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

Optimal. Leaf size=79 \[ \frac{x \left (11 x^2+9\right )}{8 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (347 x^2+547\right )}{32 \left (x^4+3 x^2+2\right )}-\frac{1}{2 x}+\frac{189}{8} \tan ^{-1}(x)-\frac{1119 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

[Out]

-1/(2*x) + (x*(9 + 11*x^2))/(8*(2 + 3*x^2 + x^4)^2) - (x*(547 + 347*x^2))/(32*(2
+ 3*x^2 + x^4)) + (189*ArcTan[x])/8 - (1119*ArcTan[x/Sqrt[2]])/(32*Sqrt[2])

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Rubi [A]  time = 0.170293, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{x \left (11 x^2+9\right )}{8 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (347 x^2+547\right )}{32 \left (x^4+3 x^2+2\right )}-\frac{1}{2 x}+\frac{189}{8} \tan ^{-1}(x)-\frac{1119 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*x) + (x*(9 + 11*x^2))/(8*(2 + 3*x^2 + x^4)^2) - (x*(547 + 347*x^2))/(32*(2
+ 3*x^2 + x^4)) + (189*ArcTan[x])/8 - (1119*ArcTan[x/Sqrt[2]])/(32*Sqrt[2])

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Rubi in Sympy [A]  time = 26.9304, size = 71, normalized size = 0.9 \[ \frac{x \left (4374 x^{2} + 7290\right )}{432 \left (x^{4} + 3 x^{2} + 2\right )^{2}} - \frac{x \left (1752516 x^{2} + 2920860\right )}{15552 \left (x^{4} + 3 x^{2} + 2\right )} + \frac{699 \operatorname{atan}{\left (x \right )}}{4} - \frac{1251 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} - \frac{405}{16 x} \]

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)**3,x)

[Out]

x*(4374*x**2 + 7290)/(432*(x**4 + 3*x**2 + 2)**2) - x*(1752516*x**2 + 2920860)/(
15552*(x**4 + 3*x**2 + 2)) + 699*atan(x)/4 - 1251*sqrt(2)*atan(sqrt(2)*x/2)/8 -
405/(16*x)

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Mathematica [A]  time = 0.11313, size = 63, normalized size = 0.8 \[ \frac{1}{64} \left (-\frac{2 \left (363 x^8+1684 x^6+2499 x^4+1250 x^2+64\right )}{x \left (x^4+3 x^2+2\right )^2}+1512 \tan ^{-1}(x)-1119 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*(64 + 1250*x^2 + 2499*x^4 + 1684*x^6 + 363*x^8))/(x*(2 + 3*x^2 + x^4)^2) +
1512*ArcTan[x] - 1119*Sqrt[2]*ArcTan[x/Sqrt[2]])/64

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Maple [A]  time = 0.021, size = 58, normalized size = 0.7 \[ -{\frac{1}{2\,x}}-{\frac{1}{2\, \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{207\,{x}^{3}}{16}}+{\frac{233\,x}{8}} \right ) }-{\frac{1119\,\sqrt{2}}{64}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{35\,{x}^{3}}{8}}-{\frac{37\,x}{8}} \right ) }+{\frac{189\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/x-1/2*(207/16*x^3+233/8*x)/(x^2+2)^2-1119/64*arctan(1/2*2^(1/2)*x)*2^(1/2)+
(-35/8*x^3-37/8*x)/(x^2+1)^2+189/8*arctan(x)

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Maxima [A]  time = 0.785337, size = 88, normalized size = 1.11 \[ -\frac{1119}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{363 \, x^{8} + 1684 \, x^{6} + 2499 \, x^{4} + 1250 \, x^{2} + 64}{32 \,{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )}} + \frac{189}{8} \, \arctan \left (x\right ) \]

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^2),x, algorithm="maxima")

[Out]

-1119/64*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/32*(363*x^8 + 1684*x^6 + 2499*x^4 + 1
250*x^2 + 64)/(x^9 + 6*x^7 + 13*x^5 + 12*x^3 + 4*x) + 189/8*arctan(x)

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Fricas [A]  time = 0.268486, size = 158, normalized size = 2. \[ \frac{\sqrt{2}{\left (756 \, \sqrt{2}{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )} \arctan \left (x\right ) - 1119 \,{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (363 \, x^{8} + 1684 \, x^{6} + 2499 \, x^{4} + 1250 \, x^{2} + 64\right )}\right )}}{64 \,{\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )}} \]

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^2),x, algorithm="fricas")

[Out]

1/64*sqrt(2)*(756*sqrt(2)*(x^9 + 6*x^7 + 13*x^5 + 12*x^3 + 4*x)*arctan(x) - 1119
*(x^9 + 6*x^7 + 13*x^5 + 12*x^3 + 4*x)*arctan(1/2*sqrt(2)*x) - sqrt(2)*(363*x^8
+ 1684*x^6 + 2499*x^4 + 1250*x^2 + 64))/(x^9 + 6*x^7 + 13*x^5 + 12*x^3 + 4*x)

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Sympy [A]  time = 0.905395, size = 70, normalized size = 0.89 \[ - \frac{363 x^{8} + 1684 x^{6} + 2499 x^{4} + 1250 x^{2} + 64}{32 x^{9} + 192 x^{7} + 416 x^{5} + 384 x^{3} + 128 x} + \frac{189 \operatorname{atan}{\left (x \right )}}{8} - \frac{1119 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{64} \]

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)**3,x)

[Out]

-(363*x**8 + 1684*x**6 + 2499*x**4 + 1250*x**2 + 64)/(32*x**9 + 192*x**7 + 416*x
**5 + 384*x**3 + 128*x) + 189*atan(x)/8 - 1119*sqrt(2)*atan(sqrt(2)*x/2)/64

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GIAC/XCAS [A]  time = 0.272404, size = 74, normalized size = 0.94 \[ -\frac{1119}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{347 \, x^{7} + 1588 \, x^{5} + 2291 \, x^{3} + 1058 \, x}{32 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{1}{2 \, x} + \frac{189}{8} \, \arctan \left (x\right ) \]

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^2),x, algorithm="giac")

[Out]

-1119/64*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/32*(347*x^7 + 1588*x^5 + 2291*x^3 + 1
058*x)/(x^4 + 3*x^2 + 2)^2 - 1/2/x + 189/8*arctan(x)