3.107 \(\int \frac{a+b x^2}{\left (1+x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=119 \[ -\frac{1}{8} (2 a-b) \log \left (x^2-x+1\right )+\frac{1}{8} (2 a-b) \log \left (x^2+x+1\right )+\frac{x \left (x^2 (-(a-2 b))+a+b\right )}{6 \left (x^4+x^2+1\right )}-\frac{(4 a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 a+b) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}} \]

[Out]

(x*(a + b - (a - 2*b)*x^2))/(6*(1 + x^2 + x^4)) - ((4*a + b)*ArcTan[(1 - 2*x)/Sq
rt[3]])/(12*Sqrt[3]) + ((4*a + b)*ArcTan[(1 + 2*x)/Sqrt[3]])/(12*Sqrt[3]) - ((2*
a - b)*Log[1 - x + x^2])/8 + ((2*a - b)*Log[1 + x + x^2])/8

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Rubi [A]  time = 0.188073, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{8} (2 a-b) \log \left (x^2-x+1\right )+\frac{1}{8} (2 a-b) \log \left (x^2+x+1\right )+\frac{x \left (x^2 (-(a-2 b))+a+b\right )}{6 \left (x^4+x^2+1\right )}-\frac{(4 a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 a+b) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(1 + x^2 + x^4)^2,x]

[Out]

(x*(a + b - (a - 2*b)*x^2))/(6*(1 + x^2 + x^4)) - ((4*a + b)*ArcTan[(1 - 2*x)/Sq
rt[3]])/(12*Sqrt[3]) + ((4*a + b)*ArcTan[(1 + 2*x)/Sqrt[3]])/(12*Sqrt[3]) - ((2*
a - b)*Log[1 - x + x^2])/8 + ((2*a - b)*Log[1 + x + x^2])/8

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Rubi in Sympy [A]  time = 30.9515, size = 109, normalized size = 0.92 \[ \frac{x \left (a + b - x^{2} \left (a - 2 b\right )\right )}{6 \left (x^{4} + x^{2} + 1\right )} - \left (\frac{a}{4} - \frac{b}{8}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{a}{4} - \frac{b}{8}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (4 a + b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{36} + \frac{\sqrt{3} \left (4 a + b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/(x**4+x**2+1)**2,x)

[Out]

x*(a + b - x**2*(a - 2*b))/(6*(x**4 + x**2 + 1)) - (a/4 - b/8)*log(x**2 - x + 1)
+ (a/4 - b/8)*log(x**2 + x + 1) + sqrt(3)*(4*a + b)*atan(sqrt(3)*(2*x/3 - 1/3))
/36 + sqrt(3)*(4*a + b)*atan(sqrt(3)*(2*x/3 + 1/3))/36

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Mathematica [C]  time = 0.412382, size = 147, normalized size = 1.24 \[ \frac{x \left (-a x^2+a+2 b x^2+b\right )}{6 \left (x^4+x^2+1\right )}-\frac{\left (\left (\sqrt{3}-11 i\right ) a-2 \left (\sqrt{3}-2 i\right ) b\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{6 \sqrt{6+6 i \sqrt{3}}}-\frac{\left (\left (\sqrt{3}+11 i\right ) a-2 \left (\sqrt{3}+2 i\right ) b\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{6 \sqrt{6-6 i \sqrt{3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^2)/(1 + x^2 + x^4)^2,x]

[Out]

(x*(a + b - a*x^2 + 2*b*x^2))/(6*(1 + x^2 + x^4)) - (((-11*I + Sqrt[3])*a - 2*(-
2*I + Sqrt[3])*b)*ArcTan[((-I + Sqrt[3])*x)/2])/(6*Sqrt[6 + (6*I)*Sqrt[3]]) - ((
(11*I + Sqrt[3])*a - 2*(2*I + Sqrt[3])*b)*ArcTan[((I + Sqrt[3])*x)/2])/(6*Sqrt[6
- (6*I)*Sqrt[3]])

