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3.48 \(\int \frac{d+e x+f x^2}{\left (1+x^2+x^4\right )^3} \, dx\)

Optimal. Leaf size=223 \[ -\frac{1}{32} (9 d-4 f) \log \left (x^2-x+1\right )+\frac{1}{32} (9 d-4 f) \log \left (x^2+x+1\right )+\frac{x \left (-7 x^2 (d-f)+2 d+3 f\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2} \]

[Out]

(e*(1 + 2*x^2))/(12*(1 + x^2 + x^4)^2) + (x*(d + f - (d - 2*f)*x^2))/(12*(1 + x^
2 + x^4)^2) + (e*(1 + 2*x^2))/(6*(1 + x^2 + x^4)) + (x*(2*d + 3*f - 7*(d - f)*x^
2))/(24*(1 + x^2 + x^4)) - ((13*d + 2*f)*ArcTan[(1 - 2*x)/Sqrt[3]])/(48*Sqrt[3])
 + ((13*d + 2*f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^
2)/Sqrt[3]])/(3*Sqrt[3]) - ((9*d - 4*f)*Log[1 - x + x^2])/32 + ((9*d - 4*f)*Log[
1 + x + x^2])/32

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Rubi [A]  time = 0.467998, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476 \[ -\frac{1}{32} (9 d-4 f) \log \left (x^2-x+1\right )+\frac{1}{32} (9 d-4 f) \log \left (x^2+x+1\right )+\frac{x \left (-7 x^2 (d-f)+2 d+3 f\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2)/(1 + x^2 + x^4)^3,x]

[Out]

(e*(1 + 2*x^2))/(12*(1 + x^2 + x^4)^2) + (x*(d + f - (d - 2*f)*x^2))/(12*(1 + x^
2 + x^4)^2) + (e*(1 + 2*x^2))/(6*(1 + x^2 + x^4)) + (x*(2*d + 3*f - 7*(d - f)*x^
2))/(24*(1 + x^2 + x^4)) - ((13*d + 2*f)*ArcTan[(1 - 2*x)/Sqrt[3]])/(48*Sqrt[3])
 + ((13*d + 2*f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^
2)/Sqrt[3]])/(3*Sqrt[3]) - ((9*d - 4*f)*Log[1 - x + x^2])/32 + ((9*d - 4*f)*Log[
1 + x + x^2])/32

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Rubi in Sympy [A]  time = 75.6237, size = 194, normalized size = 0.87 \[ \frac{2 \sqrt{3} e \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{9} + \frac{x \left (6 d - 18 e x^{3} + 6 e x + 9 f - x^{2} \left (21 d - 21 f\right )\right )}{72 \left (x^{4} + x^{2} + 1\right )} + \frac{x \left (d - e x^{3} + e x + f - x^{2} \left (d - 2 f\right )\right )}{12 \left (x^{4} + x^{2} + 1\right )^{2}} - \left (\frac{9 d}{32} - \frac{f}{8}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{9 d}{32} - \frac{f}{8}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (\frac{13 d}{2} + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{72} + \frac{\sqrt{3} \left (\frac{13 d}{2} + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{72} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**2+e*x+d)/(x**4+x**2+1)**3,x)

[Out]

2*sqrt(3)*e*atan(sqrt(3)*(2*x**2/3 + 1/3))/9 + x*(6*d - 18*e*x**3 + 6*e*x + 9*f
- x**2*(21*d - 21*f))/(72*(x**4 + x**2 + 1)) + x*(d - e*x**3 + e*x + f - x**2*(d
 - 2*f))/(12*(x**4 + x**2 + 1)**2) - (9*d/32 - f/8)*log(x**2 - x + 1) + (9*d/32
- f/8)*log(x**2 + x + 1) + sqrt(3)*(13*d/2 + f)*atan(sqrt(3)*(2*x/3 - 1/3))/72 +
 sqrt(3)*(13*d/2 + f)*atan(sqrt(3)*(2*x/3 + 1/3))/72

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Mathematica [C]  time = 1.14654, size = 235, normalized size = 1.05 \[ \frac{1}{144} \left (\frac{12 \left (x \left (-d x^2+d+2 f x^2+f\right )+2 e x^2+e\right )}{\left (x^4+x^2+1\right )^2}+\frac{6 \left (-7 d x^3+2 d x+e \left (8 x^2+4\right )+7 f x^3+3 f x\right )}{x^4+x^2+1}-\frac{\left (\left (7 \sqrt{3}-47 i\right ) d+\left (-7 \sqrt{3}+17 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\left (\left (7 \sqrt{3}+47 i\right ) d-\left (7 \sqrt{3}+17 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-32 \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x + f*x^2)/(1 + x^2 + x^4)^3,x]

