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

Optimal. Leaf size=248 \[ \frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 x \left (1-x^2\right )}{48 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (51 x^2+64\right )}{192 \left (x^4+2 x^2+3\right )}-\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

[Out]

(25*x*(1 - x^2))/(48*(3 + 2*x^2 + x^4)^2) + (x*(64 + 51*x^2))/(192*(3 + 2*x^2 +
x^4)) - (Sqrt[(-1291 + 1019*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sq
rt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(-1291 + 1019*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 +
Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(1291 + 1019*Sqrt[3])/3]*L
og[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (Sqrt[(1291 + 1019*Sqrt[3])/
3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi [A]  time = 0.704989, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{512} \sqrt{\frac{1}{3} \left (1291+1019 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{25 x \left (1-x^2\right )}{48 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (51 x^2+64\right )}{192 \left (x^4+2 x^2+3\right )}-\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{256} \sqrt{\frac{1}{3} \left (1019 \sqrt{3}-1291\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(25*x*(1 - x^2))/(48*(3 + 2*x^2 + x^4)^2) + (x*(64 + 51*x^2))/(192*(3 + 2*x^2 +
x^4)) - (Sqrt[(-1291 + 1019*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sq
rt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(-1291 + 1019*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 +
Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[(1291 + 1019*Sqrt[3])/3]*L
og[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (Sqrt[(1291 + 1019*Sqrt[3])/
3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512

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Rubi in Sympy [A]  time = 31.1256, size = 350, normalized size = 1.41 \[ \frac{x \left (- 400 x^{2} + 400\right )}{768 \left (x^{4} + 2 x^{2} + 3\right )^{2}} + \frac{x \left (9792 x^{2} + 12288\right )}{36864 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{\sqrt{6} \left (1152 + 4896 \sqrt{3}\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{442368 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{6} \left (1152 + 4896 \sqrt{3}\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{442368 \sqrt{-1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (2304 + 9792 \sqrt{3}\right )}{2} + 2304 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{221184 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} - \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (2304 + 9792 \sqrt{3}\right )}{2} + 2304 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{221184 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-400*x**2 + 400)/(768*(x**4 + 2*x**2 + 3)**2) + x*(9792*x**2 + 12288)/(36864*
(x**4 + 2*x**2 + 3)) + sqrt(6)*(1152 + 4896*sqrt(3))*log(x**2 - sqrt(2)*x*sqrt(-
1 + sqrt(3)) + sqrt(3))/(442368*sqrt(-1 + sqrt(3))) - sqrt(6)*(1152 + 4896*sqrt(
3))*log(x**2 + sqrt(2)*x*sqrt(-1 + sqrt(3)) + sqrt(3))/(442368*sqrt(-1 + sqrt(3)
)) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sqrt(3))*(2304 + 9792*sqrt(3))/2 + 2304*sqrt(2)
*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(x - sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3))
)/(221184*sqrt(-1 + sqrt(3))*sqrt(1 + sqrt(3))) - sqrt(3)*(-sqrt(2)*sqrt(-1 + sq
rt(3))*(2304 + 9792*sqrt(3))/2 + 2304*sqrt(2)*sqrt(-1 + sqrt(3)))*atan(sqrt(2)*(
x + sqrt(-2 + 2*sqrt(3))/2)/sqrt(1 + sqrt(3)))/(221184*sqrt(-1 + sqrt(3))*sqrt(1
+ sqrt(3)))

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Mathematica [C]  time = 0.615858, size = 129, normalized size = 0.52 \[ \frac{1}{768} \left (\frac{4 x \left (51 x^6+166 x^4+181 x^2+292\right )}{\left (x^4+2 x^2+3\right )^2}+\frac{3 \left (34+21 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{3 \left (34-21 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((4*x*(292 + 181*x^2 + 166*x^4 + 51*x^6))/(3 + 2*x^2 + x^4)^2 + (3*(34 + (21*I)*
Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (3*(34 - (21*I)*Sq
rt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/768

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Maple [B]  time = 0.037, size = 418, normalized size = 1.7 \[{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{17\,{x}^{7}}{64}}+{\frac{83\,{x}^{5}}{96}}+{\frac{181\,{x}^{3}}{192}}+{\frac{73\,x}{48}} \right ) }-{\frac{55\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{3072}}-{\frac{21\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -110+110\,\sqrt{3} \right ) \sqrt{3}}{1536\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-42+42\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{\sqrt{3}}{48\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{55\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{3072}}+{\frac{21\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -110+110\,\sqrt{3} \right ) \sqrt{3}}{1536\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-42+42\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{\sqrt{3}}{48\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(17/64*x^7+83/96*x^5+181/192*x^3+73/48*x)/(x^4+2*x^2+3)^2-55/3072*ln(x^2+3^(1/2)
+x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)*3^(1/2)-21/1024*ln(x^2+3^(1/2)+x*(
-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+55/1536/(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)+21/512/(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))-1/48/(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)+55/3072*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)
*3^(1/2)+21/1024*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2)+55/
1536/(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)+21/512/(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))-1/48/(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. \[ \frac{51 \, x^{7} + 166 \, x^{5} + 181 \, x^{3} + 292 \, x}{192 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{1}{64} \, \int \frac{17 \, x^{2} - 4}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

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, algorithm="maxima")

[Out]

