Optimal. Leaf size=151 \[ \frac {\sqrt {-x^2-4 x-3}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {2 x+3}{\sqrt {3} \sqrt {-x^2-4 x-3}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right ) \]
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Rubi [A] time = 0.45, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6728, 730, 724, 204, 1028, 986, 12, 1026, 1161, 618, 1027, 206} \[ \frac {\sqrt {-x^2-4 x-3}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {2 x+3}{\sqrt {3} \sqrt {-x^2-4 x-3}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {1-\frac {x+3}{\sqrt {-x^2-4 x-3}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {\frac {x+3}{\sqrt {-x^2-4 x-3}}+1}{\sqrt {2}}\right )+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 206
Rule 618
Rule 724
Rule 730
Rule 986
Rule 1026
Rule 1027
Rule 1028
Rule 1161
Rule 6728
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx &=\int \left (\frac {1}{3 x^2 \sqrt {-3-4 x-x^2}}-\frac {4}{9 x \sqrt {-3-4 x-x^2}}+\frac {2 (5+4 x)}{9 \sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )}\right ) \, dx\\ &=\frac {2}{9} \int \frac {5+4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {1}{3} \int \frac {1}{x^2 \sqrt {-3-4 x-x^2}} \, dx-\frac {4}{9} \int \frac {1}{x \sqrt {-3-4 x-x^2}} \, dx\\ &=\frac {\sqrt {-3-4 x-x^2}}{9 x}-\frac {2}{9} \int \frac {1}{x \sqrt {-3-4 x-x^2}} \, dx-\frac {2}{9} \int \frac {1}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {2}{9} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {8}{9} \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {-6-4 x}{\sqrt {-3-4 x-x^2}}\right )\\ &=\frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {4 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{9 \sqrt {3}}+\frac {1}{27} \int \frac {-6-4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {1}{27} \int -\frac {4 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx+\frac {4}{9} \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {-6-4 x}{\sqrt {-3-4 x-x^2}}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right )\\ &=\frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \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} \int \frac {x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right )\\ &=\frac {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {32}{27} \operatorname {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 {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )-\frac {4}{81} \operatorname {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 {4}{81} \operatorname {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 {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )+\frac {8}{81} \operatorname {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 {8}{81} \operatorname {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 {\sqrt {-3-4 x-x^2}}{9 x}+\frac {2 \tan ^{-1}\left (\frac {3+2 x}{\sqrt {3} \sqrt {-3-4 x-x^2}}\right )}{3 \sqrt {3}}+\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {1-\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )-\frac {2}{27} \sqrt {2} \tan ^{-1}\left (\frac {1+\frac {3+x}{\sqrt {-3-4 x-x^2}}}{\sqrt {2}}\right )+\frac {10}{27} \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.41, size = 225, normalized size = 1.49 \[ \frac {3 \left (\sqrt {-x^2-4 x-3}+2 \sqrt {3} x \tan ^{-1}\left (\frac {2 x+3}{\sqrt {3} \sqrt {-x^2-4 x-3}}\right )\right )+\sqrt {1-2 i \sqrt {2}} \left (2 \sqrt {2}+i\right ) x \tanh ^{-1}\left (\frac {-i \sqrt {2} x+2 x-2 i \sqrt {2}+2}{\sqrt {2+4 i \sqrt {2}} \sqrt {-x^2-4 x-3}}\right )+\sqrt {1+2 i \sqrt {2}} \left (2 \sqrt {2}-i\right ) x \tanh ^{-1}\left (\frac {\left (2+i \sqrt {2}\right ) x+2 i \sqrt {2}+2}{\sqrt {2-4 i \sqrt {2}} \sqrt {-x^2-4 x-3}}\right )}{27 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 194, normalized size = 1.28 \[ -\frac {12 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} \sqrt {-x^{2} - 4 \, x - 3} {\left (2 \, x + 3\right )}}{3 \, {\left (x^{2} + 4 \, x + 3\right )}}\right ) - 2 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} x + 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) - 2 \, \sqrt {2} x \arctan \left (-\frac {\sqrt {2} x - 3 \, \sqrt {2} \sqrt {-x^{2} - 4 \, x - 3}}{2 \, {\left (2 \, x + 3\right )}}\right ) + 5 \, x \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - 5 \, x \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) - 6 \, \sqrt {-x^{2} - 4 \, x - 3}}{54 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 269, normalized size = 1.78 \[ \frac {2}{27} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac {2}{27} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac {\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 2}{18 \, {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right )}} + \frac {5}{27} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac {5}{27} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 169, normalized size = 1.12 \[ -\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (-4 x -6\right ) \sqrt {3}}{6 \sqrt {-x^{2}-4 x -3}}\right )}{9}+\frac {\sqrt {-x^{2}-4 x -3}}{9 x}+\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \left (-5 \arctanh \left (\frac {3 x}{\left (-x -\frac {3}{2}\right ) \sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}}\right )+\sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {3 x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-12}\, \sqrt {2}}{6}\right )\right )}{81 \sqrt {\frac {\frac {x^{2}}{\left (-x -\frac {3}{2}\right )^{2}}-4}{\left (\frac {x}{-x -\frac {3}{2}}+1\right )^{2}}}\, \left (\frac {x}{-x -\frac {3}{2}}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt {-x^{2} - 4 \, x - 3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^2\,\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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