[Out]

((6*(2*d*x + 3*f*x - 7*d*x^3 + 7*f*x^3 + e*(4 + 8*x^2)))/(1 + x^2 + x^4) + (12*(
e + 2*e*x^2 + x*(d + f - d*x^2 + 2*f*x^2)))/(1 + x^2 + x^4)^2 - (((-47*I + 7*Sqr
t[3])*d + (17*I - 7*Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3
])/6] - (((47*I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/
2])/Sqrt[(1 - I*Sqrt[3])/6] - 32*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144

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Maple [A]  time = 0.022, size = 264, normalized size = 1.2 \[{\frac{1}{16\, \left ({x}^{2}+x+1 \right ) ^{2}} \left ( \left ( -{\frac{7\,d}{3}}+{\frac{7\,f}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f \right ){x}^{2}+ \left ( -{\frac{20\,d}{3}}+{\frac{13\,f}{3}}+{\frac{e}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}+2\,e \right ) }+{\frac{9\,d\ln \left ({x}^{2}+x+1 \right ) }{32}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{16\, \left ({x}^{2}-x+1 \right ) ^{2}} \left ( \left ({\frac{7\,d}{3}}-{\frac{7\,f}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f \right ){x}^{2}+ \left ({\frac{20\,d}{3}}-{\frac{13\,f}{3}}+{\frac{e}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}-2\,e \right ) }-{\frac{9\,d\ln \left ({x}^{2}-x+1 \right ) }{32}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^2+e*x+d)/(x^4+x^2+1)^3,x)

[Out]

1/16*((-7/3*d+7/3*f-4/3*e)*x^3+(-6*d+4*f)*x^2+(-20/3*d+13/3*f+1/3*e)*x-4*d+4/3*f
+2*e)/(x^2+x+1)^2+9/32*d*ln(x^2+x+1)-1/8*ln(x^2+x+1)*f+13/144*d*arctan(1/3*(1+2*
x)*3^(1/2))*3^(1/2)-2/9*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*e+1/72*3^(1/2)*arcta
n(1/3*(1+2*x)*3^(1/2))*f-1/16*((7/3*d-7/3*f-4/3*e)*x^3+(-6*d+4*f)*x^2+(20/3*d-13
/3*f+1/3*e)*x-4*d+4/3*f-2*e)/(x^2-x+1)^2-9/32*d*ln(x^2-x+1)+1/8*ln(x^2-x+1)*f+13
/144*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*d+2/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2
))*e+1/72*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*f

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Maxima [A]  time = 0.783577, size = 234, normalized size = 1.05 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) - \frac{7 \,{\left (d - f\right )} x^{7} - 8 \, e x^{6} + 5 \,{\left (d - 2 \, f\right )} x^{5} - 12 \, e x^{4} + 7 \,{\left (d - 2 \, f\right )} x^{3} - 16 \, e x^{2} -{\left (4 \, d + 5 \, f\right )} x - 6 \, e}{24 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="maxima")

[Out]

1/144*sqrt(3)*(13*d - 32*e + 2*f)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/144*sqrt(3)*
(13*d + 32*e + 2*f)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/32*(9*d - 4*f)*log(x^2 + x
 + 1) - 1/32*(9*d - 4*f)*log(x^2 - x + 1) - 1/24*(7*(d - f)*x^7 - 8*e*x^6 + 5*(d
 - 2*f)*x^5 - 12*e*x^4 + 7*(d - 2*f)*x^3 - 16*e*x^2 - (4*d + 5*f)*x - 6*e)/(x^8
+ 2*x^6 + 3*x^4 + 2*x^2 + 1)

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Fricas [A]  time = 0.316786, size = 531, normalized size = 2.38 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (13 \, d - 32 \, e + 2 \, f\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e + 2 \, f\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (13 \, d + 32 \, e + 2 \, f\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e + 2 \, f\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left (7 \,{\left (d - f\right )} x^{7} - 8 \, e x^{6} + 5 \,{\left (d - 2 \, f\right )} x^{5} - 12 \, e x^{4} + 7 \,{\left (d - 2 \, f\right )} x^{3} - 16 \, e x^{2} -{\left (4 \, d + 5 \, f\right )} x - 6 \, e\right )}\right )}}{288 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="fricas")

[Out]

