Optimal. Leaf size=81 \[ \frac{1}{17496 (3-2 x)}-\frac{1}{17496 (2 x+3)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748} \]
[Out]
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Rubi [A] time = 0.126895, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{1}{17496 (3-2 x)}-\frac{1}{17496 (2 x+3)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{13122 \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{2 x}{3}\right )}{8748} \]
Antiderivative was successfully verified.
[In] Int[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6)^2,x]
[Out]
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Rubi in Sympy [A] time = 21.9605, size = 65, normalized size = 0.8 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} - \frac{1}{3}\right ) \right )}}{39366} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{4 x}{9} + \frac{1}{3}\right ) \right )}}{39366} + \frac{\operatorname{atanh}{\left (\frac{2 x}{3} \right )}}{8748} - \frac{1}{17496 \left (2 x + 3\right )} + \frac{1}{17496 \left (- 2 x + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((16*x**4+36*x**2+81)/(-64*x**6+729)**2,x)
[Out]
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Mathematica [C] time = 0.771274, size = 122, normalized size = 1.51 \[ \frac{\frac{36 x}{9-4 x^2}-9 \log (3-2 x)+9 \log (2 x+3)+3 \sqrt{3} \tan ^{-1}\left (\frac{1}{3} \left (\sqrt{3}-i\right ) x\right )+4 i \sqrt{3} \tanh ^{-1}\left (\frac{1}{3} \left (1-i \sqrt{3}\right ) x\right )+\left (-3+\frac{2}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}\right ) \tanh ^{-1}\left (\frac{1}{3} \left (x+i \sqrt{3} x\right )\right )}{157464} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(81 + 36*x^2 + 16*x^4)/(729 - 64*x^6)^2,x]
[Out]
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Maple [A] time = 0.017, size = 68, normalized size = 0.8 \[ -{\frac{1}{34992\,x+52488}}+{\frac{\ln \left ( 2\,x+3 \right ) }{17496}}-{\frac{1}{-52488+34992\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{17496}}+{\frac{\sqrt{3}}{39366}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{\sqrt{3}}{39366}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((16*x^4+36*x^2+81)/(-64*x^6+729)^2,x)
[Out]
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Maxima [A] time = 1.50652, size = 82, normalized size = 1.01 \[ \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 9\right )}} + \frac{1}{17496} \, \log \left (2 \, x + 3\right ) - \frac{1}{17496} \, \log \left (2 \, x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((16*x^4 + 36*x^2 + 81)/(64*x^6 - 729)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205496, size = 131, normalized size = 1.62 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (4 \, x^{2} - 9\right )} \log \left (2 \, x + 3\right ) - 3 \, \sqrt{3}{\left (4 \, x^{2} - 9\right )} \log \left (2 \, x - 3\right ) + 4 \,{\left (4 \, x^{2} - 9\right )} \arctan \left (\frac{4}{81} \, \sqrt{3}{\left (2 \, x^{3} + 9 \, x\right )}\right ) + 4 \,{\left (4 \, x^{2} - 9\right )} \arctan \left (\frac{2}{9} \, \sqrt{3} x\right ) - 12 \, \sqrt{3} x\right )}}{157464 \,{\left (4 \, x^{2} - 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((16*x^4 + 36*x^2 + 81)/(64*x^6 - 729)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.271751, size = 70, normalized size = 0.86 \[ - \frac{x}{17496 x^{2} - 39366} + \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{9} \right )} + 2 \operatorname{atan}{\left (\frac{8 \sqrt{3} x^{3}}{81} + \frac{4 \sqrt{3} x}{9} \right )}\right )}{78732} - \frac{\log{\left (x - \frac{3}{2} \right )}}{17496} + \frac{\log{\left (x + \frac{3}{2} \right )}}{17496} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((16*x**4+36*x**2+81)/(-64*x**6+729)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.21981, size = 85, normalized size = 1.05 \[ \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{39366} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} - 9\right )}} + \frac{1}{17496} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{17496} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((16*x^4 + 36*x^2 + 81)/(64*x^6 - 729)^2,x, algorithm="giac")
[Out]