3.87 \(\int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left (2+3 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=53 \[ \frac{x \left (11 x^2+9\right )}{4 \left (x^4+3 x^2+2\right )}-\frac{1}{x}-\frac{19}{2} \tan ^{-1}(x)+\frac{45 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

[Out]

-x^(-1) + (x*(9 + 11*x^2))/(4*(2 + 3*x^2 + x^4)) - (19*ArcTan[x])/2 + (45*ArcTan
[x/Sqrt[2]])/(4*Sqrt[2])

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Rubi [A]  time = 0.118576, antiderivative size = 53, 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 \[ \frac{x \left (11 x^2+9\right )}{4 \left (x^4+3 x^2+2\right )}-\frac{1}{x}-\frac{19}{2} \tan ^{-1}(x)+\frac{45 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(2 + 3*x^2 + x^4)^2),x]

[Out]

-x^(-1) + (x*(9 + 11*x^2))/(4*(2 + 3*x^2 + x^4)) - (19*ArcTan[x])/2 + (45*ArcTan
[x/Sqrt[2]])/(4*Sqrt[2])

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Rubi in Sympy [A]  time = 19.0254, size = 48, normalized size = 0.91 \[ \frac{x \left (4374 x^{2} + 7290\right )}{216 \left (x^{4} + 3 x^{2} + 2\right )} - \frac{47 \operatorname{atan}{\left (x \right )}}{2} + \frac{199 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} + \frac{6}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**6+3*x**4+x**2+4)/x**2/(x**4+3*x**2+2)**2,x)

[Out]

x*(4374*x**2 + 7290)/(216*(x**4 + 3*x**2 + 2)) - 47*atan(x)/2 + 199*sqrt(2)*atan
(sqrt(2)*x/2)/8 + 6/x

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Mathematica [A]  time = 0.0980377, size = 51, normalized size = 0.96 \[ \frac{1}{8} \left (\frac{2 x \left (11 x^2+9\right )}{x^4+3 x^2+2}-\frac{8}{x}-76 \tan ^{-1}(x)+45 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(2 + 3*x^2 + x^4)^2),x]

[Out]

(-8/x + (2*x*(9 + 11*x^2))/(2 + 3*x^2 + x^4) - 76*ArcTan[x] + 45*Sqrt[2]*ArcTan[
x/Sqrt[2]])/8

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Maple [A]  time = 0.018, size = 43, normalized size = 0.8 \[ -{x}^{-1}+{\frac{13\,x}{4\,{x}^{2}+8}}+{\frac{45\,\sqrt{2}}{8}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }-{\frac{x}{2\,{x}^{2}+2}}-{\frac{19\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^6+3*x^4+x^2+4)/x^2/(x^4+3*x^2+2)^2,x)

[Out]

-1/x+13/4*x/(x^2+2)+45/8*arctan(1/2*2^(1/2)*x)*2^(1/2)-1/2*x/(x^2+1)-19/2*arctan
(x)

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Maxima [A]  time = 0.792478, size = 61, normalized size = 1.15 \[ \frac{45}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{7 \, x^{4} - 3 \, x^{2} - 8}{4 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} - \frac{19}{2} \, \arctan \left (x\right ) \]

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^2),x, algorithm="maxima")

[Out]

45/8*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/4*(7*x^4 - 3*x^2 - 8)/(x^5 + 3*x^3 + 2*x)
- 19/2*arctan(x)

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Fricas [A]  time = 0.27703, size = 104, normalized size = 1.96 \[ -\frac{\sqrt{2}{\left (38 \, \sqrt{2}{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )} \arctan \left (x\right ) - 45 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (7 \, x^{4} - 3 \, x^{2} - 8\right )}\right )}}{8 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} \]

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^2),x, algorithm="fricas")

[Out]

-1/8*sqrt(2)*(38*sqrt(2)*(x^5 + 3*x^3 + 2*x)*arctan(x) - 45*(x^5 + 3*x^3 + 2*x)*
arctan(1/2*sqrt(2)*x) - sqrt(2)*(7*x^4 - 3*x^2 - 8))/(x^5 + 3*x^3 + 2*x)

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Sympy [A]  time = 0.655939, size = 49, normalized size = 0.92 \[ \frac{7 x^{4} - 3 x^{2} - 8}{4 x^{5} + 12 x^{3} + 8 x} - \frac{19 \operatorname{atan}{\left (x \right )}}{2} + \frac{45 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**6+3*x**4+x**2+4)/x**2/(x**4+3*x**2+2)**2,x)

[Out]

(7*x**4 - 3*x**2 - 8)/(4*x**5 + 12*x**3 + 8*x) - 19*atan(x)/2 + 45*sqrt(2)*atan(
sqrt(2)*x/2)/8

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GIAC/XCAS [A]  time = 0.270304, size = 61, normalized size = 1.15 \[ \frac{45}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{7 \, x^{4} - 3 \, x^{2} - 8}{4 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} - \frac{19}{2} \, \arctan \left (x\right ) \]

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^2),x, algorithm="giac")

[Out]

45/8*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/4*(7*x^4 - 3*x^2 - 8)/(x^5 + 3*x^3 + 2*x)
- 19/2*arctan(x)