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

Optimal. Leaf size=185 \[ -\frac{9}{32} d \log \left (x^2-x+1\right )+\frac{9}{32} d \log \left (x^2+x+1\right )+\frac{d x \left (2-7 x^2\right )}{24 \left (x^4+x^2+1\right )}+\frac{d x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{13 d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{13 d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.258473, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688 \[ -\frac{9}{32} d \log \left (x^2-x+1\right )+\frac{9}{32} d \log \left (x^2+x+1\right )+\frac{d x \left (2-7 x^2\right )}{24 \left (x^4+x^2+1\right )}+\frac{d x \left (1-x^2\right )}{12 \left (x^4+x^2+1\right )^2}-\frac{13 d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{13 d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{48 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )}+\frac{e \left (2 x^2+1\right )}{12 \left (x^4+x^2+1\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 57.2801, size = 168, normalized size = 0.91 \[ - \frac{9 d \log{\left (x^{2} - x + 1 \right )}}{32} + \frac{9 d \log{\left (x^{2} + x + 1 \right )}}{32} + \frac{13 \sqrt{3} d \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{144} + \frac{13 \sqrt{3} d \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{144} + \frac{2 \sqrt{3} e \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{9} + \frac{x \left (- 21 d x^{2} + 6 d - 18 e x^{3} + 6 e x\right )}{72 \left (x^{4} + x^{2} + 1\right )} + \frac{x \left (- d x^{2} + d - e x^{3} + e x\right )}{12 \left (x^{4} + x^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*d*log(x**2 - x + 1)/32 + 9*d*log(x**2 + x + 1)/32 + 13*sqrt(3)*d*atan(sqrt(3)
*(2*x/3 - 1/3))/144 + 13*sqrt(3)*d*atan(sqrt(3)*(2*x/3 + 1/3))/144 + 2*sqrt(3)*e
*atan(sqrt(3)*(2*x**2/3 + 1/3))/9 + x*(-21*d*x**2 + 6*d - 18*e*x**3 + 6*e*x)/(72
*(x**4 + x**2 + 1)) + x*(-d*x**2 + d - e*x**3 + e*x)/(12*(x**4 + x**2 + 1)**2)

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Mathematica [C]  time = 2.18715, size = 186, normalized size = 1.01 \[ \frac{1}{144} \left (\frac{6 \left (d x \left (2-7 x^2\right )+e \left (8 x^2+4\right )\right )}{x^4+x^2+1}+\frac{12 \left (d \left (x-x^3\right )+2 e x^2+e\right )}{\left (x^4+x^2+1\right )^2}-\frac{\left (7 \sqrt{3}-47 i\right ) d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\left (7 \sqrt{3}+47 i\right ) d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-32 \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((6*(d*x*(2 - 7*x^2) + e*(4 + 8*x^2)))/(1 + x^2 + x^4) + (12*(e + 2*e*x^2 + d*(x
- x^3)))/(1 + x^2 + x^4)^2 - ((-47*I + 7*Sqrt[3])*d*ArcTan[((-I + Sqrt[3])*x)/2
])/Sqrt[(1 + I*Sqrt[3])/6] - ((47*I + 7*Sqrt[3])*d*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 = 0.023, size = 180, normalized size = 1. \[{\frac{1}{16\, \left ({x}^{2}+x+1 \right ) ^{2}} \left ( \left ( -{\frac{7\,d}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}-6\,d{x}^{2}+ \left ( -{\frac{20\,d}{3}}+{\frac{e}{3}} \right ) x-4\,d+2\,e \right ) }+{\frac{9\,d\ln \left ({x}^{2}+x+1 \right ) }{32}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{16\, \left ({x}^{2}-x+1 \right ) ^{2}} \left ( \left ({\frac{7\,d}{3}}-{\frac{4\,e}{3}} \right ){x}^{3}-6\,d{x}^{2}+ \left ({\frac{20\,d}{3}}+{\frac{e}{3}} \right ) x-4\,d-2\,e \right ) }-{\frac{9\,d\ln \left ({x}^{2}-x+1 \right ) }{32}}+{\frac{13\,d\sqrt{3}}{144}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(x^4+x^2+1)^3,x)

