Optimal. Leaf size=28 \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0492847, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 2*x)/(3*x + x^3),x]
[Out]
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Rubi in Sympy [A] time = 3.15747, size = 29, normalized size = 1.04 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} + 3 \right )}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+2*x)/(x**3+3*x),x)
[Out]
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Mathematica [A] time = 0.0125286, size = 28, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 2*x)/(3*x + x^3),x]
[Out]
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Maple [A] time = 0.009, size = 24, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+3 \right ) }{2}}+{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+2*x)/(x^3+3*x),x)
[Out]
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Maxima [A] time = 1.5114, size = 31, normalized size = 1.11 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 3)/(x^3 + 3*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204753, size = 43, normalized size = 1.54 \[ -\frac{1}{6} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + 3\right ) - 2 \, \sqrt{3} \log \left (x\right ) - 4 \, \arctan \left (\frac{1}{3} \, \sqrt{3} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 3)/(x^3 + 3*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.153047, size = 29, normalized size = 1.04 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} + 3 \right )}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+2*x)/(x**3+3*x),x)
[Out]
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GIAC/XCAS [A] time = 0.204067, size = 32, normalized size = 1.14 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 3\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 3)/(x^3 + 3*x),x, algorithm="giac")
[Out]