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

Optimal. Leaf size=235 \[ -\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{7 \left (58 x^2+11\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (3-x^2\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+5 x+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

5*x + (25*x*(3 - x^2))/(16*(3 + 2*x^2 + x^4)^2) + (7*x*(11 + 58*x^2))/(64*(3 + 2
*x^2 + x^4)) + (Sqrt[827621 + 1176531*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] -
2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[827621 + 1176531*Sqrt[3]]*ArcTan[(Sqrt[
2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[-827621 + 1176531*S
qrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (Sqrt[-827621 + 117
6531*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi [A]  time = 0.757798, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258 \[ -\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{1}{512} \sqrt{1176531 \sqrt{3}-827621} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{7 \left (58 x^2+11\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (3-x^2\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+5 x+\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{1}{256} \sqrt{827621+1176531 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

5*x + (25*x*(3 - x^2))/(16*(3 + 2*x^2 + x^4)^2) + (7*x*(11 + 58*x^2))/(64*(3 + 2
*x^2 + x^4)) + (Sqrt[827621 + 1176531*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] -
2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[827621 + 1176531*Sqrt[3]]*ArcTan[(Sqrt[
2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 - (Sqrt[-827621 + 1176531*S
qrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 + (Sqrt[-827621 + 117
6531*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi in Sympy [A]  time = 43.1032, size = 354, normalized size = 1.51 \[ \frac{x \left (- 9600 x^{2} + 28800\right )}{6144 \left (x^{4} + 2 x^{2} + 3\right )^{2}} + \frac{x \left (29933568 x^{2} + 5677056\right )}{4718592 \left (x^{4} + 2 x^{2} + 3\right )} + 5 x + \frac{\sqrt{6} \left (- 48734208 \sqrt{3} + 41914368\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{56623104 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (- 48734208 \sqrt{3} + 41914368\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{56623104 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 97468416 \sqrt{3} + 83828736\right )}{2} + 83828736 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{28311552 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 97468416 \sqrt{3} + 83828736\right )}{2} + 83828736 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{28311552 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

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

[Out]

x*(-9600*x**2 + 28800)/(6144*(x**4 + 2*x**2 + 3)**2) + x*(29933568*x**2 + 567705
6)/(4718592*(x**4 + 2*x**2 + 3)) + 5*x + sqrt(6)*(-48734208*sqrt(3) + 41914368)*
log(x**2 - sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(56623104*sqrt(-1 + sqrt(3)))
- sqrt(6)*(-48734208*sqrt(3) + 41914368)*log(x**2 + sqrt(2)*x*sqrt(-1 + sqrt(3)
) + sqrt(3))/(56623104*sqrt(-1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3)
)*(-97468416*sqrt(3) + 83828736)/2 + 83828736*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(s
qrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(28311552*sqrt(-1 + sqrt(
3))*sqrt(1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(-97468416*sqrt(3)
+ 83828736)/2 + 83828736*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x + sqrt(-2
+ 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(28311552*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3)
))

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Mathematica [C]  time = 0.699309, size = 138, normalized size = 0.59 \[ \frac{1}{256} \left (\frac{4 x \left (320 x^8+1686 x^6+4089 x^4+5112 x^2+3411\right )}{\left (x^4+2 x^2+3\right )^2}-\frac{i \left (185 \sqrt{2}-2644 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{i \left (185 \sqrt{2}+2644 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((4*x*(3411 + 5112*x^2 + 4089*x^4 + 1686*x^6 + 320*x^8))/(3 + 2*x^2 + x^4)^2 - (
I*(-2644*I + 185*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (
I*(2644*I + 185*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/256

