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

Optimal. Leaf size=245 \[ -\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}+\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}+\frac{25 x \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{13}{27 x}+\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296}-\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296} \]

[Out]

-4/(45*x^5) + 13/(81*x^3) - 13/(27*x) + (25*x*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^4
)) + (Sqrt[(-1139381 + 688419*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/
Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(-1139381 + 688419*Sqrt[3])/6]*ArcTan[(Sqrt
[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(1139381 + 688419
*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2592 + (Sqrt[(113938
1 + 688419*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2592

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Rubi [A]  time = 0.734498, antiderivative size = 245, 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 \[ -\frac{4}{45 x^5}+\frac{13}{81 x^3}-\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}+\frac{\sqrt{\frac{1}{6} \left (1139381+688419 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{2592}+\frac{25 x \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{13}{27 x}+\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296}-\frac{\sqrt{\frac{1}{6} \left (688419 \sqrt{3}-1139381\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{1296} \]

Antiderivative was successfully verified.

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

[Out]

-4/(45*x^5) + 13/(81*x^3) - 13/(27*x) + (25*x*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^4
)) + (Sqrt[(-1139381 + 688419*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/
Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(-1139381 + 688419*Sqrt[3])/6]*ArcTan[(Sqrt
[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(1139381 + 688419
*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2592 + (Sqrt[(113938
1 + 688419*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2592

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Rubi in Sympy [A]  time = 41.3606, size = 326, normalized size = 1.33 \[ \frac{\sqrt{6} \left (37440 + 106560 \sqrt{3}\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{1866240 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (37440 + 106560 \sqrt{3}\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{1866240 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (74880 + 213120 \sqrt{3}\right )}{2} + 74880 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{933120 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (74880 + 213120 \sqrt{3}\right )}{2} + 74880 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{933120 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} + \frac{37}{27 x} - \frac{29}{27 x^{3}} + \frac{7}{15 x^{5}} \]

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

[Out]

sqrt(6)*(37440 + 106560*sqrt(3))*log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(
3))/(1866240*sqrt(-1 + sqrt(3))) - sqrt(6)*(37440 + 106560*sqrt(3))*log(x**2 + s
qrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(1866240*sqrt(-1 + sqrt(3))) - sqrt(3)*(-
sqrt(2)*sqrt(-1 + sqrt(3))*(74880 + 213120*sqrt(3))/2 + 74880*sqrt(2)*sqrt(-1 +
sqrt(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(933120*s
qrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(748
80 + 213120*sqrt(3))/2 + 74880*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x + sqr
t(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(933120*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt
(3))) + 37/(27*x) - 29/(27*x**3) + 7/(15*x**5)

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Mathematica [C]  time = 0.608538, size = 140, normalized size = 0.57 \[ \frac{-\frac{4 \left (2435 x^8+2475 x^6+3928 x^4-984 x^2+864\right )}{x^5 \left (x^4+2 x^2+3\right )}-\frac{10 i \left (475 \sqrt{2}-487 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{10 i \left (475 \sqrt{2}+487 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}}{12960} \]

Antiderivative was successfully verified.

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

[Out]

((-4*(864 - 984*x^2 + 3928*x^4 + 2475*x^6 + 2435*x^8))/(x^5*(3 + 2*x^2 + x^4)) -
((10*I)*(-487*I + 475*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2
]] + ((10*I)*(487*I + 475*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqr
t[2]])/12960

