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

Optimal. Leaf size=243 \[ -\frac{1}{32} (9 d-4 f) \log \left (x^2-x+1\right )+\frac{1}{32} (9 d-4 f) \log \left (x^2+x+1\right )+\frac{x \left (-7 x^2 (d-f)+2 d+3 f\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g)+e-2 g}{12 \left (x^4+x^2+1\right )^2} \]

[Out]

(x*(d + f - (d - 2*f)*x^2))/(12*(1 + x^2 + x^4)^2) + (e - 2*g + (2*e - g)*x^2)/(
12*(1 + x^2 + x^4)^2) + ((2*e - g)*(1 + 2*x^2))/(12*(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)/Sqr
t[3]])/(48*Sqrt[3]) + ((13*d + 2*f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + ((
2*e - g)*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 = 0.52391, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{1}{32} (9 d-4 f) \log \left (x^2-x+1\right )+\frac{1}{32} (9 d-4 f) \log \left (x^2+x+1\right )+\frac{x \left (-7 x^2 (d-f)+2 d+3 f\right )}{24 \left (x^4+x^2+1\right )}+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{(13 d+2 f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(13 d+2 f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{(2 e-g) \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\left (2 x^2+1\right ) (2 e-g)}{12 \left (x^4+x^2+1\right )}+\frac{x^2 (2 e-g)+e-2 g}{12 \left (x^4+x^2+1\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + f - (d - 2*f)*x^2))/(12*(1 + x^2 + x^4)^2) + (e - 2*g + (2*e - g)*x^2)/(
12*(1 + x^2 + x^4)^2) + ((2*e - g)*(1 + 2*x^2))/(12*(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)/Sqr
t[3]])/(48*Sqrt[3]) + ((13*d + 2*f)*ArcTan[(1 + 2*x)/Sqrt[3]])/(48*Sqrt[3]) + ((
2*e - g)*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 = 83.973, size = 207, normalized size = 0.85 \[ \frac{x \left (6 d + 9 f - x^{3} \left (18 e - 18 g\right ) - x^{2} \left (21 d - 21 f\right ) + x \left (6 e + 6 g\right )\right )}{72 \left (x^{4} + x^{2} + 1\right )} + \frac{x \left (d + f - x^{3} \left (e - 2 g\right ) - x^{2} \left (d - 2 f\right ) + x \left (e + g\right )\right )}{12 \left (x^{4} + x^{2} + 1\right )^{2}} - \left (\frac{9 d}{32} - \frac{f}{8}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{9 d}{32} - \frac{f}{8}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (\frac{13 d}{2} + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{72} + \frac{\sqrt{3} \left (\frac{13 d}{2} + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{72} + \frac{\sqrt{3} \left (2 e - g\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(6*d + 9*f - x**3*(18*e - 18*g) - x**2*(21*d - 21*f) + x*(6*e + 6*g))/(72*(x**
4 + x**2 + 1)) + x*(d + f - x**3*(e - 2*g) - x**2*(d - 2*f) + x*(e + g))/(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 + sqrt(3)*(2*e - g)*atan(sqrt(3)*(2*x**2/3
+ 1/3))/9

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Mathematica [C]  time = 1.31532, size = 259, normalized size = 1.07 \[ \frac{1}{144} \left (\frac{12 \left (x \left (-d x^2+d+2 f x^2+f\right )+2 e x^2+e-g \left (x^2+2\right )\right )}{\left (x^4+x^2+1\right )^2}+\frac{6 \left (-7 d x^3+2 d x+e \left (8 x^2+4\right )+7 f x^3+3 f x-2 g \left (2 x^2+1\right )\right )}{x^4+x^2+1}-\frac{\left (\left (7 \sqrt{3}-47 i\right ) d+\left (-7 \sqrt{3}+17 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\left (\left (7 \sqrt{3}+47 i\right ) d-\left (7 \sqrt{3}+17 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-16 \sqrt{3} (2 e-g) \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((6*(2*d*x + 3*f*x - 7*d*x^3 + 7*f*x^3 - 2*g*(1 + 2*x^2) + e*(4 + 8*x^2)))/(1 +
x^2 + x^4) + (12*(e + 2*e*x^2 - g*(2 + x^2) + x*(d + f - d*x^2 + 2*f*x^2)))/(1 +
x^2 + x^4)^2 - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f)*ArcTan[((-I + Sq
rt[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] - 16*Sqrt[3]*(2*e -
g)*ArcTan[Sqrt[3]/(1 + 2*x^2)])/144

