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

Optimal. Leaf size=72 \[ -\frac{x \left (12 x^2+11\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (217 x^2+335\right )}{16 \left (x^4+3 x^2+2\right )}-\frac{257}{8} \tan ^{-1}(x)+\frac{731 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]

[Out]

-(x*(11 + 12*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(335 + 217*x^2))/(16*(2 + 3*x^2
+ x^4)) - (257*ArcTan[x])/8 + (731*ArcTan[x/Sqrt[2]])/(16*Sqrt[2])

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Rubi [A]  time = 0.0836836, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{x \left (12 x^2+11\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (217 x^2+335\right )}{16 \left (x^4+3 x^2+2\right )}-\frac{257}{8} \tan ^{-1}(x)+\frac{731 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(11 + 12*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(335 + 217*x^2))/(16*(2 + 3*x^2
+ x^4)) - (257*ArcTan[x])/8 + (731*ArcTan[x/Sqrt[2]])/(16*Sqrt[2])

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Rubi in Sympy [A]  time = 13.3934, size = 66, normalized size = 0.92 \[ - \frac{x \left (648 x^{2} + 594\right )}{216 \left (x^{4} + 3 x^{2} + 2\right )^{2}} + \frac{x \left (11718 x^{2} + 18090\right )}{864 \left (x^{4} + 3 x^{2} + 2\right )} - \frac{257 \operatorname{atan}{\left (x \right )}}{8} + \frac{731 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(648*x**2 + 594)/(216*(x**4 + 3*x**2 + 2)**2) + x*(11718*x**2 + 18090)/(864*(
x**4 + 3*x**2 + 2)) - 257*atan(x)/8 + 731*sqrt(2)*atan(sqrt(2)*x/2)/32

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Mathematica [A]  time = 0.104213, size = 56, normalized size = 0.78 \[ \frac{1}{32} \left (\frac{2 x \left (217 x^6+986 x^4+1391 x^2+626\right )}{\left (x^4+3 x^2+2\right )^2}-1028 \tan ^{-1}(x)+731 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*x*(626 + 1391*x^2 + 986*x^4 + 217*x^6))/(2 + 3*x^2 + x^4)^2 - 1028*ArcTan[x]
+ 731*Sqrt[2]*ArcTan[x/Sqrt[2]])/32

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Maple [A]  time = 0.016, size = 53, normalized size = 0.7 \[{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{155\,{x}^{3}}{16}}+{\frac{181\,x}{8}} \right ) }+{\frac{731\,\sqrt{2}}{32}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{31\,{x}^{3}}{8}}-{\frac{33\,x}{8}} \right ) }-{\frac{257\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(155/16*x^3+181/8*x)/(x^2+2)^2+731/32*arctan(1/2*2^(1/2)*x)*2^(1/2)-(-31/8*x^3-3
3/8*x)/(x^2+1)^2-257/8*arctan(x)

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Maxima [A]  time = 0.796606, size = 81, normalized size = 1.12 \[ \frac{731}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{217 \, x^{7} + 986 \, x^{5} + 1391 \, x^{3} + 626 \, x}{16 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac{257}{8} \, \arctan \left (x\right ) \]

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)^3,x, algorithm="maxima")

[Out]

731/32*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/16*(217*x^7 + 986*x^5 + 1391*x^3 + 626*
x)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4) - 257/8*arctan(x)

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Fricas [A]  time = 0.266398, size = 146, normalized size = 2.03 \[ -\frac{\sqrt{2}{\left (514 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 731 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (217 \, x^{7} + 986 \, x^{5} + 1391 \, x^{3} + 626 \, x\right )}\right )}}{32 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

-1/32*sqrt(2)*(514*sqrt(2)*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(x) - 731*(
x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(1/2*sqrt(2)*x) - sqrt(2)*(217*x^7 + 98
6*x^5 + 1391*x^3 + 626*x))/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)

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Sympy [A]  time = 0.805529, size = 65, normalized size = 0.9 \[ \frac{217 x^{7} + 986 x^{5} + 1391 x^{3} + 626 x}{16 x^{8} + 96 x^{6} + 208 x^{4} + 192 x^{2} + 64} - \frac{257 \operatorname{atan}{\left (x \right )}}{8} + \frac{731 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{32} \]

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)**3,x)

[Out]

(217*x**7 + 986*x**5 + 1391*x**3 + 626*x)/(16*x**8 + 96*x**6 + 208*x**4 + 192*x*
*2 + 64) - 257*atan(x)/8 + 731*sqrt(2)*atan(sqrt(2)*x/2)/32

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GIAC/XCAS [A]  time = 0.272371, size = 68, normalized size = 0.94 \[ \frac{731}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{217 \, x^{7} + 986 \, x^{5} + 1391 \, x^{3} + 626 \, x}{16 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{257}{8} \, \arctan \left (x\right ) \]

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)^3,x, algorithm="giac")

[Out]

731/32*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/16*(217*x^7 + 986*x^5 + 1391*x^3 + 626*
x)/(x^4 + 3*x^2 + 2)^2 - 257/8*arctan(x)