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Maple [B]  time = 0.037, size = 424, normalized size = 1.7 \[ -{\frac{4}{45\,{x}^{5}}}+{\frac{13}{81\,{x}^{3}}}-{\frac{13}{27\,x}}-{\frac{1}{27\,{x}^{4}+54\,{x}^{2}+81} \left ({\frac{175\,{x}^{3}}{24}}-{\frac{25\,x}{24}} \right ) }+{\frac{481\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{7776}}+{\frac{475\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{5184}}-{\frac{ \left ( -962+962\,\sqrt{3} \right ) \sqrt{3}}{3888\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-950+950\,\sqrt{3}}{2592\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{463\,\sqrt{3}}{1944\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{481\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{7776}}-{\frac{475\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{5184}}-{\frac{ \left ( -962+962\,\sqrt{3} \right ) \sqrt{3}}{3888\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-950+950\,\sqrt{3}}{2592\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{463\,\sqrt{3}}{1944\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/45/x^5+13/81/x^3-13/27/x-1/27*(175/24*x^3-25/24*x)/(x^4+2*x^2+3)+481/7776*ln(
x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+475/5184*ln(x^2
+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-481/3888/(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)
-475/2592/(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))+463/1944/(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)-481/7776*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*
(-2+2*3^(1/2))^(1/2)*3^(1/2)-475/5184*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2
+2*3^(1/2))^(1/2)-481/3888/(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)-475/2592/(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))+463/1944/(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. \[ -\frac{2435 \, x^{8} + 2475 \, x^{6} + 3928 \, x^{4} - 984 \, x^{2} + 864}{3240 \,{\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )}} - \frac{1}{648} \, \int \frac{487 \, x^{2} - 463}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

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^6),x, algorithm="maxima")

[Out]

-1/3240*(2435*x^8 + 2475*x^6 + 3928*x^4 - 984*x^2 + 864)/(x^9 + 2*x^7 + 3*x^5) -
1/648*integrate((487*x^2 - 463)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.290717, size = 1126, normalized size = 4.6 \[ \text{result too large to display} \]

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^6),x, algorithm="fricas")

[Out]