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Maple [A]  time = 0.016, size = 168, normalized size = 1.4 \[{\frac{1}{4\,{x}^{2}+4\,x+4} \left ( \left ( -{\frac{a}{3}}+{\frac{2\,b}{3}} \right ) x-{\frac{2\,a}{3}}+{\frac{b}{3}} \right ) }+{\frac{\ln \left ({x}^{2}+x+1 \right ) a}{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) b}{8}}+{\frac{\sqrt{3}a}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}b}{36}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{2}-4\,x+4} \left ( \left ({\frac{a}{3}}-{\frac{2\,b}{3}} \right ) x-{\frac{2\,a}{3}}+{\frac{b}{3}} \right ) }-{\frac{\ln \left ({x}^{2}-x+1 \right ) a}{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) b}{8}}+{\frac{\sqrt{3}a}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}b}{36}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(x^4+x^2+1)^2,x)

[Out]

1/4*((-1/3*a+2/3*b)*x-2/3*a+1/3*b)/(x^2+x+1)+1/4*ln(x^2+x+1)*a-1/8*ln(x^2+x+1)*b
+1/9*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*a+1/36*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/
2))*b-1/4*((1/3*a-2/3*b)*x-2/3*a+1/3*b)/(x^2-x+1)-1/4*ln(x^2-x+1)*a+1/8*ln(x^2-x
+1)*b+1/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*a+1/36*3^(1/2)*arctan(1/3*(2*x-1)*
3^(1/2))*b

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Maxima [A]  time = 0.842297, size = 142, normalized size = 1.19 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) - \frac{{\left (a - 2 \, b\right )} x^{3} -{\left (a + b\right )} x}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(x^4 + x^2 + 1)^2,x, algorithm="maxima")

[Out]

1/36*sqrt(3)*(4*a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/36*sqrt(3)*(4*a + b)*ar
ctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*(2*a - b)*log(x^2 + x + 1) - 1/8*(2*a - b)*log
(x^2 - x + 1) - 1/6*((a - 2*b)*x^3 - (a + b)*x)/(x^4 + x^2 + 1)

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Fricas [A]  time = 0.298028, size = 261, normalized size = 2.19 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left ({\left (2 \, a - b\right )} x^{4} +{\left (2 \, a - b\right )} x^{2} + 2 \, a - b\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left ({\left (2 \, a - b\right )} x^{4} +{\left (2 \, a - b\right )} x^{2} + 2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (4 \, a + b\right )} x^{4} +{\left (4 \, a + b\right )} x^{2} + 4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (4 \, a + b\right )} x^{4} +{\left (4 \, a + b\right )} x^{2} + 4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left ({\left (a - 2 \, b\right )} x^{3} -{\left (a + b\right )} x\right )}\right )}}{72 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(x^4 + x^2 + 1)^2,x, algorithm="fricas")

[Out]

1/72*sqrt(3)*(3*sqrt(3)*((2*a - b)*x^4 + (2*a - b)*x^2 + 2*a - b)*log(x^2 + x +
1) - 3*sqrt(3)*((2*a - b)*x^4 + (2*a - b)*x^2 + 2*a - b)*log(x^2 - x + 1) + 2*((
4*a + b)*x^4 + (4*a + b)*x^2 + 4*a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 2*((4*a
+ b)*x^4 + (4*a + b)*x^2 + 4*a + b)*arctan(1/3*sqrt(3)*(2*x - 1)) - 4*sqrt(3)*((
a - 2*b)*x^3 - (a + b)*x))/(x^4 + x^2 + 1)

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Sympy [A]  time = 4.76565, size = 876, normalized size = 7.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/(x**4+x**2+1)**2,x)

[Out]

