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

Optimal. Leaf size=238 \[ \frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac{25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

(-25*x*(3 + x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(238 - 59*x^2))/(64*(3 + 2*x^2 +
x^4)) - (Sqrt[3*(-48835 + 32827*Sqrt[3])]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)
/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[3*(-48835 + 32827*Sqrt[3])]*ArcTan[(Sqrt[2*
(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[3*(48835 + 32827*Sqrt
[3])]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (Sqrt[3*(48835 + 3282
7*Sqrt[3])]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi [A]  time = 0.725481, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{3 \left (48835+32827 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac{25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{3 \left (32827 \sqrt{3}-48835\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-25*x*(3 + x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(238 - 59*x^2))/(64*(3 + 2*x^2 +
x^4)) - (Sqrt[3*(-48835 + 32827*Sqrt[3])]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)
/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[3*(-48835 + 32827*Sqrt[3])]*ArcTan[(Sqrt[2*
(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[3*(48835 + 32827*Sqrt
[3])]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (Sqrt[3*(48835 + 3282
7*Sqrt[3])]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi in Sympy [A]  time = 38.5947, size = 350, normalized size = 1.47 \[ \frac{x \left (- 1087488 x^{2} + 4386816\right )}{1179648 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{x \left (4800 x^{2} + 14400\right )}{3072 \left (x^{4} + 2 x^{2} + 3\right )^{2}} + \frac{\sqrt{6} \left (1271808 + 2405376 \sqrt{3}\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{14155776 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (1271808 + 2405376 \sqrt{3}\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{14155776 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (2543616 + 4810752 \sqrt{3}\right )}{2} + 2543616 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{7077888 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (2543616 + 4810752 \sqrt{3}\right )}{2} + 2543616 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{7077888 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-1087488*x**2 + 4386816)/(1179648*(x**4 + 2*x**2 + 3)) - x*(4800*x**2 + 14400
)/(3072*(x**4 + 2*x**2 + 3)**2) + sqrt(6)*(1271808 + 2405376*sqrt(3))*log(x**2 -
sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(14155776*sqrt(-1 + sqrt(3))) - sqrt(6)
*(1271808 + 2405376*sqrt(3))*log(x**2 + sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/
(14155776*sqrt(-1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(2543616 +
4810752*sqrt(3))/2 + 2543616*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x - sqrt(
-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(7077888*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(
3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(2543616 + 4810752*sqrt(3))/2 + 2543
616*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x + sqrt(-2 + 2*sqrt(3))/2)/sqrt(1
+ sqrt(3)))/(7077888*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3)))

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Mathematica [C]  time = 0.664537, size = 129, normalized size = 0.54 \[ \frac{1}{256} \left (\frac{4 x \left (-59 x^6+120 x^4+199 x^2+414\right )}{\left (x^4+2 x^2+3\right )^2}+\frac{3 \left (174+133 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{3 \left (174-133 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((4*x*(414 + 199*x^2 + 120*x^4 - 59*x^6))/(3 + 2*x^2 + x^4)^2 + (3*(174 + (133*I
)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (3*(174 - (133*I
)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/256

