Optimal. Leaf size=17 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right ) \]
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Rubi [A] time = 0.0665603, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(3 + 2*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]
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Rubi in Sympy [A] time = 18.129, size = 15, normalized size = 0.88 \[ \operatorname{atanh}{\left (\frac{x}{\sqrt{- x^{2} - 4 x - 3}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+2*x)/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)
[Out]
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Mathematica [C] time = 6.28269, size = 1057, normalized size = 62.18 \[ -\frac{i \left (-i+\sqrt{2}\right ) \tan ^{-1}\left (\frac{6 i \sqrt{2} x^4+8 x^4+6 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+20 i \sqrt{2} x^3+52 x^3+24 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2+11 i \sqrt{2} x^2+112 x^2+33 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x-16 i \sqrt{2} x+92 x+18 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-12 i \sqrt{2}+24}{8 \sqrt{2} x^4+6 i x^4+40 \sqrt{2} x^3+16 i x^3+58 \sqrt{2} x^2+19 i x^2+32 \sqrt{2} x+28 i x+6 \sqrt{2}+21 i}\right )}{2 \sqrt{1-2 i \sqrt{2}}}+\frac{\left (i+\sqrt{2}\right ) \tanh ^{-1}\left (\frac{6 \sqrt{2} x^4+8 i x^4+6 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+20 \sqrt{2} x^3+52 i x^3+24 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2+11 \sqrt{2} x^2+112 i x^2+33 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x-16 \sqrt{2} x+92 i x+18 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-12 \sqrt{2}+24 i}{8 \sqrt{2} x^4-6 i x^4+40 \sqrt{2} x^3-16 i x^3+58 \sqrt{2} x^2-19 i x^2+32 \sqrt{2} x-28 i x+6 \sqrt{2}-21 i}\right )}{2 \sqrt{1+2 i \sqrt{2}}}+\frac{\left (i+\sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{4 \sqrt{1+2 i \sqrt{2}}}+\frac{\left (-i+\sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{4 \sqrt{1-2 i \sqrt{2}}}-\frac{\left (-i+\sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x+8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}+6 i \sqrt{2}+3\right )\right )}{4 \sqrt{1-2 i \sqrt{2}}}-\frac{\left (i+\sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (-2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x-8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+3\right )\right )}{4 \sqrt{1+2 i \sqrt{2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 2*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]
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Maple [B] time = 0.015, size = 94, normalized size = 5.5 \[ -{\frac{\sqrt{3}\sqrt{4}}{6}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}{\it Artanh} \left ( 3\,{\frac{x}{-3/2-x}{\frac{1}{\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}}}} \right ){\frac{1}{\sqrt{{1 \left ({{x}^{2} \left ( -{\frac{3}{2}}-x \right ) ^{-2}}-4 \right ) \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-2}}}}} \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+2*x)/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x + 3}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt{-x^{2} - 4 \, x - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 3)/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="maxima")
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Fricas [A] time = 0.288099, size = 76, normalized size = 4.47 \[ -\frac{1}{4} \, \log \left (-\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) + \frac{1}{4} \, \log \left (\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 3)/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 x + 3}{\sqrt{- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+2*x)/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.274404, size = 132, normalized size = 7.76 \[ \frac{1}{2} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac{1}{2} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 3)/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="giac")
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