-(x**3*(a - 2*b) + x*(-a - b))/(6*x**4 + 6*x**2 + 6) + (-a/4 + b/8 - sqrt(3)*I*(
4*a + b)/72)*log(x + (76*a**3*(-a/4 + b/8 - sqrt(3)*I*(4*a + b)/72) + 948*a**2*b
*(-a/4 + b/8 - sqrt(3)*I*(4*a + b)/72) - 816*a*b**2*(-a/4 + b/8 - sqrt(3)*I*(4*a
+ b)/72) + 12096*a*(-a/4 + b/8 - sqrt(3)*I*(4*a + b)/72)**3 + 148*b**3*(-a/4 +
b/8 - sqrt(3)*I*(4*a + b)/72) - 8640*b*(-a/4 + b/8 - sqrt(3)*I*(4*a + b)/72)**3)
/(248*a**4 - 262*a**3*b + 75*a**2*b**2 + 11*a*b**3 - 7*b**4)) + (-a/4 + b/8 + sq
rt(3)*I*(4*a + b)/72)*log(x + (76*a**3*(-a/4 + b/8 + sqrt(3)*I*(4*a + b)/72) + 9
48*a**2*b*(-a/4 + b/8 + sqrt(3)*I*(4*a + b)/72) - 816*a*b**2*(-a/4 + b/8 + sqrt(
3)*I*(4*a + b)/72) + 12096*a*(-a/4 + b/8 + sqrt(3)*I*(4*a + b)/72)**3 + 148*b**3
*(-a/4 + b/8 + sqrt(3)*I*(4*a + b)/72) - 8640*b*(-a/4 + b/8 + sqrt(3)*I*(4*a + b
)/72)**3)/(248*a**4 - 262*a**3*b + 75*a**2*b**2 + 11*a*b**3 - 7*b**4)) + (a/4 -
b/8 - sqrt(3)*I*(4*a + b)/72)*log(x + (76*a**3*(a/4 - b/8 - sqrt(3)*I*(4*a + b)/
72) + 948*a**2*b*(a/4 - b/8 - sqrt(3)*I*(4*a + b)/72) - 816*a*b**2*(a/4 - b/8 -
sqrt(3)*I*(4*a + b)/72) + 12096*a*(a/4 - b/8 - sqrt(3)*I*(4*a + b)/72)**3 + 148*
b**3*(a/4 - b/8 - sqrt(3)*I*(4*a + b)/72) - 8640*b*(a/4 - b/8 - sqrt(3)*I*(4*a +
b)/72)**3)/(248*a**4 - 262*a**3*b + 75*a**2*b**2 + 11*a*b**3 - 7*b**4)) + (a/4
- b/8 + sqrt(3)*I*(4*a + b)/72)*log(x + (76*a**3*(a/4 - b/8 + sqrt(3)*I*(4*a + b
)/72) + 948*a**2*b*(a/4 - b/8 + sqrt(3)*I*(4*a + b)/72) - 816*a*b**2*(a/4 - b/8
+ sqrt(3)*I*(4*a + b)/72) + 12096*a*(a/4 - b/8 + sqrt(3)*I*(4*a + b)/72)**3 + 14
8*b**3*(a/4 - b/8 + sqrt(3)*I*(4*a + b)/72) - 8640*b*(a/4 - b/8 + sqrt(3)*I*(4*a
+ b)/72)**3)/(248*a**4 - 262*a**3*b + 75*a**2*b**2 + 11*a*b**3 - 7*b**4))

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GIAC/XCAS [A]  time = 0.271991, size = 147, normalized size = 1.24 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, a - b\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, a - b\right )}{\rm ln}\left (x^{2} - x + 1\right ) - \frac{a x^{3} - 2 \, b x^{3} - a x - b x}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(x^4 + x^2 + 1)^2,x, algorithm="giac")

[Out]

1/36*sqrt(3)*(4*a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/36*sqrt(3)*(4*a + b)*ar
ctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*(2*a - b)*ln(x^2 + x + 1) - 1/8*(2*a - b)*ln(x
^2 - x + 1) - 1/6*(a*x^3 - 2*b*x^3 - a*x - b*x)/(x^4 + x^2 + 1)