1/288*sqrt(3)*(3*sqrt(3)*((9*d - 4*f)*x^8 + 2*(9*d - 4*f)*x^6 + 3*(9*d - 4*f)*x^
4 + 2*(9*d - 4*f)*x^2 + 9*d - 4*f)*log(x^2 + x + 1) - 3*sqrt(3)*((9*d - 4*f)*x^8
 + 2*(9*d - 4*f)*x^6 + 3*(9*d - 4*f)*x^4 + 2*(9*d - 4*f)*x^2 + 9*d - 4*f)*log(x^
2 - x + 1) + 2*((13*d - 32*e + 2*f)*x^8 + 2*(13*d - 32*e + 2*f)*x^6 + 3*(13*d -
32*e + 2*f)*x^4 + 2*(13*d - 32*e + 2*f)*x^2 + 13*d - 32*e + 2*f)*arctan(1/3*sqrt
(3)*(2*x + 1)) + 2*((13*d + 32*e + 2*f)*x^8 + 2*(13*d + 32*e + 2*f)*x^6 + 3*(13*
d + 32*e + 2*f)*x^4 + 2*(13*d + 32*e + 2*f)*x^2 + 13*d + 32*e + 2*f)*arctan(1/3*
sqrt(3)*(2*x - 1)) - 4*sqrt(3)*(7*(d - f)*x^7 - 8*e*x^6 + 5*(d - 2*f)*x^5 - 12*e
*x^4 + 7*(d - 2*f)*x^3 - 16*e*x^2 - (4*d + 5*f)*x - 6*e))/(x^8 + 2*x^6 + 3*x^4 +
 2*x^2 + 1)

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Sympy [A]  time = 106.84, size = 4498, normalized size = 20.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**2+e*x+d)/(x**4+x**2+1)**3,x)

[Out]