1/2973970080*sqrt(8499)*27^(3/4)*(29828280*8025889^(1/4)*sqrt(3)*(x^9 + 2*x^7 +
3*x^5)*arctan(54*8025889^(1/4)*(475*sqrt(3) + 962)/(sqrt(8499)*27^(1/4)*sqrt(1/2
833)*(1139381*sqrt(3)*sqrt(2) + 2065257*sqrt(2))*sqrt(sqrt(3)*(2*8025889^(1/4)*s
qrt(8499)*27^(1/4)*(2070494440200369267506*sqrt(3)*x + 3586177043470224593529*x)
*sqrt((1139381*sqrt(3) + 2065257)/(784371528639*sqrt(3) + 1359975610922)) + 8499
*sqrt(3)*(3169459161670314505*sqrt(3)*x^2 + 5489793200302634331*x^2) + 808117002
45108008933985*sqrt(3) + 139973257228116267537507)/(3169459161670314505*sqrt(3)
+ 5489793200302634331))*sqrt((1139381*sqrt(3) + 2065257)/(784371528639*sqrt(3) +
1359975610922)) + 3*sqrt(8499)*27^(1/4)*(1139381*sqrt(3)*sqrt(2)*x + 2065257*sq
rt(2)*x)*sqrt((1139381*sqrt(3) + 2065257)/(784371528639*sqrt(3) + 1359975610922)
) + 27*8025889^(1/4)*(487*sqrt(3)*sqrt(2) + 463*sqrt(2)))) + 29828280*8025889^(1
/4)*sqrt(3)*(x^9 + 2*x^7 + 3*x^5)*arctan(54*8025889^(1/4)*(475*sqrt(3) + 962)/(s
qrt(8499)*27^(1/4)*sqrt(1/2833)*(1139381*sqrt(3)*sqrt(2) + 2065257*sqrt(2))*sqrt
(-sqrt(3)*(2*8025889^(1/4)*sqrt(8499)*27^(1/4)*(2070494440200369267506*sqrt(3)*x
+ 3586177043470224593529*x)*sqrt((1139381*sqrt(3) + 2065257)/(784371528639*sqrt
(3) + 1359975610922)) - 8499*sqrt(3)*(3169459161670314505*sqrt(3)*x^2 + 54897932
00302634331*x^2) - 80811700245108008933985*sqrt(3) - 139973257228116267537507)/(
3169459161670314505*sqrt(3) + 5489793200302634331))*sqrt((1139381*sqrt(3) + 2065
257)/(784371528639*sqrt(3) + 1359975610922)) + 3*sqrt(8499)*27^(1/4)*(1139381*sq
rt(3)*sqrt(2)*x + 2065257*sqrt(2)*x)*sqrt((1139381*sqrt(3) + 2065257)/(784371528
639*sqrt(3) + 1359975610922)) - 27*8025889^(1/4)*(487*sqrt(3)*sqrt(2) + 463*sqrt
(2)))) - 4*sqrt(8499)*27^(1/4)*(1139381*sqrt(3)*sqrt(2)*(2435*x^8 + 2475*x^6 + 3
928*x^4 - 984*x^2 + 864) + 2065257*sqrt(2)*(2435*x^8 + 2475*x^6 + 3928*x^4 - 984
*x^2 + 864))*sqrt((1139381*sqrt(3) + 2065257)/(784371528639*sqrt(3) + 1359975610
922)) + 15*8025889^(1/4)*(1139381*sqrt(3)*sqrt(2)*(x^9 + 2*x^7 + 3*x^5) + 206525
7*sqrt(2)*(x^9 + 2*x^7 + 3*x^5))*log(18*8025889^(1/4)*sqrt(8499)*27^(1/4)*(20704
94440200369267506*sqrt(3)*x + 3586177043470224593529*x)*sqrt((1139381*sqrt(3) +
2065257)/(784371528639*sqrt(3) + 1359975610922)) + 76491*sqrt(3)*(31694591616703
14505*sqrt(3)*x^2 + 5489793200302634331*x^2) + 727305302205972080405865*sqrt(3)
+ 1259759315053046407837563) - 15*8025889^(1/4)*(1139381*sqrt(3)*sqrt(2)*(x^9 +
2*x^7 + 3*x^5) + 2065257*sqrt(2)*(x^9 + 2*x^7 + 3*x^5))*log(-18*8025889^(1/4)*sq
rt(8499)*27^(1/4)*(2070494440200369267506*sqrt(3)*x + 3586177043470224593529*x)*
sqrt((1139381*sqrt(3) + 2065257)/(784371528639*sqrt(3) + 1359975610922)) + 76491
*sqrt(3)*(3169459161670314505*sqrt(3)*x^2 + 5489793200302634331*x^2) + 727305302
205972080405865*sqrt(3) + 1259759315053046407837563))/((1139381*sqrt(3)*sqrt(2)*
(x^9 + 2*x^7 + 3*x^5) + 2065257*sqrt(2)*(x^9 + 2*x^7 + 3*x^5))*sqrt((1139381*sqr
t(3) + 2065257)/(784371528639*sqrt(3) + 1359975610922)))

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Sympy [A]  time = 2.17976, size = 65, normalized size = 0.27 \[ \operatorname{RootSum}{\left (20639121408 t^{4} - 2333452288 t^{2} + 72233001, \left ( t \mapsto t \log{\left (- \frac{206821195776 t^{3}}{704195977} + \frac{38757503008 t}{2112587931} + x \right )} \right )\right )} - \frac{2435 x^{8} + 2475 x^{6} + 3928 x^{4} - 984 x^{2} + 864}{3240 x^{9} + 6480 x^{7} + 9720 x^{5}} \]

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

[Out]

RootSum(20639121408*_t**4 - 2333452288*_t**2 + 72233001, Lambda(_t, _t*log(-2068
21195776*_t**3/704195977 + 38757503008*_t/2112587931 + x))) - (2435*x**8 + 2475*
x**6 + 3928*x**4 - 984*x**2 + 864)/(3240*x**9 + 6480*x**7 + 9720*x**5)

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

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^6),x, algorithm="giac")

[Out]

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