3.90 \(\int \frac{4+x^2+3 x^4+5 x^6}{x^8 \left (2+3 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac{1}{7 x^7}+\frac{11}{20 x^5}-\frac{23}{12 x^3}+\frac{x \left (3 x^2+19\right )}{32 \left (x^4+3 x^2+2\right )}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

[Out]

-1/(7*x^7) + 11/(20*x^5) - 23/(12*x^3) + 137/(16*x) + (x*(19 + 3*x^2))/(32*(2 +
3*x^2 + x^4)) + (25*ArcTan[x])/2 - (123*ArcTan[x/Sqrt[2]])/(32*Sqrt[2])

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Rubi [A]  time = 0.154096, antiderivative size = 76, 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 \[ -\frac{1}{7 x^7}+\frac{11}{20 x^5}-\frac{23}{12 x^3}+\frac{x \left (3 x^2+19\right )}{32 \left (x^4+3 x^2+2\right )}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^8*(2 + 3*x^2 + x^4)^2),x]

[Out]

-1/(7*x^7) + 11/(20*x^5) - 23/(12*x^3) + 137/(16*x) + (x*(19 + 3*x^2))/(32*(2 +
3*x^2 + x^4)) + (25*ArcTan[x])/2 - (123*ArcTan[x/Sqrt[2]])/(32*Sqrt[2])

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Rubi in Sympy [A]  time = 25.9088, size = 48, normalized size = 0.63 \[ - 17 \operatorname{atan}{\left (x \right )} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} - \frac{125}{8 x} + \frac{19}{4 x^{3}} - \frac{23}{10 x^{5}} + \frac{6}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**6+3*x**4+x**2+4)/x**8/(x**4+3*x**2+2)**2,x)

[Out]

-17*atan(x) + 11*sqrt(2)*atan(sqrt(2)*x/2)/16 - 125/(8*x) + 19/(4*x**3) - 23/(10
*x**5) + 6/(7*x**7)

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Mathematica [A]  time = 0.126884, size = 77, normalized size = 1.01 \[ -\frac{1}{7 x^7}+\frac{11}{20 x^5}-\frac{23}{12 x^3}+\frac{3 x^3+19 x}{32 \left (x^4+3 x^2+2\right )}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^8*(2 + 3*x^2 + x^4)^2),x]

[Out]

-1/(7*x^7) + 11/(20*x^5) - 23/(12*x^3) + 137/(16*x) + (19*x + 3*x^3)/(32*(2 + 3*
x^2 + x^4)) + (25*ArcTan[x])/2 - (123*ArcTan[x/Sqrt[2]])/(32*Sqrt[2])

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Maple [A]  time = 0.023, size = 58, normalized size = 0.8 \[ -{\frac{1}{7\,{x}^{7}}}+{\frac{11}{20\,{x}^{5}}}-{\frac{23}{12\,{x}^{3}}}+{\frac{137}{16\,x}}-{\frac{13\,x}{32\,{x}^{2}+64}}-{\frac{123\,\sqrt{2}}{64}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }+{\frac{x}{2\,{x}^{2}+2}}+{\frac{25\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^6+3*x^4+x^2+4)/x^8/(x^4+3*x^2+2)^2,x)

[Out]

-1/7/x^7+11/20/x^5-23/12/x^3+137/16/x-13/32*x/(x^2+2)-123/64*arctan(1/2*2^(1/2)*
x)*2^(1/2)+1/2*x/(x^2+1)+25/2*arctan(x)

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Maxima [A]  time = 0.794469, size = 84, normalized size = 1.11 \[ -\frac{123}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{29085 \, x^{10} + 81865 \, x^{8} + 40068 \, x^{6} - 7816 \, x^{4} + 2256 \, x^{2} - 960}{3360 \,{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )}} + \frac{25}{2} \, \arctan \left (x\right ) \]

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^8),x, algorithm="maxima")

[Out]

-123/64*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/3360*(29085*x^10 + 81865*x^8 + 40068*x
^6 - 7816*x^4 + 2256*x^2 - 960)/(x^11 + 3*x^9 + 2*x^7) + 25/2*arctan(x)

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Fricas [A]  time = 0.291305, size = 131, normalized size = 1.72 \[ \frac{\sqrt{2}{\left (42000 \, \sqrt{2}{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )} \arctan \left (x\right ) - 12915 \,{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \sqrt{2}{\left (29085 \, x^{10} + 81865 \, x^{8} + 40068 \, x^{6} - 7816 \, x^{4} + 2256 \, x^{2} - 960\right )}\right )}}{6720 \,{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )}} \]

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^8),x, algorithm="fricas")

[Out]

1/6720*sqrt(2)*(42000*sqrt(2)*(x^11 + 3*x^9 + 2*x^7)*arctan(x) - 12915*(x^11 + 3
*x^9 + 2*x^7)*arctan(1/2*sqrt(2)*x) + sqrt(2)*(29085*x^10 + 81865*x^8 + 40068*x^
6 - 7816*x^4 + 2256*x^2 - 960))/(x^11 + 3*x^9 + 2*x^7)

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Sympy [A]  time = 0.904628, size = 66, normalized size = 0.87 \[ \frac{25 \operatorname{atan}{\left (x \right )}}{2} - \frac{123 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{64} + \frac{29085 x^{10} + 81865 x^{8} + 40068 x^{6} - 7816 x^{4} + 2256 x^{2} - 960}{3360 x^{11} + 10080 x^{9} + 6720 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**6+3*x**4+x**2+4)/x**8/(x**4+3*x**2+2)**2,x)

[Out]

25*atan(x)/2 - 123*sqrt(2)*atan(sqrt(2)*x/2)/64 + (29085*x**10 + 81865*x**8 + 40
068*x**6 - 7816*x**4 + 2256*x**2 - 960)/(3360*x**11 + 10080*x**9 + 6720*x**7)

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GIAC/XCAS [A]  time = 0.269846, size = 84, normalized size = 1.11 \[ -\frac{123}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{3 \, x^{3} + 19 \, x}{32 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{14385 \, x^{6} - 3220 \, x^{4} + 924 \, x^{2} - 240}{1680 \, x^{7}} + \frac{25}{2} \, \arctan \left (x\right ) \]

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^8),x, algorithm="giac")

[Out]

-123/64*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/32*(3*x^3 + 19*x)/(x^4 + 3*x^2 + 2) +
1/1680*(14385*x^6 - 3220*x^4 + 924*x^2 - 240)/x^7 + 25/2*arctan(x)