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

Optimal. Leaf size=238 \[ -\frac{4}{27 x^3}+\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 x \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}+\frac{13}{27 x}-\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

-4/(27*x^3) + 13/(27*x) + (25*x*(7 + 5*x^2))/(216*(3 + 2*x^2 + x^4)) - (Sqrt[(60
73 + 56673*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3]
)]])/432 + (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)
/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] - Sq
rt[2*(-1 + Sqrt[3])]*x + x^2])/864 - (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3
] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/864

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Rubi [A]  time = 0.744989, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ -\frac{4}{27 x^3}+\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{864} \sqrt{\frac{1}{6} \left (56673 \sqrt{3}-6073\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 x \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}+\frac{13}{27 x}-\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{432} \sqrt{\frac{1}{6} \left (6073+56673 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-4/(27*x^3) + 13/(27*x) + (25*x*(7 + 5*x^2))/(216*(3 + 2*x^2 + x^4)) - (Sqrt[(60
73 + 56673*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3]
)]])/432 + (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)
/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] - Sq
rt[2*(-1 + Sqrt[3])]*x + x^2])/864 - (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3
] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/864

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Rubi in Sympy [A]  time = 37.298, size = 320, normalized size = 1.34 \[ \frac{\sqrt{6} \left (- 16704 \sqrt{3} + 21312\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{124416 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (- 16704 \sqrt{3} + 21312\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{124416 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 33408 \sqrt{3} + 42624\right )}{2} + 42624 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{62208 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 33408 \sqrt{3} + 42624\right )}{2} + 42624 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{62208 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{29}{9 x} + \frac{7}{9 x^{3}} \]

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

[Out]

sqrt(6)*(-16704*sqrt(3) + 21312)*log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(
3))/(124416*sqrt(-1 + sqrt(3))) - sqrt(6)*(-16704*sqrt(3) + 21312)*log(x**2 + sq
rt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(124416*sqrt(-1 + sqrt(3))) - sqrt(3)*(-sq
rt(2)*sqrt(-1 + sqrt(3))*(-33408*sqrt(3) + 42624)/2 + 42624*sqrt(2)*sqrt(-1 + sq
rt(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(62208*sqrt
(-1 + sqrt(3))*sqrt(1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(-33408
*sqrt(3) + 42624)/2 + 42624*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x + sqrt(-
2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(62208*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))
) - 29/(9*x) + 7/(9*x**3)

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Mathematica [C]  time = 0.555454, size = 131, normalized size = 0.55 \[ \frac{1}{864} \left (\frac{4 \left (229 x^6+351 x^4+248 x^2-96\right )}{x^3 \left (x^4+2 x^2+3\right )}+\frac{2 \left (229+46 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{2 \left (229-46 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((4*(-96 + 248*x^2 + 351*x^4 + 229*x^6))/(x^3*(3 + 2*x^2 + x^4)) + (2*(229 + (46
*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (2*(229 - (46*
I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/864

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Maple [B]  time = 0.038, size = 419, normalized size = 1.8 \[ -{\frac{4}{27\,{x}^{3}}}+{\frac{13}{27\,x}}+{\frac{1}{27\,{x}^{4}+54\,{x}^{2}+81} \left ({\frac{125\,{x}^{3}}{8}}+{\frac{175\,x}{8}} \right ) }-{\frac{275\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{5184}}-{\frac{23\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{864}}+{\frac{ \left ( -550+550\,\sqrt{3} \right ) \sqrt{3}}{2592\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-46+46\,\sqrt{3}}{432\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{137\,\sqrt{3}}{648\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{275\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{5184}}+{\frac{23\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{864}}+{\frac{ \left ( -550+550\,\sqrt{3} \right ) \sqrt{3}}{2592\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-46+46\,\sqrt{3}}{432\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{137\,\sqrt{3}}{648\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/27/x^3+13/27/x+1/27*(125/8*x^3+175/8*x)/(x^4+2*x^2+3)-275/5184*ln(x^2+3^(1/2)
+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-23/864*ln(x^2+3^(1/2)+x*(-
2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+275/2592/(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)+23/432/(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))+137/648/(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)+275/5184*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(
1/2)*3^(1/2)+23/864*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+
275/2592/(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)+23/432/(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))+137/648/(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. \[ \frac{229 \, x^{6} + 351 \, x^{4} + 248 \, x^{2} - 96}{216 \,{\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\right )}} + \frac{1}{216} \, \int \frac{229 \, x^{2} + 137}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

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)^2*x^4),x, algorithm="maxima")

[Out]

1/216*(229*x^6 + 351*x^4 + 248*x^2 - 96)/(x^7 + 2*x^5 + 3*x^3) + 1/216*integrate
((229*x^2 + 137)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.300086, size = 1112, normalized size = 4.67 \[ \text{result too large to display} \]

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)^2*x^4),x, algorithm="fricas")

[Out]