(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)*log(x + (-1025428432*d**5*e
- 334752912*d**5*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 200896136
0*d**4*e*f + 1151575920*d**4*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/28
8) - 431308800*d**3*e**3 - 3143688192*d**3*e**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d
 + 32*e + 2*f)/288) - 1598857120*d**3*e*f**2 + 9917005824*d**3*e*(-9*d/32 + f/8
- sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 - 944300160*d**3*f**2*(-9*d/32 + f/8 - s
qrt(3)*I*(13*d + 32*e + 2*f)/288) + 11878244352*d**3*(-9*d/32 + f/8 - sqrt(3)*I*
(13*d + 32*e + 2*f)/288)**3 + 233164800*d**2*e**3*f + 4409634816*d**2*e**2*f*(-9
*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 662937520*d**2*e*f**3 - 13004
623872*d**2*e*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 + 2317960
80*d**2*f**3*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 10089639936*d
**2*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 + 142606336*d*e**5
+ 754974720*d*e**4*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 1843200
*d*e**3*f**2 + 3850371072*d*e**3*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/
288)**2 - 1926291456*d*e**2*f**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/
288) + 20384317440*d*e**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3
 - 146756960*d*e*f**4 + 5813379072*d*e*f**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 3
2*e + 2*f)/288)**2 + 12679200*d*f**4*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2
*f)/288) + 1116758016*d*f**2*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/288)
**3 - 79691776*e**5*f - 188743680*e**4*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e
 + 2*f)/288) - 7372800*e**3*f**3 - 2151677952*e**3*f*(-9*d/32 + f/8 - sqrt(3)*I*
(13*d + 32*e + 2*f)/288)**2 + 287096832*e**2*f**3*(-9*d/32 + f/8 - sqrt(3)*I*(13
*d + 32*e + 2*f)/288) - 5096079360*e**2*f*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*
e + 2*f)/288)**3 + 14093632*e*f**5 - 859521024*e*f**3*(-9*d/32 + f/8 - sqrt(3)*I
*(13*d + 32*e + 2*f)/288)**2 - 7648128*f**5*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 3
2*e + 2*f)/288) + 453869568*f**3*(-9*d/32 + f/8 - sqrt(3)*I*(13*d + 32*e + 2*f)/
288)**3)/(217696167*d**6 - 301346487*d**5*f - 1217128448*d**4*e**2 + 130506255*d
**4*f**2 + 2181281792*d**3*e**2*f - 5619240*d**3*f**3 - 617611264*d**2*e**4 - 14
50149888*d**2*e**2*f**2 - 8036820*d**2*f**4 + 495976448*d*e**4*f + 430088192*d*e
**2*f**3 + 783648*d*f**5 - 114294784*e**4*f**2 - 47771648*e**2*f**4 + 188352*f**
6)) + (-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)*log(x + (-1025428432*d
**5*e - 334752912*d**5*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 200
8961360*d**4*e*f + 1151575920*d**4*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2
*f)/288) - 431308800*d**3*e**3 - 3143688192*d**3*e**2*(-9*d/32 + f/8 + sqrt(3)*I
*(13*d + 32*e + 2*f)/288) - 1598857120*d**3*e*f**2 + 9917005824*d**3*e*(-9*d/32
+ f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 - 944300160*d**3*f**2*(-9*d/32 + f
/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 11878244352*d**3*(-9*d/32 + f/8 + sqrt
(3)*I*(13*d + 32*e + 2*f)/288)**3 + 233164800*d**2*e**3*f + 4409634816*d**2*e**2
*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) + 662937520*d**2*e*f**3 -
 13004623872*d**2*e*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**2 + 2
31796080*d**2*f**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 1008963
9936*d**2*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288)**3 + 142606336*d
*e**5 + 754974720*d*e**4*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/288) - 1
843200*d*e**3*f**2 + 3850371072*d*e**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e +
 2*f)/288)**2 - 1926291456*d*e**2*f**2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e +
 2*f)/288) + 20384317440*d*e**2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f)/2
88)**3 - 146756960*d*e*f**4 + 5813379072*d*e*f**2*(-9*d/32 + f/8 + sqrt(3)*I*(13
*d + 32*e + 2*f)/288)**2 + 12679200*d*f**4*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32
*e + 2*f)/288) + 1116758016*d*f**2*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e + 2*f
)/288)**3 - 79691776*e**5*f - 188743680*e**4*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d
+ 32*e + 2*f)/288) - 7372800*e**3*f**3 - 2151677952*e**3*f*(-9*d/32 + f/8 + sqrt
(3)*I*(13*d + 32*e + 2*f)/288)**2 + 287096832*e**2*f**3*(-9*d/32 + f/8 + sqrt(3)
*I*(13*d + 32*e + 2*f)/288) - 5096079360*e**2*f*(-9*d/32 + f/8 + sqrt(3)*I*(13*d
 + 32*e + 2*f)/288)**3 + 14093632*e*f**5 - 859521024*e*f**3*(-9*d/32 + f/8 + sqr
t(3)*I*(13*d + 32*e + 2*f)/288)**2 - 7648128*f**5*(-9*d/32 + f/8 + sqrt(3)*I*(13
*d + 32*e + 2*f)/288) + 453869568*f**3*(-9*d/32 + f/8 + sqrt(3)*I*(13*d + 32*e +
 2*f)/288)**3)/(217696167*d**6 - 301346487*d**5*f - 1217128448*d**4*e**2 + 13050
6255*d**4*f**2 + 2181281792*d**3*e**2*f - 5619240*d**3*f**3 - 617611264*d**2*e**
4 - 1450149888*d**2*e**2*f**2 - 8036820*d**2*f**4 + 495976448*d*e**4*f + 4300881
92*d*e**2*f**3 + 783648*d*f**5 - 114294784*e**4*f**2 - 47771648*e**2*f**4 + 1883
52*f**6)) + (9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)*log(x + (-1025428
432*d**5*e - 334752912*d**5*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) +
 2008961360*d**4*e*f + 1151575920*d**4*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e
+ 2*f)/288) - 431308800*d**3*e**3 - 3143688192*d**3*e**2*(9*d/32 - f/8 - sqrt(3)
*I*(13*d - 32*e + 2*f)/288) - 1598857120*d**3*e*f**2 + 9917005824*d**3*e*(9*d/32
 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 - 944300160*d**3*f**2*(9*d/32 - f
