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

Optimal. Leaf size=93 \[ -\frac{1}{10 x^5}+\frac{17}{24 x^3}-\frac{x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (999 x^2+1771\right )}{128 \left (x^4+3 x^2+2\right )}-\frac{93}{16 x}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}} \]

[Out]

-1/(10*x^5) + 17/(24*x^3) - 93/(16*x) - (x*(3 - 5*x^2))/(32*(2 + 3*x^2 + x^4)^2)
- (x*(1771 + 999*x^2))/(128*(2 + 3*x^2 + x^4)) + (29*ArcTan[x])/8 - (2207*ArcTa
n[x/Sqrt[2]])/(128*Sqrt[2])

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Rubi [A]  time = 0.208624, antiderivative size = 93, 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 \[ -\frac{1}{10 x^5}+\frac{17}{24 x^3}-\frac{x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (999 x^2+1771\right )}{128 \left (x^4+3 x^2+2\right )}-\frac{93}{16 x}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(10*x^5) + 17/(24*x^3) - 93/(16*x) - (x*(3 - 5*x^2))/(32*(2 + 3*x^2 + x^4)^2)
- (x*(1771 + 999*x^2))/(128*(2 + 3*x^2 + x^4)) + (29*ArcTan[x])/8 - (2207*ArcTa
n[x/Sqrt[2]])/(128*Sqrt[2])

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Rubi in Sympy [A]  time = 26.4956, size = 65, normalized size = 0.7 \[ \frac{41 \operatorname{atan}{\left (x \right )}}{2} + \frac{155 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{32} + \frac{483}{16 x} - \frac{319}{24 x^{3}} - \frac{12096 x^{2} + 19440}{864 x^{5} \left (x^{4} + 3 x^{2} + 2\right )} + \frac{237}{20 x^{5}} \]

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)**3,x)

[Out]

41*atan(x)/2 + 155*sqrt(2)*atan(sqrt(2)*x/2)/32 + 483/(16*x) - 319/(24*x**3) - (
12096*x**2 + 19440)/(864*x**5*(x**4 + 3*x**2 + 2)) + 237/(20*x**5)

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Mathematica [A]  time = 0.146521, size = 73, normalized size = 0.78 \[ \frac{-\frac{2 \left (26145 x^{12}+137120 x^{10}+246477 x^8+170702 x^6+30816 x^4-3136 x^2+768\right )}{x^5 \left (x^4+3 x^2+2\right )^2}+13920 \tan ^{-1}(x)-33105 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{3840} \]

Antiderivative was successfully verified.

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

[Out]

((-2*(768 - 3136*x^2 + 30816*x^4 + 170702*x^6 + 246477*x^8 + 137120*x^10 + 26145
*x^12))/(x^5*(2 + 3*x^2 + x^4)^2) + 13920*ArcTan[x] - 33105*Sqrt[2]*ArcTan[x/Sqr
t[2]])/3840

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Maple [A]  time = 0.024, size = 68, normalized size = 0.7 \[ -{\frac{1}{10\,{x}^{5}}}+{\frac{17}{24\,{x}^{3}}}-{\frac{93}{16\,x}}-{\frac{1}{16\, \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{311\,{x}^{3}}{8}}+{\frac{337\,x}{4}} \right ) }-{\frac{2207\,\sqrt{2}}{256}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{43\,{x}^{3}}{8}}-{\frac{45\,x}{8}} \right ) }+{\frac{29\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10/x^5+17/24/x^3-93/16/x-1/16*(311/8*x^3+337/4*x)/(x^2+2)^2-2207/256*arctan(1
/2*2^(1/2)*x)*2^(1/2)+(-43/8*x^3-45/8*x)/(x^2+1)^2+29/8*arctan(x)

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Maxima [A]  time = 0.775917, size = 104, normalized size = 1.12 \[ -\frac{2207}{256} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{26145 \, x^{12} + 137120 \, x^{10} + 246477 \, x^{8} + 170702 \, x^{6} + 30816 \, x^{4} - 3136 \, x^{2} + 768}{1920 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )}} + \frac{29}{8} \, \arctan \left (x\right ) \]

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^6),x, algorithm="maxima")

[Out]

-2207/256*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/1920*(26145*x^12 + 137120*x^10 + 246
477*x^8 + 170702*x^6 + 30816*x^4 - 3136*x^2 + 768)/(x^13 + 6*x^11 + 13*x^9 + 12*
x^7 + 4*x^5) + 29/8*arctan(x)

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Fricas [A]  time = 0.2784, size = 180, normalized size = 1.94 \[ \frac{\sqrt{2}{\left (6960 \, \sqrt{2}{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (x\right ) - 33105 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (26145 \, x^{12} + 137120 \, x^{10} + 246477 \, x^{8} + 170702 \, x^{6} + 30816 \, x^{4} - 3136 \, x^{2} + 768\right )}\right )}}{3840 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )}} \]

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)^3*x^6),x, algorithm="fricas")

[Out]

1/3840*sqrt(2)*(6960*sqrt(2)*(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)*arctan(x)
- 33105*(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)*arctan(1/2*sqrt(2)*x) - sqrt(
2)*(26145*x^12 + 137120*x^10 + 246477*x^8 + 170702*x^6 + 30816*x^4 - 3136*x^2 +
768))/(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)

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Sympy [A]  time = 1.05169, size = 82, normalized size = 0.88 \[ \frac{29 \operatorname{atan}{\left (x \right )}}{8} - \frac{2207 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{256} - \frac{26145 x^{12} + 137120 x^{10} + 246477 x^{8} + 170702 x^{6} + 30816 x^{4} - 3136 x^{2} + 768}{1920 x^{13} + 11520 x^{11} + 24960 x^{9} + 23040 x^{7} + 7680 x^{5}} \]

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)**3,x)

[Out]

29*atan(x)/8 - 2207*sqrt(2)*atan(sqrt(2)*x/2)/256 - (26145*x**12 + 137120*x**10
+ 246477*x**8 + 170702*x**6 + 30816*x**4 - 3136*x**2 + 768)/(1920*x**13 + 11520*
x**11 + 24960*x**9 + 23040*x**7 + 7680*x**5)

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GIAC/XCAS [A]  time = 0.270573, size = 90, normalized size = 0.97 \[ -\frac{2207}{256} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{999 \, x^{7} + 4768 \, x^{5} + 7291 \, x^{3} + 3554 \, x}{128 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{1395 \, x^{4} - 170 \, x^{2} + 24}{240 \, x^{5}} + \frac{29}{8} \, \arctan \left (x\right ) \]

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^6),x, algorithm="giac")

[Out]

-2207/256*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/128*(999*x^7 + 4768*x^5 + 7291*x^3 +
3554*x)/(x^4 + 3*x^2 + 2)^2 - 1/240*(1395*x^4 - 170*x^2 + 24)/x^5 + 29/8*arctan
(x)