[Out]

1/16*((-7/3*d-4/3*e)*x^3-6*d*x^2+(-20/3*d+1/3*e)*x-4*d+2*e)/(x^2+x+1)^2+9/32*d*l
n(x^2+x+1)+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-1/16*((7/3*d-4/3*e)*x^3-6*d*x^2+(20/3*d+1/3*e)*x-4*d-2*e)/(x^2
-x+1)^2-9/32*d*ln(x^2-x+1)+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

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Maxima [A]  time = 0.775462, size = 185, normalized size = 1. \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{9}{32} \, d \log \left (x^{2} + x + 1\right ) - \frac{9}{32} \, d \log \left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 8 \, e x^{6} + 5 \, d x^{5} - 12 \, e x^{4} + 7 \, d x^{3} - 16 \, e x^{2} - 4 \, d x - 6 \, e}{24 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.277566, size = 387, normalized size = 2.09 \[ \frac{\sqrt{3}{\left (27 \, \sqrt{3}{\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} + x + 1\right ) - 27 \, \sqrt{3}{\left (d x^{8} + 2 \, d x^{6} + 3 \, d x^{4} + 2 \, d x^{2} + d\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (13 \, d - 32 \, e\right )} x^{8} + 2 \,{\left (13 \, d - 32 \, e\right )} x^{6} + 3 \,{\left (13 \, d - 32 \, e\right )} x^{4} + 2 \,{\left (13 \, d - 32 \, e\right )} x^{2} + 13 \, d - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (13 \, d + 32 \, e\right )} x^{8} + 2 \,{\left (13 \, d + 32 \, e\right )} x^{6} + 3 \,{\left (13 \, d + 32 \, e\right )} x^{4} + 2 \,{\left (13 \, d + 32 \, e\right )} x^{2} + 13 \, d + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left (7 \, d x^{7} - 8 \, e x^{6} + 5 \, d x^{5} - 12 \, e x^{4} + 7 \, d x^{3} - 16 \, e x^{2} - 4 \, d x - 6 \, e\right )}\right )}}{288 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/288*sqrt(3)*(27*sqrt(3)*(d*x^8 + 2*d*x^6 + 3*d*x^4 + 2*d*x^2 + d)*log(x^2 + x
+ 1) - 27*sqrt(3)*(d*x^8 + 2*d*x^6 + 3*d*x^4 + 2*d*x^2 + d)*log(x^2 - x + 1) + 2
*((13*d - 32*e)*x^8 + 2*(13*d - 32*e)*x^6 + 3*(13*d - 32*e)*x^4 + 2*(13*d - 32*e
)*x^2 + 13*d - 32*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 2*((13*d + 32*e)*x^8 + 2*(1
3*d + 32*e)*x^6 + 3*(13*d + 32*e)*x^4 + 2*(13*d + 32*e)*x^2 + 13*d + 32*e)*arcta
n(1/3*sqrt(3)*(2*x - 1)) - 4*sqrt(3)*(7*d*x^7 - 8*e*x^6 + 5*d*x^5 - 12*e*x^4 + 7
*d*x^3 - 16*e*x^2 - 4*d*x - 6*e))/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)