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Maple [B]  time = 0.039, size = 418, normalized size = 1.8 \[{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{59\,{x}^{7}}{64}}+{\frac{15\,{x}^{5}}{8}}+{\frac{199\,{x}^{3}}{64}}+{\frac{207\,x}{32}} \right ) }-{\frac{307\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{399\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -614+614\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-798+798\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{23\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{307\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{399\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -614+614\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-798+798\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{23\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-59/64*x^7+15/8*x^5+199/64*x^3+207/32*x)/(x^4+2*x^2+3)^2-307/1024*ln(x^2+3^(1/2
)+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-399/1024*ln(x^2+3^(1/2)+x
*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+307/512/(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)+399/512/(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))-23/32/(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)+307/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^
(1/2)*3^(1/2)+399/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/
2)+307/512/(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)+399/512/(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))-23/32/(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. \[ -\frac{59 \, x^{7} - 120 \, x^{5} - 199 \, x^{3} - 414 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{3}{64} \, \int \frac{87 \, x^{2} - 46}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(59*x^7 - 120*x^5 - 199*x^3 - 414*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) +
3/64*integrate((87*x^2 - 46)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.302172, size = 1165, normalized size = 4.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/33614848*sqrt(32827)*4^(3/4)*(4*sqrt(32827)*4^(1/4)*(32827*sqrt(3)*sqrt(2)*(5
9*x^7 - 120*x^5 - 199*x^3 - 414*x) + 48835*sqrt(2)*(59*x^7 - 120*x^5 - 199*x^3 -
414*x))*sqrt((48835*sqrt(3) + 98481)/(1603106545*sqrt(3) + 2808846506)) + 16472
8*29095522083^(1/4)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*arctan(2*29095522083^(1/
4)*(307*sqrt(3) + 399)/(3*sqrt(32827)*4^(1/4)*sqrt(1/98481)*(32827*sqrt(3)*sqrt(
2) + 48835*sqrt(2))*sqrt((29056381280156723055*sqrt(3)*x^2 + 29095522083^(1/4)*s
qrt(32827)*4^(1/4)*(58070954450355287*sqrt(3)*x + 100535022105329061*x)*sqrt((48
835*sqrt(3) + 98481)/(1603106545*sqrt(3) + 2808846506)) + 50371173865956663891*x
^2 + 98481*sqrt(3)*(295045554778655*sqrt(3) + 511481137132611))/(295045554778655
*sqrt(3) + 511481137132611))*sqrt((48835*sqrt(3) + 98481)/(1603106545*sqrt(3) +
2808846506)) + 3*sqrt(32827)*4^(1/4)*(32827*sqrt(3)*sqrt(2)*x + 48835*sqrt(2)*x)
*sqrt((48835*sqrt(3) + 98481)/(1603106545*sqrt(3) + 2808846506)) + 2*29095522083
^(1/4)*(46*sqrt(3)*sqrt(2) + 261*sqrt(2)))) + 164728*29095522083^(1/4)*(x^8 + 4*
x^6 + 10*x^4 + 12*x^2 + 9)*arctan(2*29095522083^(1/4)*(307*sqrt(3) + 399)/(3*sqr
t(32827)*4^(1/4)*sqrt(1/98481)*(32827*sqrt(3)*sqrt(2) + 48835*sqrt(2))*sqrt((290
56381280156723055*sqrt(3)*x^2 - 29095522083^(1/4)*sqrt(32827)*4^(1/4)*(580709544
50355287*sqrt(3)*x + 100535022105329061*x)*sqrt((48835*sqrt(3) + 98481)/(1603106
545*sqrt(3) + 2808846506)) + 50371173865956663891*x^2 + 98481*sqrt(3)*(295045554
778655*sqrt(3) + 511481137132611))/(295045554778655*sqrt(3) + 511481137132611))*
sqrt((48835*sqrt(3) + 98481)/(1603106545*sqrt(3) + 2808846506)) + 3*sqrt(32827)*
4^(1/4)*(32827*sqrt(3)*sqrt(2)*x + 48835*sqrt(2)*x)*sqrt((48835*sqrt(3) + 98481)
/(1603106545*sqrt(3) + 2808846506)) - 2*29095522083^(1/4)*(46*sqrt(3)*sqrt(2) +
261*sqrt(2)))) + 29095522083^(1/4)*(32827*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4
+ 12*x^2 + 9) + 48835*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*log(581127625
60313446110*sqrt(3)*x^2 + 2*29095522083^(1/4)*sqrt(32827)*4^(1/4)*(5807095445035
5287*sqrt(3)*x + 100535022105329061*x)*sqrt((48835*sqrt(3) + 98481)/(1603106545*
sqrt(3) + 2808846506)) + 100742347731913327782*x^2 + 196962*sqrt(3)*(29504555477
8655*sqrt(3) + 511481137132611)) - 29095522083^(1/4)*(32827*sqrt(3)*sqrt(2)*(x^8
+ 4*x^6 + 10*x^4 + 12*x^2 + 9) + 48835*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 +
9))*log(58112762560313446110*sqrt(3)*x^2 - 2*29095522083^(1/4)*sqrt(32827)*4^(1
/4)*(58070954450355287*sqrt(3)*x + 100535022105329061*x)*sqrt((48835*sqrt(3) + 9
8481)/(1603106545*sqrt(3) + 2808846506)) + 100742347731913327782*x^2 + 196962*sq
rt(3)*(295045554778655*sqrt(3) + 511481137132611)))/((32827*sqrt(3)*sqrt(2)*(x^8
+ 4*x^6 + 10*x^4 + 12*x^2 + 9) + 48835*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 +
9))*sqrt((48835*sqrt(3) + 98481)/(1603106545*sqrt(3) + 2808846506)))

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Sympy [A]  time = 2.12792, size = 68, normalized size = 0.29 \[ - \frac{59 x^{7} - 120 x^{5} - 199 x^{3} - 414 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + \operatorname{RootSum}{\left (17179869184 t^{4} - 38405406720 t^{2} + 29095522083, \left ( t \mapsto t \log{\left (\frac{10301210624 t^{3}}{6083466813} - \frac{4322999552 t}{2027822271} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(59*x**7 - 120*x**5 - 199*x**3 - 414*x)/(64*x**8 + 256*x**6 + 640*x**4 + 768*x*
*2 + 576) + RootSum(17179869184*_t**4 - 38405406720*_t**2 + 29095522083, Lambda(
_t, _t*log(10301210624*_t**3/6083466813 - 4322999552*_t/2027822271 + x)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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