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

Optimal. Leaf size=253 \[ -\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}-\frac{25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac{4}{27 x}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304} \]

[Out]

-4/(27*x) - (25*x*(5 + x^2))/(144*(3 + 2*x^2 + x^4)^2) - (x*(325 + 242*x^2))/(17
28*(3 + 2*x^2 + x^4)) + (Sqrt[(59711 + 55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sq
rt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(59711 + 55161*Sqrt[3])/3]*A
rcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(-5971
1 + 55161*Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608 + (Sqr
t[(-59711 + 55161*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/460
8

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Rubi [A]  time = 0.832625, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ -\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}-\frac{25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac{4}{27 x}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304} \]

Antiderivative was successfully verified.

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

[Out]

-4/(27*x) - (25*x*(5 + x^2))/(144*(3 + 2*x^2 + x^4)^2) - (x*(325 + 242*x^2))/(17
28*(3 + 2*x^2 + x^4)) + (Sqrt[(59711 + 55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sq
rt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(59711 + 55161*Sqrt[3])/3]*A
rcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(-5971
1 + 55161*Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608 + (Sqr
t[(-59711 + 55161*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/460
8

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Rubi in Sympy [A]  time = 38.1595, size = 333, normalized size = 1.32 \[ - \frac{\sqrt{6} \left (- 36864 \sqrt{3} + 225792\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{1327104 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{6} \left (- 36864 \sqrt{3} + 225792\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{1327104 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 73728 \sqrt{3} + 451584\right )}{2} + 451584 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{663552 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 73728 \sqrt{3} + 451584\right )}{2} + 451584 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{663552 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} + \frac{33792 x^{2} + 12288}{36864 x \left (x^{4} + 2 x^{2} + 3\right )} + \frac{2}{3 x} \]

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+2*x**2+3)**3,x)

[Out]

-sqrt(6)*(-36864*sqrt(3) + 225792)*log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqr
t(3))/(1327104*sqrt(-1 + sqrt(3))) + sqrt(6)*(-36864*sqrt(3) + 225792)*log(x**2
+ sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(1327104*sqrt(-1 + sqrt(3))) + sqrt(3)
*(-sqrt(2)*sqrt(-1 + sqrt(3))*(-73728*sqrt(3) + 451584)/2 + 451584*sqrt(2)*sqrt(
-1 + sqrt(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(663
552*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))) + sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))
*(-73728*sqrt(3) + 451584)/2 + 451584*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(
x + sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(663552*sqrt(-1 + sqrt(3))*sqrt(1
+ sqrt(3))) + (33792*x**2 + 12288)/(36864*x*(x**4 + 2*x**2 + 3)) + 2/(3*x)

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Mathematica [C]  time = 0.785148, size = 140, normalized size = 0.55 \[ \frac{-\frac{12 \left (166 x^8+611 x^6+1412 x^4+1849 x^2+768\right )}{x \left (x^4+2 x^2+3\right )^2}+\frac{3 i \left (7 \sqrt{2}+332 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}-\frac{3 i \left (7 \sqrt{2}-332 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}}{6912} \]

Antiderivative was successfully verified.

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

[Out]

((-12*(768 + 1849*x^2 + 1412*x^4 + 611*x^6 + 166*x^8))/(x*(3 + 2*x^2 + x^4)^2) +
((3*I)*(332*I + 7*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] -
((3*I)*(-332*I + 7*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])
/6912

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Maple [B]  time = 0.037, size = 424, normalized size = 1.7 \[ -{\frac{4}{27\,x}}-{\frac{1}{27\, \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{121\,{x}^{7}}{32}}+{\frac{809\,{x}^{5}}{64}}+{\frac{419\,{x}^{3}}{16}}+{\frac{2475\,x}{64}} \right ) }+{\frac{325\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{27648}}-{\frac{7\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{9216}}-{\frac{ \left ( -650+650\,\sqrt{3} \right ) \sqrt{3}}{13824\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-14+14\,\sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{173\,\sqrt{3}}{1728\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{325\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{27648}}+{\frac{7\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{9216}}-{\frac{ \left ( -650+650\,\sqrt{3} \right ) \sqrt{3}}{13824\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-14+14\,\sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{173\,\sqrt{3}}{1728\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

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+2*x^2+3)^3,x)

[Out]