/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 11878244352*d**3*(9*d/32 - f/8 - sqrt(
3)*I*(13*d - 32*e + 2*f)/288)**3 + 233164800*d**2*e**3*f + 4409634816*d**2*e**2*
f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 662937520*d**2*e*f**3 - 1
3004623872*d**2*e*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 + 2317
96080*d**2*f**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 10089639936
*d**2*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 + 142606336*d*e**5
 + 754974720*d*e**4*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 1843200
*d*e**3*f**2 + 3850371072*d*e**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/2
88)**2 - 1926291456*d*e**2*f**2*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/28
8) + 20384317440*d*e**2*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 -
146756960*d*e*f**4 + 5813379072*d*e*f**2*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e
+ 2*f)/288)**2 + 12679200*d*f**4*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/2
88) + 1116758016*d*f**2*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 -
79691776*e**5*f - 188743680*e**4*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)
/288) - 7372800*e**3*f**3 - 2151677952*e**3*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d -
32*e + 2*f)/288)**2 + 287096832*e**2*f**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e
 + 2*f)/288) - 5096079360*e**2*f*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/2
88)**3 + 14093632*e*f**5 - 859521024*e*f**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32
*e + 2*f)/288)**2 - 7648128*f**5*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/2
88) + 453869568*f**3*(9*d/32 - f/8 - sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3)/(217
696167*d**6 - 301346487*d**5*f - 1217128448*d**4*e**2 + 130506255*d**4*f**2 + 21
81281792*d**3*e**2*f - 5619240*d**3*f**3 - 617611264*d**2*e**4 - 1450149888*d**2
*e**2*f**2 - 8036820*d**2*f**4 + 495976448*d*e**4*f + 430088192*d*e**2*f**3 + 78
3648*d*f**5 - 114294784*e**4*f**2 - 47771648*e**2*f**4 + 188352*f**6)) + (9*d/32
 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)*log(x + (-1025428432*d**5*e - 334752
912*d**5*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 2008961360*d**4*e*
f + 1151575920*d**4*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 43130
8800*d**3*e**3 - 3143688192*d**3*e**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2
*f)/288) - 1598857120*d**3*e*f**2 + 9917005824*d**3*e*(9*d/32 - f/8 + sqrt(3)*I*
(13*d - 32*e + 2*f)/288)**2 - 944300160*d**3*f**2*(9*d/32 - f/8 + sqrt(3)*I*(13*
d - 32*e + 2*f)/288) + 11878244352*d**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e +
 2*f)/288)**3 + 233164800*d**2*e**3*f + 4409634816*d**2*e**2*f*(9*d/32 - f/8 + s
qrt(3)*I*(13*d - 32*e + 2*f)/288) + 662937520*d**2*e*f**3 - 13004623872*d**2*e*f
*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 + 231796080*d**2*f**3*(9*
d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 10089639936*d**2*f*(9*d/32 - f
/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 + 142606336*d*e**5 + 754974720*d*e**4
*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 1843200*d*e**3*f**2 + 3850
371072*d*e**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 - 1926291456
*d*e**2*f**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 20384317440*d*
e**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 - 146756960*d*e*f**4
+ 5813379072*d*e*f**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 + 12
679200*d*f**4*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 1116758016*d*
f**2*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 - 79691776*e**5*f - 1
88743680*e**4*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 7372800*e**
3*f**3 - 2151677952*e**3*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2
 + 287096832*e**2*f**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) - 5096
079360*e**2*f*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3 + 14093632*e
*f**5 - 859521024*e*f**3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**2 -
 7648128*f**5*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288) + 453869568*f**
3*(9*d/32 - f/8 + sqrt(3)*I*(13*d - 32*e + 2*f)/288)**3)/(217696167*d**6 - 30134
6487*d**5*f - 1217128448*d**4*e**2 + 130506255*d**4*f**2 + 2181281792*d**3*e**2*
f - 5619240*d**3*f**3 - 617611264*d**2*e**4 - 1450149888*d**2*e**2*f**2 - 803682
0*d**2*f**4 + 495976448*d*e**4*f + 430088192*d*e**2*f**3 + 783648*d*f**5 - 11429
4784*e**4*f**2 - 47771648*e**2*f**4 + 188352*f**6)) - (-8*e*x**6 - 12*e*x**4 - 1
6*e*x**2 - 6*e + x**7*(7*d - 7*f) + x**5*(5*d - 10*f) + x**3*(7*d - 14*f) + x*(-
4*d - 5*f))/(24*x**8 + 48*x**6 + 72*x**4 + 48*x**2 + 24)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.265839, size = 231, normalized size = 1.04 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )}{\rm ln}\left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 7 \, f x^{7} - 8 \, x^{6} e + 5 \, d x^{5} - 10 \, f x^{5} - 12 \, x^{4} e + 7 \, d x^{3} - 14 \, f x^{3} - 16 \, x^{2} e - 4 \, d x - 5 \, f x - 6 \, e}{24 \,{\left (x^{4} + x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 + x^2 + 1)^3,x, algorithm="giac")

[Out]

1/144*sqrt(3)*(13*d + 2*f - 32*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/144*sqrt(3)*
(13*d + 2*f + 32*e)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/32*(9*d - 4*f)*ln(x^2 + x
+ 1) - 1/32*(9*d - 4*f)*ln(x^2 - x + 1) - 1/24*(7*d*x^7 - 7*f*x^7 - 8*x^6*e + 5*
d*x^5 - 10*f*x^5 - 12*x^4*e + 7*d*x^3 - 14*f*x^3 - 16*x^2*e - 4*d*x - 5*f*x - 6*
e)/(x^4 + x^2 + 1)^2

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