Optimal. Leaf size=89 \[ \frac{2 x}{3 \left (3-4 x^2\right )}+\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (x+1)}-\frac{1}{18 (2 x+1)}+\frac{67}{54} \tanh ^{-1}(x)-\frac{7}{27} \tanh ^{-1}(2 x)-\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.122869, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 x}{3 \left (3-4 x^2\right )}+\frac{1}{18 (1-2 x)}+\frac{1}{36 (1-x)}-\frac{1}{36 (x+1)}-\frac{1}{18 (2 x+1)}+\frac{67}{54} \tanh ^{-1}(x)-\frac{7}{27} \tanh ^{-1}(2 x)-\frac{5 \tanh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-16*x**6+32*x**4-19*x**2+3)**2,x)
[Out]
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Mathematica [A] time = 0.0878126, size = 103, normalized size = 1.16 \[ \frac{1}{108} \left (-\frac{6 x \left (80 x^4-104 x^2+27\right )}{16 x^6-32 x^4+19 x^2-3}+14 \log (1-2 x)+30 \sqrt{3} \log \left (\sqrt{3}-2 x\right )-67 \log (1-x)+67 \log (x+1)-14 \log (2 x+1)-30 \sqrt{3} \log \left (2 x+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-2),x]
[Out]
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Maple [A] time = 0.031, size = 84, normalized size = 0.9 \[ -{\frac{1}{-36+36\,x}}-{\frac{67\,\ln \left ( -1+x \right ) }{108}}-{\frac{x}{6} \left ({x}^{2}-{\frac{3}{4}} \right ) ^{-1}}-{\frac{5\,\sqrt{3}}{9}{\it Artanh} \left ({\frac{2\,x\sqrt{3}}{3}} \right ) }-{\frac{1}{36+36\,x}}+{\frac{67\,\ln \left ( 1+x \right ) }{108}}-{\frac{1}{36\,x-18}}+{\frac{7\,\ln \left ( 2\,x-1 \right ) }{54}}-{\frac{1}{18+36\,x}}-{\frac{7\,\ln \left ( 1+2\,x \right ) }{54}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-16*x^6+32*x^4-19*x^2+3)^2,x)
[Out]
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Maxima [A] time = 0.894588, size = 120, normalized size = 1.35 \[ \frac{5}{18} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3}}{2 \, x + \sqrt{3}}\right ) - \frac{80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac{7}{54} \, \log \left (2 \, x + 1\right ) + \frac{7}{54} \, \log \left (2 \, x - 1\right ) + \frac{67}{108} \, \log \left (x + 1\right ) - \frac{67}{108} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((16*x^6 - 32*x^4 + 19*x^2 - 3)^(-2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.259509, size = 266, normalized size = 2.99 \[ -\frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x + 1\right ) - 14 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x - 1\right ) - 67 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x + 1\right ) + 67 \, \sqrt{3}{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x - 1\right ) - 90 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (\frac{\sqrt{3}{\left (4 \, x^{2} + 3\right )} - 12 \, x}{4 \, x^{2} - 3}\right ) + 6 \, \sqrt{3}{\left (80 \, x^{5} - 104 \, x^{3} + 27 \, x\right )}\right )}}{324 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((16*x^6 - 32*x^4 + 19*x^2 - 3)^(-2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.85285, size = 104, normalized size = 1.17 \[ - \frac{80 x^{5} - 104 x^{3} + 27 x}{288 x^{6} - 576 x^{4} + 342 x^{2} - 54} - \frac{67 \log{\left (x - 1 \right )}}{108} + \frac{7 \log{\left (x - \frac{1}{2} \right )}}{54} - \frac{7 \log{\left (x + \frac{1}{2} \right )}}{54} + \frac{67 \log{\left (x + 1 \right )}}{108} + \frac{5 \sqrt{3} \log{\left (x - \frac{\sqrt{3}}{2} \right )}}{18} - \frac{5 \sqrt{3} \log{\left (x + \frac{\sqrt{3}}{2} \right )}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-16*x**6+32*x**4-19*x**2+3)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.263432, size = 131, normalized size = 1.47 \[ \frac{5}{18} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} \right |}}\right ) - \frac{80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \,{\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac{7}{54} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) + \frac{7}{54} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) + \frac{67}{108} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{67}{108} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((16*x^6 - 32*x^4 + 19*x^2 - 3)^(-2),x, algorithm="giac")
[Out]