-4/27/x-1/27*(121/32*x^7+809/64*x^5+419/16*x^3+2475/64*x)/(x^4+2*x^2+3)^2+325/27
648*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-7/9216*l
n(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-325/13824/(2+2*3^(1/2
))^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3
^(1/2)+7/4608/(2+2*3^(1/2))^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2)
)^(1/2))*(-2+2*3^(1/2))-173/1728/(2+2*3^(1/2))^(1/2)*arctan((2*x+(-2+2*3^(1/2))^
(1/2))/(2+2*3^(1/2))^(1/2))*3^(1/2)-325/27648*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1
/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+7/9216*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*
(-2+2*3^(1/2))^(1/2)-325/13824/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1
/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)+7/4608/(2+2*3^(1/2))^(1/2)*arct
an((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))-173/1728/(2+2*
3^(1/2))^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{166 \, x^{8} + 611 \, x^{6} + 1412 \, x^{4} + 1849 \, x^{2} + 768}{576 \,{\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )}} - \frac{1}{576} \, \int \frac{166 \, x^{2} + 173}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/576*(166*x^8 + 611*x^6 + 1412*x^4 + 1849*x^2 + 768)/(x^9 + 4*x^7 + 10*x^5 + 1
2*x^3 + 9*x) - 1/576*integrate((166*x^2 + 173)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.296996, size = 1189, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6276096*sqrt(681)*4^(3/4)*(4*sqrt(681)*4^(1/4)*(55161*sqrt(3)*sqrt(2)*(166*x^
8 + 611*x^6 + 1412*x^4 + 1849*x^2 + 768) - 59711*sqrt(2)*(166*x^8 + 611*x^6 + 14
12*x^4 + 1849*x^2 + 768))*sqrt((59711*sqrt(3) - 165483)/(3293718471*sqrt(3) - 63
46805642)) - 421912*154587^(1/4)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x)*arctan(2*
154587^(1/4)*(325*sqrt(3) - 21)/(sqrt(681)*4^(1/4)*sqrt(1/681)*(55161*sqrt(3)*sq
rt(2) - 59711*sqrt(2))*sqrt((629263425815075355*sqrt(3)*x^2 + 154587^(1/4)*sqrt(
681)*4^(1/4)*(432147084979531229*sqrt(3)*x - 743938470505411707*x)*sqrt((59711*s
qrt(3) - 165483)/(3293718471*sqrt(3) - 6346805642)) - 1117045736175016449*x^2 +
681*sqrt(3)*(924028525425955*sqrt(3) - 1640302108920729))/(924028525425955*sqrt(
3) - 1640302108920729))*sqrt((59711*sqrt(3) - 165483)/(3293718471*sqrt(3) - 6346
805642)) + sqrt(681)*4^(1/4)*(55161*sqrt(3)*sqrt(2)*x - 59711*sqrt(2)*x)*sqrt((5
9711*sqrt(3) - 165483)/(3293718471*sqrt(3) - 6346805642)) + 2*154587^(1/4)*(173*
sqrt(3)*sqrt(2) - 498*sqrt(2)))) - 421912*154587^(1/4)*(x^9 + 4*x^7 + 10*x^5 + 1
2*x^3 + 9*x)*arctan(2*154587^(1/4)*(325*sqrt(3) - 21)/(sqrt(681)*4^(1/4)*sqrt(1/
681)*(55161*sqrt(3)*sqrt(2) - 59711*sqrt(2))*sqrt((629263425815075355*sqrt(3)*x^
2 - 154587^(1/4)*sqrt(681)*4^(1/4)*(432147084979531229*sqrt(3)*x - 7439384705054
11707*x)*sqrt((59711*sqrt(3) - 165483)/(3293718471*sqrt(3) - 6346805642)) - 1117
045736175016449*x^2 + 681*sqrt(3)*(924028525425955*sqrt(3) - 1640302108920729))/
(924028525425955*sqrt(3) - 1640302108920729))*sqrt((59711*sqrt(3) - 165483)/(329
3718471*sqrt(3) - 6346805642)) + sqrt(681)*4^(1/4)*(55161*sqrt(3)*sqrt(2)*x - 59
711*sqrt(2)*x)*sqrt((59711*sqrt(3) - 165483)/(3293718471*sqrt(3) - 6346805642))
- 2*154587^(1/4)*(173*sqrt(3)*sqrt(2) - 498*sqrt(2)))) + 154587^(1/4)*(55161*sqr
t(3)*sqrt(2)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) - 59711*sqrt(2)*(x^9 + 4*x^7
+ 10*x^5 + 12*x^3 + 9*x))*log(11326741664671356390*sqrt(3)*x^2 + 18*154587^(1/4)
*sqrt(681)*4^(1/4)*(432147084979531229*sqrt(3)*x - 743938470505411707*x)*sqrt((5
9711*sqrt(3) - 165483)/(3293718471*sqrt(3) - 6346805642)) - 20106823251150296082
*x^2 + 12258*sqrt(3)*(924028525425955*sqrt(3) - 1640302108920729)) - 154587^(1/4
)*(55161*sqrt(3)*sqrt(2)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) - 59711*sqrt(2)*(
x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x))*log(11326741664671356390*sqrt(3)*x^2 - 18*
154587^(1/4)*sqrt(681)*4^(1/4)*(432147084979531229*sqrt(3)*x - 74393847050541170
7*x)*sqrt((59711*sqrt(3) - 165483)/(3293718471*sqrt(3) - 6346805642)) - 20106823
251150296082*x^2 + 12258*sqrt(3)*(924028525425955*sqrt(3) - 1640302108920729)))/
((55161*sqrt(3)*sqrt(2)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) - 59711*sqrt(2)*(x
^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x))*sqrt((59711*sqrt(3) - 165483)/(3293718471*s
qrt(3) - 6346805642)))

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Sympy [A]  time = 2.26729, size = 73, normalized size = 0.29 \[ - \frac{166 x^{8} + 611 x^{6} + 1412 x^{4} + 1849 x^{2} + 768}{576 x^{9} + 2304 x^{7} + 5760 x^{5} + 6912 x^{3} + 5184 x} + \operatorname{RootSum}{\left (4174708211712 t^{4} + 15652880384 t^{2} + 37564641, \left ( t \mapsto t \log{\left (- \frac{98146713600 t^{3}}{11971753} - \frac{9639364864 t}{323237331} + x \right )} \right )\right )} \]

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+2*x**2+3)**3,x)

[Out]

-(166*x**8 + 611*x**6 + 1412*x**4 + 1849*x**2 + 768)/(576*x**9 + 2304*x**7 + 576
0*x**5 + 6912*x**3 + 5184*x) + RootSum(4174708211712*_t**4 + 15652880384*_t**2 +
37564641, Lambda(_t, _t*log(-98146713600*_t**3/11971753 - 9639364864*_t/3232373
31 + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{5 \, x^{6} + 3 \, x^{4} + x^{2} + 4}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((5*x^6 + 3*x^4 + x^2 + 4)/((x^4 + 2*x^2 + 3)^3*x^2), x)