)/(-x^2-4*x-3)^(1/2))+1/9*(-x^2-4*x -3)^(1/2)/x+1/81*3^(1/2)*4^(1/2)*(3*x^2/(-3/2-x)^2-12)^(1/2)*(2^(1/2)*arct an(1/6*(3*x^2/(-3/2-x)^2-12)^(1/2)*2^(1/2))-5*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 = 194, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {12 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} \sqrt {-x^{2} - 4 \, x - 3} {\left (2 \, x + 3\right )}}{3 \, {\left (x^{2} + 4 \, x + 3\right )}}\right ) - 2 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - 2 \, \sqrt {2} x \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + 5 \, x \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - 5 \, x \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) - 6 \, \sqrt {-x^{2} - 4 \, x - 3}}{54 \, x} \]
-1/54*(12*sqrt(3)*x*arctan(1/3*sqrt(3)*sqrt(-x^2 - 4*x - 3)*(2*x + 3)/(x^2 + 4*x + 3)) - 2*sqrt(2)*x*arctan(1/2*(sqrt(2)*x + 3*sqrt(2)*sqrt(-x^2 - 4 *x - 3))/(2*x + 3)) - 2*sqrt(2)*x*arctan(-1/2*(sqrt(2)*x - 3*sqrt(2)*sqrt( -x^2 - 4*x - 3))/(2*x + 3)) + 5*x*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3 )/x^2) - 5*x*log((2*sqrt(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2) - 6*sqrt(-x^2 - 4*x - 3))/x
\[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {1}{x^{2} \sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
\[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt {-x^{2} - 4 \, x - 3} x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (121) = 242\).
Time = 0.31 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.78 \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {2}{27} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {2}{27} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 2}{18 \, {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right )}} + \frac {5}{27} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac {5}{27} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
2/27*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1) ) - 4/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 2/27*sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) - 1/18*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 2)/((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) + 5/27*log( 2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3*(sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) - 5/27*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. \[ \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {1}{x^2\,\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]