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

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]

-1/(5*x^5) + 11/(12*x^3) - 23/(4*x) - (x*(3 - 5*x^2))/(16*(2 + 3*x^2 + x^4)) - (
23*ArcTan[x])/2 + (97*ArcTan[x/Sqrt[2]])/(16*Sqrt[2])

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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]

-1/(5*x^5) + 11/(12*x^3) - 23/(4*x) - (x*(3 - 5*x^2))/(16*(2 + 3*x^2 + x^4)) - (
23*ArcTan[x])/2 + (97*ArcTan[x/Sqrt[2]])/(16*Sqrt[2])

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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]

17*atan(x) - 11*sqrt(2)*atan(sqrt(2)*x/2)/8 + 57/(4*x) - 23/(6*x**3) + 6/(5*x**5
)

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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]

(-96/x^5 + 440/x^3 - 2760/x + (30*x*(-3 + 5*x^2))/(2 + 3*x^2 + x^4) - 5520*ArcTa
n[x] + 1455*Sqrt[2]*ArcTan[x/Sqrt[2]])/480

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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]

-1/5/x^5+11/12/x^3-23/4/x+13/16*x/(x^2+2)+97/32*arctan(1/2*2^(1/2)*x)*2^(1/2)-1/
2*x/(x^2+1)-23/2*arctan(x)

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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]

97/32*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/240*(1305*x^8 + 3965*x^6 + 2148*x^4 - 29
6*x^2 + 96)/(x^9 + 3*x^7 + 2*x^5) - 23/2*arctan(x)

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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]

-1/480*sqrt(2)*(2760*sqrt(2)*(x^9 + 3*x^7 + 2*x^5)*arctan(x) - 1455*(x^9 + 3*x^7
+ 2*x^5)*arctan(1/2*sqrt(2)*x) + sqrt(2)*(1305*x^8 + 3965*x^6 + 2148*x^4 - 296*
x^2 + 96))/(x^9 + 3*x^7 + 2*x^5)

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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]

-23*atan(x)/2 + 97*sqrt(2)*atan(sqrt(2)*x/2)/32 - (1305*x**8 + 3965*x**6 + 2148*
x**4 - 296*x**2 + 96)/(240*x**9 + 720*x**7 + 480*x**5)

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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]

97/32*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/16*(5*x^3 - 3*x)/(x^4 + 3*x^2 + 2) - 1/6
0*(345*x^4 - 55*x^2 + 12)/x^5 - 23/2*arctan(x)