-1/146896416*sqrt(6297)*27^(3/4)*(1662648*4405801^(1/4)*sqrt(3)*(x^7 + 2*x^5 + 3
*x^3)*arctan(54*4405801^(1/4)*(46*sqrt(3) + 275)/(sqrt(6297)*27^(1/4)*sqrt(1/209
9)*(6073*sqrt(3)*sqrt(2) - 170019*sqrt(2))*sqrt(sqrt(3)*(2*4405801^(1/4)*sqrt(62
97)*27^(1/4)*(100885041419059841*sqrt(3)*x - 86942186720034978*x)*sqrt((6073*sqr
t(3) - 170019)/(344175129*sqrt(3) - 4836184058)) + 6297*sqrt(3)*(87886457041685*
sqrt(3)*x^2 - 828513704032353*x^2) + 1660263059974471335*sqrt(3) - 1565145238287
5180523)/(87886457041685*sqrt(3) - 828513704032353))*sqrt((6073*sqrt(3) - 170019
)/(344175129*sqrt(3) - 4836184058)) + 3*sqrt(6297)*27^(1/4)*(6073*sqrt(3)*sqrt(2
)*x - 170019*sqrt(2)*x)*sqrt((6073*sqrt(3) - 170019)/(344175129*sqrt(3) - 483618
4058)) + 27*4405801^(1/4)*(229*sqrt(3)*sqrt(2) - 137*sqrt(2)))) + 1662648*440580
1^(1/4)*sqrt(3)*(x^7 + 2*x^5 + 3*x^3)*arctan(54*4405801^(1/4)*(46*sqrt(3) + 275)
/(sqrt(6297)*27^(1/4)*sqrt(1/2099)*(6073*sqrt(3)*sqrt(2) - 170019*sqrt(2))*sqrt(
-sqrt(3)*(2*4405801^(1/4)*sqrt(6297)*27^(1/4)*(100885041419059841*sqrt(3)*x - 86
942186720034978*x)*sqrt((6073*sqrt(3) - 170019)/(344175129*sqrt(3) - 4836184058)
) - 6297*sqrt(3)*(87886457041685*sqrt(3)*x^2 - 828513704032353*x^2) - 1660263059
974471335*sqrt(3) + 15651452382875180523)/(87886457041685*sqrt(3) - 828513704032
353))*sqrt((6073*sqrt(3) - 170019)/(344175129*sqrt(3) - 4836184058)) + 3*sqrt(62
97)*27^(1/4)*(6073*sqrt(3)*sqrt(2)*x - 170019*sqrt(2)*x)*sqrt((6073*sqrt(3) - 17
0019)/(344175129*sqrt(3) - 4836184058)) - 27*4405801^(1/4)*(229*sqrt(3)*sqrt(2)
- 137*sqrt(2)))) - 4*sqrt(6297)*27^(1/4)*(6073*sqrt(3)*sqrt(2)*(229*x^6 + 351*x^
4 + 248*x^2 - 96) - 170019*sqrt(2)*(229*x^6 + 351*x^4 + 248*x^2 - 96))*sqrt((607
3*sqrt(3) - 170019)/(344175129*sqrt(3) - 4836184058)) - 3*4405801^(1/4)*(6073*sq
rt(3)*sqrt(2)*(x^7 + 2*x^5 + 3*x^3) - 170019*sqrt(2)*(x^7 + 2*x^5 + 3*x^3))*log(
6*4405801^(1/4)*sqrt(6297)*27^(1/4)*(100885041419059841*sqrt(3)*x - 869421867200
34978*x)*sqrt((6073*sqrt(3) - 170019)/(344175129*sqrt(3) - 4836184058)) + 18891*
sqrt(3)*(87886457041685*sqrt(3)*x^2 - 828513704032353*x^2) + 4980789179923414005
*sqrt(3) - 46954357148625541569) + 3*4405801^(1/4)*(6073*sqrt(3)*sqrt(2)*(x^7 +
2*x^5 + 3*x^3) - 170019*sqrt(2)*(x^7 + 2*x^5 + 3*x^3))*log(-6*4405801^(1/4)*sqrt
(6297)*27^(1/4)*(100885041419059841*sqrt(3)*x - 86942186720034978*x)*sqrt((6073*
sqrt(3) - 170019)/(344175129*sqrt(3) - 4836184058)) + 18891*sqrt(3)*(87886457041
685*sqrt(3)*x^2 - 828513704032353*x^2) + 4980789179923414005*sqrt(3) - 469543571
48625541569))/((6073*sqrt(3)*sqrt(2)*(x^7 + 2*x^5 + 3*x^3) - 170019*sqrt(2)*(x^7
+ 2*x^5 + 3*x^3))*sqrt((6073*sqrt(3) - 170019)/(344175129*sqrt(3) - 4836184058)
))

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Sympy [A]  time = 2.17939, size = 60, normalized size = 0.25 \[ \operatorname{RootSum}{\left (2293235712 t^{4} + 12437504 t^{2} + 4405801, \left ( t \mapsto t \log{\left (\frac{19707494400 t^{3}}{145412423} + \frac{357152768 t}{145412423} + x \right )} \right )\right )} + \frac{229 x^{6} + 351 x^{4} + 248 x^{2} - 96}{216 x^{7} + 432 x^{5} + 648 x^{3}} \]

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

[Out]

RootSum(2293235712*_t**4 + 12437504*_t**2 + 4405801, Lambda(_t, _t*log(197074944
00*_t**3/145412423 + 357152768*_t/145412423 + x))) + (229*x**6 + 351*x**4 + 248*
x**2 - 96)/(216*x**7 + 432*x**5 + 648*x**3)

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

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)^2*x^4),x, algorithm="giac")

[Out]

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