Integrand size = 30, antiderivative size = 130 \[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {\arctan \left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {1}{9} \sqrt {2} \arctan \left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {1}{9} \sqrt {2} \arctan \left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {4}{9} \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {6860, 738, 210, 1042, 1000, 12, 1040, 1175, 632, 1041, 212} \[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {\arctan \left (\frac {2 x+3}{\sqrt {3} \sqrt {-x^2-4 x-3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \sqrt {2} \arctan \left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )-\frac {1}{9} \sqrt {2} \arctan \left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )-\frac {4}{9} \text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right ) \]
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Rule 12
Rule 210
Rule 212
Rule 632
Rule 738
Rule 1000
Rule 1040
Rule 1041
Rule 1042
Rule 1175
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{3 x \sqrt {-3-4 x-x^2}}-\frac {2 (2+x)}{3 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )}\right ) \, dx \\ & = \frac {1}{3} \int \frac {1}{x \sqrt {-3-4 x-x^2}} \, dx-\frac {2}{3} \int \frac {2+x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx \\ & = \frac {1}{6} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {1}{3} \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {-6-4 x}{\sqrt {-3-4 x-x^2}}\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {1}{18} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {1}{18} \int -\frac {4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\text {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {2}{9} \int \frac {x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {1}{3} \text {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}-\frac {4}{9} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {16}{9} \text {Subst}\left (\int \frac {1+3 x^2}{-4-8 x^2-36 x^4} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}-\frac {4}{9} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\frac {2}{27} \text {Subst}\left (\int \frac {1}{\frac {1}{3}-\frac {2 x}{3}+x^2} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right )-\frac {2}{27} \text {Subst}\left (\int \frac {1}{\frac {1}{3}+\frac {2 x}{3}+x^2} \, dx,x,\frac {1+\frac {x}{3}}{\sqrt {-3-4 x-x^2}}\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}-\frac {4}{9} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {4}{27} \text {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3} \left (-1+\frac {3+x}{\sqrt {-3-4 x-x^2}}\right )\right )+\frac {4}{27} \text {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3} \left (1+\frac {3+x}{\sqrt {-3-4 x-x^2}}\right )\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {1}{9} \sqrt {2} \tan ^{-1}\left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {1}{9} \sqrt {2} \tan ^{-1}\left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {4}{9} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{9} \left (-\sqrt {2} \arctan \left (\frac {3+2 x}{\sqrt {2} \sqrt {-3-4 x-x^2}}\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3-4 x-x^2}}{3+x}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\right ) \]
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Time = 0.77 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {\sqrt {3}\, \arctan \left (\frac {\left (-6-4 x \right ) \sqrt {3}}{6 \sqrt {-x^{2}-4 x -3}}\right )}{9}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \sqrt {2}}{6}\right )+4 \,\operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )\right )}{54 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\) | \(152\) |
trager | \(\operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \ln \left (\frac {40500 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x -40500 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}+4680 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+3636 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -41886 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-287 \sqrt {-x^{2}-4 x -3}-1731 x -8078}{18 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -18 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+8 x +1}\right )-\frac {4 \ln \left (\frac {4500 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x -4500 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}-520 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+3596 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x +654 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-263 \sqrt {-x^{2}-4 x -3}+517 x +282}{2 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -2 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-1}\right )}{9}-\ln \left (\frac {4500 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x -4500 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}-520 \sqrt {-x^{2}-4 x -3}\, \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )+3596 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x +654 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-263 \sqrt {-x^{2}-4 x -3}+517 x +282}{2 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x -2 \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-1}\right ) \operatorname {RootOf}\left (18 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )-3 \sqrt {-x^{2}-4 x -3}}{x}\right )}{9}\) | \(483\) |
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Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-x^{2} - 4 \, x - 3} {\left (2 \, x + 3\right )}}{3 \, {\left (x^{2} + 4 \, x + 3\right )}}\right ) + \frac {1}{18} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{18} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + \frac {1}{9} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \frac {1}{9} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]
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\[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {1}{x \sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
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\[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int { \frac {1}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt {-x^{2} - 4 \, x - 3} x} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.53 \[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{9} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {1}{9} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {2}{9} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac {2}{9} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
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Timed out. \[ \int \frac {1}{x \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {1}{x\,\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]
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