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Maple [A]  time = 0.023, size = 322, normalized size = 1.3 \[{\frac{1}{16\, \left ({x}^{2}+x+1 \right ) ^{2}} \left ( \left ( -{\frac{7\,d}{3}}+{\frac{7\,f}{3}}-{\frac{4\,e}{3}}-{\frac{g}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f-2\,g \right ){x}^{2}+ \left ( -{\frac{20\,d}{3}}+{\frac{13\,f}{3}}+{\frac{e}{3}}-{\frac{8\,g}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}+2\,e-2\,g \right ) }+{\frac{9\,d\ln \left ({x}^{2}+x+1 \right ) }{32}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}g}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{16\, \left ({x}^{2}-x+1 \right ) ^{2}} \left ( \left ({\frac{7\,d}{3}}-{\frac{7\,f}{3}}-{\frac{4\,e}{3}}-{\frac{g}{3}} \right ){x}^{3}+ \left ( -6\,d+4\,f+2\,g \right ){x}^{2}+ \left ({\frac{20\,d}{3}}-{\frac{13\,f}{3}}+{\frac{e}{3}}-{\frac{8\,g}{3}} \right ) x-4\,d+{\frac{4\,f}{3}}-2\,e+2\,g \right ) }-{\frac{9\,d\ln \left ({x}^{2}-x+1 \right ) }{32}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{72}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}g}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x^3+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-1/3*g)*x^3+(-6*d+4*f-2*g)*x^2+(-20/3*d+13/3*f+1/3*e-8/
3*g)*x-4*d+4/3*f+2*e-2*g)/(x^2+x+1)^2+9/32*d*ln(x^2+x+1)-1/8*ln(x^2+x+1)*f+13/14
4*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)*arctan(1/3*(1+2*x)*3^(1/2))*f+1/9*3^(1/2)*arctan(1/3*(1+2*x)*3^(1
/2))*g-1/16*((7/3*d-7/3*f-4/3*e-1/3*g)*x^3+(-6*d+4*f+2*g)*x^2+(20/3*d-13/3*f+1/3
*e-8/3*g)*x-4*d+4/3*f-2*e+2*g)/(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-1/9*3^(1/2)*arctan(1/3*(2*x-1)
*3^(1/2))*g

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Maxima [A]  time = 0.785153, size = 270, normalized size = 1.11 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) - \frac{7 \,{\left (d - f\right )} x^{7} - 4 \,{\left (2 \, e - g\right )} x^{6} + 5 \,{\left (d - 2 \, f\right )} x^{5} - 6 \,{\left (2 \, e - g\right )} x^{4} + 7 \,{\left (d - 2 \, f\right )} x^{3} - 8 \,{\left (2 \, e - g\right )} x^{2} -{\left (4 \, d + 5 \, f\right )} x - 6 \, e + 6 \, g}{24 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^3 + 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 + 16*g)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/144*s
qrt(3)*(13*d + 32*e + 2*f - 16*g)*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 -
4*(2*e - g)*x^6 + 5*(d - 2*f)*x^5 - 6*(2*e - g)*x^4 + 7*(d - 2*f)*x^3 - 8*(2*e -
g)*x^2 - (4*d + 5*f)*x - 6*e + 6*g)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)

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Fricas [A]  time = 0.503587, size = 599, normalized size = 2.47 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left ({\left (9 \, d - 4 \, f\right )} x^{8} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{6} + 3 \,{\left (9 \, d - 4 \, f\right )} x^{4} + 2 \,{\left (9 \, d - 4 \, f\right )} x^{2} + 9 \, d - 4 \, f\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f + 16 \, g\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f - 16 \, g\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left (7 \,{\left (d - f\right )} x^{7} - 4 \,{\left (2 \, e - g\right )} x^{6} + 5 \,{\left (d - 2 \, f\right )} x^{5} - 6 \,{\left (2 \, e - g\right )} x^{4} + 7 \,{\left (d - 2 \, f\right )} x^{3} - 8 \,{\left (2 \, e - g\right )} x^{2} -{\left (4 \, d + 5 \, f\right )} x - 6 \, e + 6 \, g\right )}\right )}}{288 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^3 + 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 + 16*g)*x^8 + 2*(13*d - 32*e + 2*f + 16*g)*x^
6 + 3*(13*d - 32*e + 2*f + 16*g)*x^4 + 2*(13*d - 32*e + 2*f + 16*g)*x^2 + 13*d -
32*e + 2*f + 16*g)*arctan(1/3*sqrt(3)*(2*x + 1)) + 2*((13*d + 32*e + 2*f - 16*g
)*x^8 + 2*(13*d + 32*e + 2*f - 16*g)*x^6 + 3*(13*d + 32*e + 2*f - 16*g)*x^4 + 2*
(13*d + 32*e + 2*f - 16*g)*x^2 + 13*d + 32*e + 2*f - 16*g)*arctan(1/3*sqrt(3)*(2
*x - 1)) - 4*sqrt(3)*(7*(d - f)*x^7 - 4*(2*e - g)*x^6 + 5*(d - 2*f)*x^5 - 6*(2*e
- g)*x^4 + 7*(d - 2*f)*x^3 - 8*(2*e - g)*x^2 - (4*d + 5*f)*x - 6*e + 6*g))/(x^8
+ 2*x^6 + 3*x^4 + 2*x^2 + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273037, size = 267, normalized size = 1.1 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f + 16 \, g - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 2 \, f - 16 \, g + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{32} \,{\left (9 \, d - 4 \, f\right )}{\rm ln}\left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 7 \, f x^{7} + 4 \, g x^{6} - 8 \, x^{6} e + 5 \, d x^{5} - 10 \, f x^{5} + 6 \, g x^{4} - 12 \, x^{4} e + 7 \, d x^{3} - 14 \, f x^{3} + 8 \, g x^{2} - 16 \, x^{2} e - 4 \, d x - 5 \, f x + 6 \, g - 6 \, e}{24 \,{\left (x^{4} + x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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