Optimal. Leaf size=69 \[ -\frac{1}{5 x^5}+\frac{11}{12 x^3}-\frac{x \left (3-5 x^2\right )}{16 \left (x^4+3 x^2+2\right )}-\frac{23}{4 x}-\frac{23}{2} \tan ^{-1}(x)+\frac{97 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]
[Out]
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Rubi [A] time = 0.143821, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{1}{5 x^5}+\frac{11}{12 x^3}-\frac{x \left (3-5 x^2\right )}{16 \left (x^4+3 x^2+2\right )}-\frac{23}{4 x}-\frac{23}{2} \tan ^{-1}(x)+\frac{97 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^6*(2 + 3*x^2 + x^4)^2),x]
[Out]
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Rubi in Sympy [A] time = 22.4563, size = 41, normalized size = 0.59 \[ 17 \operatorname{atan}{\left (x \right )} - \frac{11 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} + \frac{57}{4 x} - \frac{23}{6 x^{3}} + \frac{6}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**6+3*x**4+x**2+4)/x**6/(x**4+3*x**2+2)**2,x)
[Out]
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Mathematica [A] time = 0.111535, size = 61, normalized size = 0.88 \[ \frac{1}{480} \left (-\frac{96}{x^5}+\frac{440}{x^3}+\frac{30 x \left (5 x^2-3\right )}{x^4+3 x^2+2}-\frac{2760}{x}-5520 \tan ^{-1}(x)+1455 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^6*(2 + 3*x^2 + x^4)^2),x]
[Out]
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Maple [A] time = 0.023, size = 53, normalized size = 0.8 \[ -{\frac{1}{5\,{x}^{5}}}+{\frac{11}{12\,{x}^{3}}}-{\frac{23}{4\,x}}+{\frac{13\,x}{16\,{x}^{2}+32}}+{\frac{97\,\sqrt{2}}{32}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }-{\frac{x}{2\,{x}^{2}+2}}-{\frac{23\,\arctan \left ( x \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^6+3*x^4+x^2+4)/x^6/(x^4+3*x^2+2)^2,x)
[Out]
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Maxima [A] time = 0.797223, size = 77, normalized size = 1.12 \[ \frac{97}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1305 \, x^{8} + 3965 \, x^{6} + 2148 \, x^{4} - 296 \, x^{2} + 96}{240 \,{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )}} - \frac{23}{2} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 3*x^2 + 2)^2*x^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282272, size = 124, normalized size = 1.8 \[ -\frac{\sqrt{2}{\left (2760 \, \sqrt{2}{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )} \arctan \left (x\right ) - 1455 \,{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \sqrt{2}{\left (1305 \, x^{8} + 3965 \, x^{6} + 2148 \, x^{4} - 296 \, x^{2} + 96\right )}\right )}}{480 \,{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 3*x^2 + 2)^2*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.806293, size = 61, normalized size = 0.88 \[ - \frac{23 \operatorname{atan}{\left (x \right )}}{2} + \frac{97 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{32} - \frac{1305 x^{8} + 3965 x^{6} + 2148 x^{4} - 296 x^{2} + 96}{240 x^{9} + 720 x^{7} + 480 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**6+3*x**4+x**2+4)/x**6/(x**4+3*x**2+2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.270049, size = 77, normalized size = 1.12 \[ \frac{97}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{5 \, x^{3} - 3 \, x}{16 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{345 \, x^{4} - 55 \, x^{2} + 12}{60 \, x^{5}} - \frac{23}{2} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 3*x^2 + 2)^2*x^6),x, algorithm="giac")
[Out]