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

Optimal. Leaf size=62 \[ -\frac{1}{3 x^3}-\frac{x \left (9 x^2+5\right )}{8 \left (x^4+3 x^2+2\right )}+\frac{11}{4 x}+\frac{21}{2} \tan ^{-1}(x)-\frac{71 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

[Out]

-1/(3*x^3) + 11/(4*x) - (x*(5 + 9*x^2))/(8*(2 + 3*x^2 + x^4)) + (21*ArcTan[x])/2
- (71*ArcTan[x/Sqrt[2]])/(8*Sqrt[2])

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Rubi [A]  time = 0.132133, antiderivative size = 62, 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 \[ -\frac{1}{3 x^3}-\frac{x \left (9 x^2+5\right )}{8 \left (x^4+3 x^2+2\right )}+\frac{11}{4 x}+\frac{21}{2} \tan ^{-1}(x)-\frac{71 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*x^3) + 11/(4*x) - (x*(5 + 9*x^2))/(8*(2 + 3*x^2 + x^4)) + (21*ArcTan[x])/2
- (71*ArcTan[x/Sqrt[2]])/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 18.9232, size = 32, normalized size = 0.52 \[ - 17 \operatorname{atan}{\left (x \right )} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} - \frac{23}{2 x} + \frac{2}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17*atan(x) + 11*sqrt(2)*atan(sqrt(2)*x/2)/4 - 23/(2*x) + 2/x**3

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Mathematica [A]  time = 0.111851, size = 56, normalized size = 0.9 \[ \frac{1}{48} \left (-\frac{16}{x^3}-\frac{6 x \left (9 x^2+5\right )}{x^4+3 x^2+2}+\frac{132}{x}+504 \tan ^{-1}(x)-213 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-16/x^3 + 132/x - (6*x*(5 + 9*x^2))/(2 + 3*x^2 + x^4) + 504*ArcTan[x] - 213*Sqr
t[2]*ArcTan[x/Sqrt[2]])/48

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Maple [A]  time = 0.021, size = 48, normalized size = 0.8 \[ -{\frac{1}{3\,{x}^{3}}}+{\frac{11}{4\,x}}-{\frac{13\,x}{8\,{x}^{2}+16}}-{\frac{71\,\sqrt{2}}{16}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }+{\frac{x}{2\,{x}^{2}+2}}+{\frac{21\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/x^3+11/4/x-13/8*x/(x^2+2)-71/16*arctan(1/2*2^(1/2)*x)*2^(1/2)+1/2*x/(x^2+1)
+21/2*arctan(x)

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Maxima [A]  time = 0.794324, size = 70, normalized size = 1.13 \[ -\frac{71}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{39 \, x^{6} + 175 \, x^{4} + 108 \, x^{2} - 16}{24 \,{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )}} + \frac{21}{2} \, \arctan \left (x\right ) \]

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^4),x, algorithm="maxima")

[Out]

-71/16*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/24*(39*x^6 + 175*x^4 + 108*x^2 - 16)/(x
^7 + 3*x^5 + 2*x^3) + 21/2*arctan(x)

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Fricas [A]  time = 0.271057, size = 117, normalized size = 1.89 \[ \frac{\sqrt{2}{\left (252 \, \sqrt{2}{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )} \arctan \left (x\right ) - 213 \,{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \sqrt{2}{\left (39 \, x^{6} + 175 \, x^{4} + 108 \, x^{2} - 16\right )}\right )}}{48 \,{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )}} \]

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^4),x, algorithm="fricas")

[Out]

1/48*sqrt(2)*(252*sqrt(2)*(x^7 + 3*x^5 + 2*x^3)*arctan(x) - 213*(x^7 + 3*x^5 + 2
*x^3)*arctan(1/2*sqrt(2)*x) + sqrt(2)*(39*x^6 + 175*x^4 + 108*x^2 - 16))/(x^7 +
3*x^5 + 2*x^3)

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Sympy [A]  time = 0.731493, size = 56, normalized size = 0.9 \[ \frac{21 \operatorname{atan}{\left (x \right )}}{2} - \frac{71 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} + \frac{39 x^{6} + 175 x^{4} + 108 x^{2} - 16}{24 x^{7} + 72 x^{5} + 48 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

21*atan(x)/2 - 71*sqrt(2)*atan(sqrt(2)*x/2)/16 + (39*x**6 + 175*x**4 + 108*x**2
- 16)/(24*x**7 + 72*x**5 + 48*x**3)

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GIAC/XCAS [A]  time = 0.27081, size = 70, normalized size = 1.13 \[ -\frac{71}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{9 \, x^{3} + 5 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{33 \, x^{2} - 4}{12 \, x^{3}} + \frac{21}{2} \, \arctan \left (x\right ) \]

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^4),x, algorithm="giac")

[Out]

-71/16*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/8*(9*x^3 + 5*x)/(x^4 + 3*x^2 + 2) + 1/1
2*(33*x^2 - 4)/x^3 + 21/2*arctan(x)