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Maple [B]  time = 0.033, size = 422, normalized size = 1.8 \[ 5\,x-{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{203\,{x}^{7}}{32}}-{\frac{889\,{x}^{5}}{64}}-{\frac{159\,{x}^{3}}{8}}-{\frac{531\,x}{64}} \right ) }+{\frac{943\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{185\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}-{\frac{ \left ( -1886+1886\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-370+370\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{379\,\sqrt{3}}{64\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{943\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{185\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}-{\frac{ \left ( -1886+1886\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-370+370\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{379\,\sqrt{3}}{64\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x-(-203/32*x^7-889/64*x^5-159/8*x^3-531/64*x)/(x^4+2*x^2+3)^2+943/1024*ln(x^2+
3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)+185/1024*ln(x^2+3^(
1/2)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)-943/512/(2+2*3^(1/2))^(1/2)*ar
ctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))*3^(1/2)-185/
512/(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))-379/64/(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)-943/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^
(1/2))^(1/2)*3^(1/2)-185/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/
2))^(1/2)-943/512/(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)-185/512/(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))-379/64/(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. \[ 5 \, x + \frac{406 \, x^{7} + 889 \, x^{5} + 1272 \, x^{3} + 531 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} - \frac{1}{64} \, \int \frac{1322 \, x^{2} + 1137}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x + 1/64*(406*x^7 + 889*x^5 + 1272*x^3 + 531*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2
+ 9) - 1/64*integrate((1322*x^2 + 1137)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.306692, size = 1173, normalized size = 4.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1204767744*sqrt(1176531)*4^(3/4)*(4*sqrt(1176531)*4^(1/4)*(1176531*sqrt(3)*sqr
t(2)*(320*x^9 + 1686*x^7 + 4089*x^5 + 5112*x^3 + 3411*x) - 827621*sqrt(2)*(320*x
^9 + 1686*x^7 + 4089*x^5 + 5112*x^3 + 3411*x))*sqrt((827621*sqrt(3) - 3529593)/(
973721762751*sqrt(3) - 2418816050762)) + 10534088*4152675581883^(1/4)*(x^8 + 4*x
^6 + 10*x^4 + 12*x^2 + 9)*arctan(2*4152675581883^(1/4)*(943*sqrt(3) + 185)/(sqrt
(1176531)*4^(1/4)*sqrt(1/1176531)*(1176531*sqrt(3)*sqrt(2) - 827621*sqrt(2))*sqr
t((6398804416442536336606395*sqrt(3)*x^2 + 4152675581883^(1/4)*sqrt(1176531)*4^(
1/4)*(6888796098644077434301*sqrt(3)*x - 11341932760469461370531*x)*sqrt((827621
*sqrt(3) - 3529593)/(973721762751*sqrt(3) - 2418816050762)) - 128889605648307726
08808849*x^2 + 1176531*sqrt(3)*(5438704476501287545*sqrt(3) - 109550539380864359
79))/(5438704476501287545*sqrt(3) - 10955053938086435979))*sqrt((827621*sqrt(3)
- 3529593)/(973721762751*sqrt(3) - 2418816050762)) + sqrt(1176531)*4^(1/4)*(1176
531*sqrt(3)*sqrt(2)*x - 827621*sqrt(2)*x)*sqrt((827621*sqrt(3) - 3529593)/(97372
1762751*sqrt(3) - 2418816050762)) + 2*4152675581883^(1/4)*(379*sqrt(3)*sqrt(2) -
1322*sqrt(2)))) + 10534088*4152675581883^(1/4)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 +
9)*arctan(2*4152675581883^(1/4)*(943*sqrt(3) + 185)/(sqrt(1176531)*4^(1/4)*sqrt
(1/1176531)*(1176531*sqrt(3)*sqrt(2) - 827621*sqrt(2))*sqrt((6398804416442536336
606395*sqrt(3)*x^2 - 4152675581883^(1/4)*sqrt(1176531)*4^(1/4)*(6888796098644077
434301*sqrt(3)*x - 11341932760469461370531*x)*sqrt((827621*sqrt(3) - 3529593)/(9
73721762751*sqrt(3) - 2418816050762)) - 12888960564830772608808849*x^2 + 1176531
*sqrt(3)*(5438704476501287545*sqrt(3) - 10955053938086435979))/(5438704476501287
545*sqrt(3) - 10955053938086435979))*sqrt((827621*sqrt(3) - 3529593)/(9737217627
51*sqrt(3) - 2418816050762)) + sqrt(1176531)*4^(1/4)*(1176531*sqrt(3)*sqrt(2)*x
- 827621*sqrt(2)*x)*sqrt((827621*sqrt(3) - 3529593)/(973721762751*sqrt(3) - 2418
816050762)) - 2*4152675581883^(1/4)*(379*sqrt(3)*sqrt(2) - 1322*sqrt(2)))) - 415
2675581883^(1/4)*(1176531*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) -
827621*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*log(127976088328850726732127
90*sqrt(3)*x^2 + 2*4152675581883^(1/4)*sqrt(1176531)*4^(1/4)*(688879609864407743
4301*sqrt(3)*x - 11341932760469461370531*x)*sqrt((827621*sqrt(3) - 3529593)/(973
721762751*sqrt(3) - 2418816050762)) - 25777921129661545217617698*x^2 + 2353062*s
qrt(3)*(5438704476501287545*sqrt(3) - 10955053938086435979)) + 4152675581883^(1/
4)*(1176531*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 827621*sqrt(2)
*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*log(12797608832885072673212790*sqrt(3)*x^2
- 2*4152675581883^(1/4)*sqrt(1176531)*4^(1/4)*(6888796098644077434301*sqrt(3)*x
- 11341932760469461370531*x)*sqrt((827621*sqrt(3) - 3529593)/(973721762751*sqrt
(3) - 2418816050762)) - 25777921129661545217617698*x^2 + 2353062*sqrt(3)*(543870
4476501287545*sqrt(3) - 10955053938086435979)))/((1176531*sqrt(3)*sqrt(2)*(x^8 +
4*x^6 + 10*x^4 + 12*x^2 + 9) - 827621*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 +
9))*sqrt((827621*sqrt(3) - 3529593)/(973721762751*sqrt(3) - 2418816050762)))

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Sympy [A]  time = 2.21824, size = 71, normalized size = 0.3 \[ 5 x + \frac{406 x^{7} + 889 x^{5} + 1272 x^{3} + 531 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + \operatorname{RootSum}{\left (17179869184 t^{4} + 216955879424 t^{2} + 4152675581883, \left ( t \mapsto t \log{\left (- \frac{31641829376 t^{3}}{1549210136091} - \frac{455309168896 t}{1549210136091} + x \right )} \right )\right )} \]

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

[Out]

5*x + (406*x**7 + 889*x**5 + 1272*x**3 + 531*x)/(64*x**8 + 256*x**6 + 640*x**4 +
768*x**2 + 576) + RootSum(17179869184*_t**4 + 216955879424*_t**2 + 415267558188
3, Lambda(_t, _t*log(-31641829376*_t**3/1549210136091 - 455309168896*_t/15492101
36091 + x)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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