1/192*(51*x^7 + 166*x^5 + 181*x^3 + 292*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) +
1/64*integrate((17*x^2 - 4)/(x^4 + 2*x^2 + 3), x)

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Fricas [A]  time = 0.315774, size = 1177, normalized size = 4.75 \[ \text{result too large to display} \]

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, algorithm="fricas")

[Out]

-1/9391104*sqrt(1019)*12^(3/4)*(20424*1038361^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^
4 + 12*x^2 + 9)*arctan(6*1038361^(1/4)*(21*sqrt(3) + 55)/(sqrt(1019)*12^(1/4)*sq
rt(1/3057)*(1291*sqrt(3)*sqrt(2) + 3057*sqrt(2))*sqrt(sqrt(3)*(1038361^(1/4)*sqr
t(1019)*12^(1/4)*(239300294807*sqrt(3)*x + 412134121929*x)*sqrt((1291*sqrt(3) +
3057)/(1315529*sqrt(3) + 2390882)) + 1019*sqrt(3)*(7108200815*sqrt(3)*x^2 + 1240
3970091*x^2) + 21729769891455*sqrt(3) + 37918936568187)/(7108200815*sqrt(3) + 12
403970091))*sqrt((1291*sqrt(3) + 3057)/(1315529*sqrt(3) + 2390882)) + sqrt(1019)
*12^(1/4)*(1291*sqrt(3)*sqrt(2)*x + 3057*sqrt(2)*x)*sqrt((1291*sqrt(3) + 3057)/(
1315529*sqrt(3) + 2390882)) + 6*1038361^(1/4)*(17*sqrt(3)*sqrt(2) + 4*sqrt(2))))
+ 20424*1038361^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*arctan(6*1038
361^(1/4)*(21*sqrt(3) + 55)/(sqrt(1019)*12^(1/4)*sqrt(1/3057)*(1291*sqrt(3)*sqrt
(2) + 3057*sqrt(2))*sqrt(-sqrt(3)*(1038361^(1/4)*sqrt(1019)*12^(1/4)*(2393002948
07*sqrt(3)*x + 412134121929*x)*sqrt((1291*sqrt(3) + 3057)/(1315529*sqrt(3) + 239
0882)) - 1019*sqrt(3)*(7108200815*sqrt(3)*x^2 + 12403970091*x^2) - 2172976989145
5*sqrt(3) - 37918936568187)/(7108200815*sqrt(3) + 12403970091))*sqrt((1291*sqrt(
3) + 3057)/(1315529*sqrt(3) + 2390882)) + sqrt(1019)*12^(1/4)*(1291*sqrt(3)*sqrt
(2)*x + 3057*sqrt(2)*x)*sqrt((1291*sqrt(3) + 3057)/(1315529*sqrt(3) + 2390882))
- 6*1038361^(1/4)*(17*sqrt(3)*sqrt(2) + 4*sqrt(2)))) - 4*sqrt(1019)*12^(1/4)*(12
91*sqrt(3)*sqrt(2)*(51*x^7 + 166*x^5 + 181*x^3 + 292*x) + 3057*sqrt(2)*(51*x^7 +
166*x^5 + 181*x^3 + 292*x))*sqrt((1291*sqrt(3) + 3057)/(1315529*sqrt(3) + 23908
82)) + 3*1038361^(1/4)*(1291*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)
+ 3057*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*log(2*1038361^(1/4)*sqrt(10
19)*12^(1/4)*(239300294807*sqrt(3)*x + 412134121929*x)*sqrt((1291*sqrt(3) + 3057
)/(1315529*sqrt(3) + 2390882)) + 2038*sqrt(3)*(7108200815*sqrt(3)*x^2 + 12403970
091*x^2) + 43459539782910*sqrt(3) + 75837873136374) - 3*1038361^(1/4)*(1291*sqrt
(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 3057*sqrt(2)*(x^8 + 4*x^6 + 10
*x^4 + 12*x^2 + 9))*log(-2*1038361^(1/4)*sqrt(1019)*12^(1/4)*(239300294807*sqrt(
3)*x + 412134121929*x)*sqrt((1291*sqrt(3) + 3057)/(1315529*sqrt(3) + 2390882)) +
2038*sqrt(3)*(7108200815*sqrt(3)*x^2 + 12403970091*x^2) + 43459539782910*sqrt(3
) + 75837873136374))/((1291*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)
+ 3057*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9))*sqrt((1291*sqrt(3) + 3057)/(
1315529*sqrt(3) + 2390882)))

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Sympy [A]  time = 2.09213, size = 68, normalized size = 0.27 \[ \frac{51 x^{7} + 166 x^{5} + 181 x^{3} + 292 x}{192 x^{8} + 768 x^{6} + 1920 x^{4} + 2304 x^{2} + 1728} + \operatorname{RootSum}{\left (51539607552 t^{4} - 338427904 t^{2} + 1038361, \left ( t \mapsto t \log{\left (\frac{5536481280 t^{3}}{867169} - \frac{19920128 t}{867169} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(51*x**7 + 166*x**5 + 181*x**3 + 292*x)/(192*x**8 + 768*x**6 + 1920*x**4 + 2304*
x**2 + 1728) + RootSum(51539607552*_t**4 - 338427904*_t**2 + 1038361, Lambda(_t,
_t*log(5536481280*_t**3/867169 - 19920128*_t/867169 + x)))

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

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, algorithm="giac")

[Out]

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