Optimal. Leaf size=28 \[ \frac{5}{2} \tanh ^{-1}(x)-\frac{7}{2} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} x\right ) \]
[Out]
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Rubi [A] time = 0.0313043, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{5}{2} \tanh ^{-1}(x)-\frac{7}{2} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} x\right ) \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x^2)/(5 - 8*x^2 + 3*x^4),x]
[Out]
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Rubi in Sympy [A] time = 8.6189, size = 24, normalized size = 0.86 \[ \frac{5 \operatorname{atanh}{\left (x \right )}}{2} - \frac{7 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} x}{5} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3*x**2+2)/(3*x**4-8*x**2+5),x)
[Out]
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Mathematica [A] time = 0.0304294, size = 53, normalized size = 1.89 \[ \frac{1}{20} \left (7 \sqrt{15} \log \left (\sqrt{15}-3 x\right )-25 \log (1-x)+25 \log (x+1)-7 \sqrt{15} \log \left (3 x+\sqrt{15}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x^2)/(5 - 8*x^2 + 3*x^4),x]
[Out]
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Maple [A] time = 0.01, size = 26, normalized size = 0.9 \[ -{\frac{5\,\ln \left ( -1+x \right ) }{4}}-{\frac{7\,\sqrt{15}}{10}{\it Artanh} \left ({\frac{x\sqrt{15}}{5}} \right ) }+{\frac{5\,\ln \left ( 1+x \right ) }{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3*x^2+2)/(3*x^4-8*x^2+5),x)
[Out]
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Maxima [A] time = 0.847103, size = 51, normalized size = 1.82 \[ \frac{7}{20} \, \sqrt{15} \log \left (\frac{3 \, x - \sqrt{15}}{3 \, x + \sqrt{15}}\right ) + \frac{5}{4} \, \log \left (x + 1\right ) - \frac{5}{4} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)/(3*x^4 - 8*x^2 + 5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286182, size = 78, normalized size = 2.79 \[ \frac{1}{20} \, \sqrt{5}{\left (5 \, \sqrt{5} \log \left (x + 1\right ) - 5 \, \sqrt{5} \log \left (x - 1\right ) + 7 \, \sqrt{3} \log \left (\frac{\sqrt{5}{\left (3 \, x^{2} + 5\right )} - 10 \, \sqrt{3} x}{3 \, x^{2} - 5}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)/(3*x^4 - 8*x^2 + 5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.85561, size = 53, normalized size = 1.89 \[ - \frac{5 \log{\left (x - 1 \right )}}{4} + \frac{5 \log{\left (x + 1 \right )}}{4} + \frac{7 \sqrt{15} \log{\left (x - \frac{\sqrt{15}}{3} \right )}}{20} - \frac{7 \sqrt{15} \log{\left (x + \frac{\sqrt{15}}{3} \right )}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x**2+2)/(3*x**4-8*x**2+5),x)
[Out]
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GIAC/XCAS [A] time = 0.269819, size = 59, normalized size = 2.11 \[ \frac{7}{20} \, \sqrt{15}{\rm ln}\left (\frac{{\left | 6 \, x - 2 \, \sqrt{15} \right |}}{{\left | 6 \, x + 2 \, \sqrt{15} \right |}}\right ) + \frac{5}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{5}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x^2 + 2)/(3*x^4 - 8*x^2 + 5),x, algorithm="giac")
[Out]