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

Optimal. Leaf size=75 \[ -\frac{\left (15 x^2+244\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (103 x^2+102\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+5 x+\frac{413}{8} \tan ^{-1}(x)-\frac{191 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}} \]

[Out]

5*x + (x*(102 + 103*x^2))/(4*(2 + 3*x^2 + x^4)^2) - (x*(244 + 15*x^2))/(8*(2 + 3
*x^2 + x^4)) + (413*ArcTan[x])/8 - (191*ArcTan[x/Sqrt[2]])/(2*Sqrt[2])

_______________________________________________________________________________________

Rubi [A]  time = 0.156005, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{\left (15 x^2+244\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (103 x^2+102\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+5 x+\frac{413}{8} \tan ^{-1}(x)-\frac{191 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

5*x + (x*(102 + 103*x^2))/(4*(2 + 3*x^2 + x^4)^2) - (x*(244 + 15*x^2))/(8*(2 + 3
*x^2 + x^4)) + (413*ArcTan[x])/8 - (191*ArcTan[x/Sqrt[2]])/(2*Sqrt[2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.4393, size = 70, normalized size = 0.93 \[ \frac{x \left (150174 x^{2} + 148716\right )}{5832 \left (x^{4} + 3 x^{2} + 2\right )^{2}} - \frac{x \left (3542940 x^{2} + 57631824\right )}{1889568 \left (x^{4} + 3 x^{2} + 2\right )} + 5 x + \frac{413 \operatorname{atan}{\left (x \right )}}{8} - \frac{191 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(150174*x**2 + 148716)/(5832*(x**4 + 3*x**2 + 2)**2) - x*(3542940*x**2 + 57631
824)/(1889568*(x**4 + 3*x**2 + 2)) + 5*x + 413*atan(x)/8 - 191*sqrt(2)*atan(sqrt
(2)*x/2)/4

_______________________________________________________________________________________

Mathematica [A]  time = 0.112113, size = 60, normalized size = 0.8 \[ \frac{1}{8} \left (\frac{x \left (40 x^8+225 x^6+231 x^4-76 x^2-124\right )}{\left (x^4+3 x^2+2\right )^2}+413 \tan ^{-1}(x)-382 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((x*(-124 - 76*x^2 + 231*x^4 + 225*x^6 + 40*x^8))/(2 + 3*x^2 + x^4)^2 + 413*ArcT
an[x] - 382*Sqrt[2]*ArcTan[x/Sqrt[2]])/8

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 56, normalized size = 0.8 \[ 5\,x-16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -1/32\,{x}^{3}+{\frac{25\,x}{16}} \right ) }-{\frac{191\,\sqrt{2}}{4}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{19\,{x}^{3}}{8}}-{\frac{21\,x}{8}} \right ) }+{\frac{413\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x-16*(-1/32*x^3+25/16*x)/(x^2+2)^2-191/4*arctan(1/2*2^(1/2)*x)*2^(1/2)+(-19/8*
x^3-21/8*x)/(x^2+1)^2+413/8*arctan(x)

_______________________________________________________________________________________

Maxima [A]  time = 0.785785, size = 85, normalized size = 1.13 \[ -\frac{191}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x - \frac{15 \, x^{7} + 289 \, x^{5} + 556 \, x^{3} + 284 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac{413}{8} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^6/(x^4 + 3*x^2 + 2)^3,x, algorithm="maxima")

[Out]

-191/4*sqrt(2)*arctan(1/2*sqrt(2)*x) + 5*x - 1/8*(15*x^7 + 289*x^5 + 556*x^3 + 2
84*x)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4) + 413/8*arctan(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.269459, size = 151, normalized size = 2.01 \[ \frac{\sqrt{2}{\left (413 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 764 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \sqrt{2}{\left (40 \, x^{9} + 225 \, x^{7} + 231 \, x^{5} - 76 \, x^{3} - 124 \, x\right )}\right )}}{16 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^6/(x^4 + 3*x^2 + 2)^3,x, algorithm="fricas")

[Out]

1/16*sqrt(2)*(413*sqrt(2)*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(x) - 764*(x
^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(1/2*sqrt(2)*x) + sqrt(2)*(40*x^9 + 225*
x^7 + 231*x^5 - 76*x^3 - 124*x))/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)

_______________________________________________________________________________________

Sympy [A]  time = 0.79104, size = 68, normalized size = 0.91 \[ 5 x - \frac{15 x^{7} + 289 x^{5} + 556 x^{3} + 284 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac{413 \operatorname{atan}{\left (x \right )}}{8} - \frac{191 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x - (15*x**7 + 289*x**5 + 556*x**3 + 284*x)/(8*x**8 + 48*x**6 + 104*x**4 + 96*
x**2 + 32) + 413*atan(x)/8 - 191*sqrt(2)*atan(sqrt(2)*x/2)/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270034, size = 72, normalized size = 0.96 \[ -\frac{191}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x - \frac{15 \, x^{7} + 289 \, x^{5} + 556 \, x^{3} + 284 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{413}{8} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^6/(x^4 + 3*x^2 + 2)^3,x, algorithm="giac")

[Out]

-191/4*sqrt(2)*arctan(1/2*sqrt(2)*x) + 5*x - 1/8*(15*x^7 + 289*x^5 + 556*x^3 + 2
84*x)/(x^4 + 3*x^2 + 2)^2 + 413/8*arctan(x)