Optimal. Leaf size=99 \[ \frac{5}{576} \log \left (4 x^2-6 x+9\right )+\frac{1}{192} \log \left (4 x^2+6 x+9\right )-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (2 x+3)-\frac{5 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{96 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{32 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.136423, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{5}{576} \log \left (4 x^2-6 x+9\right )+\frac{1}{192} \log \left (4 x^2+6 x+9\right )-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (2 x+3)-\frac{5 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{96 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{32 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(x*(27 - 2*x^3))/(729 - 64*x^6),x]
[Out]
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Rubi in Sympy [A] time = 19.2437, size = 92, normalized size = 0.93 \[ - \frac{\log{\left (- 2 x + 3 \right )}}{96} - \frac{5 \log{\left (2 x + 3 \right )}}{288} + \frac{5 \log{\left (16 x^{2} - 24 x + 36 \right )}}{576} + \frac{\log{\left (16 x^{2} + 24 x + 36 \right )}}{192} + \frac{5 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{288} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{96} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(-2*x**3+27)/(-64*x**6+729),x)
[Out]
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Mathematica [A] time = 0.0317212, size = 91, normalized size = 0.92 \[ \frac{1}{576} \left (5 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-6 \log (3-2 x)-10 \log (2 x+3)+10 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )-6 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x*(27 - 2*x^3))/(729 - 64*x^6),x]
[Out]
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Maple [A] time = 0.013, size = 76, normalized size = 0.8 \[ -{\frac{5\,\ln \left ( 2\,x+3 \right ) }{288}}-{\frac{\ln \left ( -3+2\,x \right ) }{96}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{192}}-{\frac{\sqrt{3}}{96}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{5\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{576}}+{\frac{5\,\sqrt{3}}{288}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(-2*x^3+27)/(-64*x^6+729),x)
[Out]
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Maxima [A] time = 1.52351, size = 101, normalized size = 1.02 \[ -\frac{1}{96} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{5}{288} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{192} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{576} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac{5}{288} \, \log \left (2 \, x + 3\right ) - \frac{1}{96} \, \log \left (2 \, x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^3 - 27)*x/(64*x^6 - 729),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213124, size = 116, normalized size = 1.17 \[ \frac{1}{1728} \, \sqrt{3}{\left (3 \, \sqrt{3} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 5 \, \sqrt{3} \log \left (4 \, x^{2} - 6 \, x + 9\right ) - 10 \, \sqrt{3} \log \left (2 \, x + 3\right ) - 6 \, \sqrt{3} \log \left (2 \, x - 3\right ) - 18 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 30 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^3 - 27)*x/(64*x^6 - 729),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.573778, size = 102, normalized size = 1.03 \[ - \frac{\log{\left (x - \frac{3}{2} \right )}}{96} - \frac{5 \log{\left (x + \frac{3}{2} \right )}}{288} + \frac{5 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{576} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{192} + \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{288} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{96} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(-2*x**3+27)/(-64*x**6+729),x)
[Out]
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GIAC/XCAS [A] time = 0.220171, size = 93, normalized size = 0.94 \[ -\frac{1}{96} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{5}{288} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{192} \,{\rm ln}\left (x^{2} + \frac{3}{2} \, x + \frac{9}{4}\right ) + \frac{5}{576} \,{\rm ln}\left (x^{2} - \frac{3}{2} \, x + \frac{9}{4}\right ) - \frac{5}{288} \,{\rm ln}\left ({\left | x + \frac{3}{2} \right |}\right ) - \frac{1}{96} \,{\rm ln}\left ({\left | x - \frac{3}{2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^3 - 27)*x/(64*x^6 - 729),x, algorithm="giac")
[Out]