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 = 0.453625, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 6728
Rule 730
Rule 724
Rule 204
Rule 1028
Rule 986
Rule 12
Rule 1026
Rule 1161
Rule 618
Rule 1027
Rule 206
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*}
Mathematica [C] time = 0.455593, 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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Maple [A] time = 0.111, size = 169, normalized size = 1.1 \begin{align*} -{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( -6-4\,x \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{-{x}^{2}-4\,x-3}}}} \right ) }+{\frac{\sqrt{4}\sqrt{3}}{81}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{6}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}} \right ) -5\,{\it Artanh} \left ( 3\,{\frac{x}{-3/2-x}{\frac{1}{\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}}}} \right ) \right ){\frac{1}{\sqrt{{ \left ({{x}^{2} \left ( -{\frac{3}{2}}-x \right ) ^{-2}}-4 \right ) \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-2}}}}} \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-1}}+{\frac{1}{9\,x}\sqrt{-{x}^{2}-4\,x-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt{-x^{2} - 4 \, x - 3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63239, size = 518, normalized size = 3.43 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \sqrt{- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2789, size = 363, normalized size = 2.4 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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