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

Optimal. Leaf size=86 \[ -\frac{1}{6 x^3}-\frac{x \left (9 x^2+5\right )}{16 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (571 x^2+951\right )}{64 \left (x^4+3 x^2+2\right )}+\frac{17}{8 x}-\frac{113}{8} \tan ^{-1}(x)+\frac{1611 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{64 \sqrt{2}} \]

[Out]

-1/(6*x^3) + 17/(8*x) - (x*(5 + 9*x^2))/(16*(2 + 3*x^2 + x^4)^2) + (x*(951 + 571
*x^2))/(64*(2 + 3*x^2 + x^4)) - (113*ArcTan[x])/8 + (1611*ArcTan[x/Sqrt[2]])/(64
*Sqrt[2])

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Rubi [A]  time = 0.191496, antiderivative size = 86, 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 \[ -\frac{1}{6 x^3}-\frac{x \left (9 x^2+5\right )}{16 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (571 x^2+951\right )}{64 \left (x^4+3 x^2+2\right )}+\frac{17}{8 x}-\frac{113}{8} \tan ^{-1}(x)+\frac{1611 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{64 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(6*x^3) + 17/(8*x) - (x*(5 + 9*x^2))/(16*(2 + 3*x^2 + x^4)^2) + (x*(951 + 571
*x^2))/(64*(2 + 3*x^2 + x^4)) - (113*ArcTan[x])/8 + (1611*ArcTan[x/Sqrt[2]])/(64
*Sqrt[2])

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Rubi in Sympy [A]  time = 22.9406, size = 58, normalized size = 0.67 \[ - \frac{7 \operatorname{atan}{\left (x \right )}}{2} - \frac{133 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} - \frac{161}{8 x} - \frac{12096 x^{2} + 19440}{864 x^{3} \left (x^{4} + 3 x^{2} + 2\right )} + \frac{49}{4 x^{3}} \]

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)**3,x)

[Out]

-7*atan(x)/2 - 133*sqrt(2)*atan(sqrt(2)*x/2)/16 - 161/(8*x) - (12096*x**2 + 1944
0)/(864*x**3*(x**4 + 3*x**2 + 2)) + 49/(4*x**3)

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Mathematica [A]  time = 0.11641, size = 78, normalized size = 0.91 \[ \frac{1}{384} \left (-\frac{64}{x^3}-\frac{24 x \left (9 x^2+5\right )}{\left (x^4+3 x^2+2\right )^2}+\frac{6 x \left (571 x^2+951\right )}{x^4+3 x^2+2}+\frac{816}{x}-5424 \tan ^{-1}(x)+4833 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-64/x^3 + 816/x - (24*x*(5 + 9*x^2))/(2 + 3*x^2 + x^4)^2 + (6*x*(951 + 571*x^2)
)/(2 + 3*x^2 + x^4) - 5424*ArcTan[x] + 4833*Sqrt[2]*ArcTan[x/Sqrt[2]])/384

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Maple [A]  time = 0.024, size = 64, normalized size = 0.7 \[ -{\frac{1}{6\,{x}^{3}}}+{\frac{17}{8\,x}}+{\frac{1}{8\, \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{259\,{x}^{3}}{8}}+{\frac{285\,x}{4}} \right ) }+{\frac{1611\,\sqrt{2}}{128}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{39\,{x}^{3}}{8}}-{\frac{41\,x}{8}} \right ) }-{\frac{113\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/x^3+17/8/x+1/8*(259/8*x^3+285/4*x)/(x^2+2)^2+1611/128*arctan(1/2*2^(1/2)*x)
*2^(1/2)-(-39/8*x^3-41/8*x)/(x^2+1)^2-113/8*arctan(x)

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Maxima [A]  time = 0.788658, size = 97, normalized size = 1.13 \[ \frac{1611}{128} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{2121 \, x^{10} + 10408 \, x^{8} + 16989 \, x^{6} + 10126 \, x^{4} + 1248 \, x^{2} - 128}{192 \,{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )}} - \frac{113}{8} \, \arctan \left (x\right ) \]

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^4),x, algorithm="maxima")

[Out]

1611/128*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/192*(2121*x^10 + 10408*x^8 + 16989*x^
6 + 10126*x^4 + 1248*x^2 - 128)/(x^11 + 6*x^9 + 13*x^7 + 12*x^5 + 4*x^3) - 113/8
*arctan(x)

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Fricas [A]  time = 0.272864, size = 173, normalized size = 2.01 \[ -\frac{\sqrt{2}{\left (2712 \, \sqrt{2}{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )} \arctan \left (x\right ) - 4833 \,{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (2121 \, x^{10} + 10408 \, x^{8} + 16989 \, x^{6} + 10126 \, x^{4} + 1248 \, x^{2} - 128\right )}\right )}}{384 \,{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )}} \]

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)^3*x^4),x, algorithm="fricas")

[Out]

-1/384*sqrt(2)*(2712*sqrt(2)*(x^11 + 6*x^9 + 13*x^7 + 12*x^5 + 4*x^3)*arctan(x)
- 4833*(x^11 + 6*x^9 + 13*x^7 + 12*x^5 + 4*x^3)*arctan(1/2*sqrt(2)*x) - sqrt(2)*
(2121*x^10 + 10408*x^8 + 16989*x^6 + 10126*x^4 + 1248*x^2 - 128))/(x^11 + 6*x^9
+ 13*x^7 + 12*x^5 + 4*x^3)

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Sympy [A]  time = 0.952064, size = 76, normalized size = 0.88 \[ - \frac{113 \operatorname{atan}{\left (x \right )}}{8} + \frac{1611 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{128} + \frac{2121 x^{10} + 10408 x^{8} + 16989 x^{6} + 10126 x^{4} + 1248 x^{2} - 128}{192 x^{11} + 1152 x^{9} + 2496 x^{7} + 2304 x^{5} + 768 x^{3}} \]

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)**3,x)

[Out]

-113*atan(x)/8 + 1611*sqrt(2)*atan(sqrt(2)*x/2)/128 + (2121*x**10 + 10408*x**8 +
16989*x**6 + 10126*x**4 + 1248*x**2 - 128)/(192*x**11 + 1152*x**9 + 2496*x**7 +
2304*x**5 + 768*x**3)

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GIAC/XCAS [A]  time = 0.270553, size = 84, normalized size = 0.98 \[ \frac{1611}{128} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{571 \, x^{7} + 2664 \, x^{5} + 3959 \, x^{3} + 1882 \, x}{64 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{51 \, x^{2} - 4}{24 \, x^{3}} - \frac{113}{8} \, \arctan \left (x\right ) \]

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^4),x, algorithm="giac")

[Out]

1611/128*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/64*(571*x^7 + 2664*x^5 + 3959*x^3 + 1
882*x)/(x^4 + 3*x^2 + 2)^2 + 1/24*(51*x^2 - 4)/x^3 - 113/8*arctan(x)