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Sympy [A]  time = 9.10324, size = 1103, normalized size = 5.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288)*log(x + (-1025428432*d**4*e - 334752912*
d**4*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288) - 431308800*d**2*e**3 - 3143688192*
d**2*e**2*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288) + 9917005824*d**2*e*(-9*d/32 -
sqrt(3)*I*(13*d + 32*e)/288)**2 + 11878244352*d**2*(-9*d/32 - sqrt(3)*I*(13*d +
32*e)/288)**3 + 142606336*e**5 + 754974720*e**4*(-9*d/32 - sqrt(3)*I*(13*d + 32
*e)/288) + 3850371072*e**3*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288)**2 + 20384317
440*e**2*(-9*d/32 - sqrt(3)*I*(13*d + 32*e)/288)**3)/(217696167*d**5 - 121712844
8*d**3*e**2 - 617611264*d*e**4)) + (-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)*log(x
+ (-1025428432*d**4*e - 334752912*d**4*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)
- 431308800*d**2*e**3 - 3143688192*d**2*e**2*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/
288) + 9917005824*d**2*e*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)**2 + 1187824435
2*d**2*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288)**3 + 142606336*e**5 + 754974720*e
**4*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/288) + 3850371072*e**3*(-9*d/32 + sqrt(3)
*I*(13*d + 32*e)/288)**2 + 20384317440*e**2*(-9*d/32 + sqrt(3)*I*(13*d + 32*e)/2
88)**3)/(217696167*d**5 - 1217128448*d**3*e**2 - 617611264*d*e**4)) + (9*d/32 -
sqrt(3)*I*(13*d - 32*e)/288)*log(x + (-1025428432*d**4*e - 334752912*d**4*(9*d/3
2 - sqrt(3)*I*(13*d - 32*e)/288) - 431308800*d**2*e**3 - 3143688192*d**2*e**2*(9
*d/32 - sqrt(3)*I*(13*d - 32*e)/288) + 9917005824*d**2*e*(9*d/32 - sqrt(3)*I*(13
*d - 32*e)/288)**2 + 11878244352*d**2*(9*d/32 - sqrt(3)*I*(13*d - 32*e)/288)**3
+ 142606336*e**5 + 754974720*e**4*(9*d/32 - sqrt(3)*I*(13*d - 32*e)/288) + 38503
71072*e**3*(9*d/32 - sqrt(3)*I*(13*d - 32*e)/288)**2 + 20384317440*e**2*(9*d/32
- sqrt(3)*I*(13*d - 32*e)/288)**3)/(217696167*d**5 - 1217128448*d**3*e**2 - 6176
11264*d*e**4)) + (9*d/32 + sqrt(3)*I*(13*d - 32*e)/288)*log(x + (-1025428432*d**
4*e - 334752912*d**4*(9*d/32 + sqrt(3)*I*(13*d - 32*e)/288) - 431308800*d**2*e**
3 - 3143688192*d**2*e**2*(9*d/32 + sqrt(3)*I*(13*d - 32*e)/288) + 9917005824*d**
2*e*(9*d/32 + sqrt(3)*I*(13*d - 32*e)/288)**2 + 11878244352*d**2*(9*d/32 + sqrt(
3)*I*(13*d - 32*e)/288)**3 + 142606336*e**5 + 754974720*e**4*(9*d/32 + sqrt(3)*I
*(13*d - 32*e)/288) + 3850371072*e**3*(9*d/32 + sqrt(3)*I*(13*d - 32*e)/288)**2
+ 20384317440*e**2*(9*d/32 + sqrt(3)*I*(13*d - 32*e)/288)**3)/(217696167*d**5 -
1217128448*d**3*e**2 - 617611264*d*e**4)) - (7*d*x**7 + 5*d*x**5 + 7*d*x**3 - 4*
d*x - 8*e*x**6 - 12*e*x**4 - 16*e*x**2 - 6*e)/(24*x**8 + 48*x**6 + 72*x**4 + 48*
x**2 + 24)

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GIAC/XCAS [A]  time = 0.265535, size = 177, normalized size = 0.96 \[ \frac{1}{144} \, \sqrt{3}{\left (13 \, d - 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{144} \, \sqrt{3}{\left (13 \, d + 32 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{9}{32} \, d{\rm ln}\left (x^{2} + x + 1\right ) - \frac{9}{32} \, d{\rm ln}\left (x^{2} - x + 1\right ) - \frac{7 \, d x^{7} - 8 \, x^{6} e + 5 \, d x^{5} - 12 \, x^{4} e + 7 \, d x^{3} - 16 \, x^{2} e - 4 \, d x - 6 \, e}{24 \,{\left (x^{4} + x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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