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

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

[Out]

(x*(a + b - (a - 2*b)*x^2))/(6*(1 + x^2 + x^4)) - ((4*a + b)*ArcTan[(1 - 2*x)/Sqrt[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 = 0.0910057, 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, Rules used = {1178, 1169, 634, 618, 204, 628} \[ \frac{x \left (x^2 (-(a-2 b))+a+b\right )}{6 \left (x^4+x^2+1\right )}-\frac{1}{8} (2 a-b) \log \left (x^2-x+1\right )+\frac{1}{8} (2 a-b) \log \left (x^2+x+1\right )-\frac{(4 a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 a+b) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}} \]

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)/Sqrt[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

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{a+b x^2}{\left (1+x^2+x^4\right )^2} \, dx &=\frac{x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{6} \int \frac{5 a-b+(-a+2 b) x^2}{1+x^2+x^4} \, dx\\ &=\frac{x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{12} \int \frac{5 a-b-(6 a-3 b) x}{1-x+x^2} \, dx+\frac{1}{12} \int \frac{5 a-b+(6 a-3 b) x}{1+x+x^2} \, dx\\ &=\frac{x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac{1}{8} (2 a-b) \int \frac{1+2 x}{1+x+x^2} \, dx+\frac{1}{8} (-2 a+b) \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{1}{24} (4 a+b) \int \frac{1}{1-x+x^2} \, dx+\frac{1}{24} (4 a+b) \int \frac{1}{1+x+x^2} \, dx\\ &=\frac{x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac{1}{8} (2 a-b) \log \left (1-x+x^2\right )+\frac{1}{8} (2 a-b) \log \left (1+x+x^2\right )+\frac{1}{12} (-4 a-b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{12} (-4 a-b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac{x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\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{1+2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}-\frac{1}{8} (2 a-b) \log \left (1-x+x^2\right )+\frac{1}{8} (2 a-b) \log \left (1+x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.243995, 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 = 0.056, size = 168, normalized size = 1.4 \begin{align*} -{\frac{1}{4\,{x}^{2}-4\,x+4} \left ( \left ( -{\frac{2\,b}{3}}+{\frac{a}{3}} \right ) x+{\frac{b}{3}}-{\frac{2\,a}{3}} \right ) }-{\frac{\ln \left ({x}^{2}-x+1 \right ) a}{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) b}{8}}+{\frac{a\sqrt{3}}{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 ) }+{\frac{1}{4\,{x}^{2}+4\,x+4} \left ( \left ({\frac{2\,b}{3}}-{\frac{a}{3}} \right ) x+{\frac{b}{3}}-{\frac{2\,a}{3}} \right ) }+{\frac{\ln \left ({x}^{2}+x+1 \right ) a}{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) b}{8}}+{\frac{a\sqrt{3}}{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 ) } \end{align*}

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*((-2/3*b+1/3*a)*x+1/3*b-2/3*a)/(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+1/4*((2/3*b-1/3*a)*x+1/3*b-2/3*a)/(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

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Maxima [A]  time = 1.46076, size = 142, normalized size = 1.19 \begin{align*} \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 )}} \end{align*}

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)*arctan(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 = 1.38824, size = 474, normalized size = 3.98 \begin{align*} -\frac{12 \,{\left (a - 2 \, b\right )} x^{3} - 2 \, \sqrt{3}{\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 \, \sqrt{3}{\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 ) - 12 \,{\left (a + b\right )} x - 9 \,{\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 ) + 9 \,{\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 )}{72 \,{\left (x^{4} + x^{2} + 1\right )}} \end{align*}

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*(12*(a - 2*b)*x^3 - 2*sqrt(3)*((4*a + b)*x^4 + (4*a + b)*x^2 + 4*a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) -
2*sqrt(3)*((4*a + b)*x^4 + (4*a + b)*x^2 + 4*a + b)*arctan(1/3*sqrt(3)*(2*x - 1)) - 12*(a + b)*x - 9*((2*a - b
)*x^4 + (2*a - b)*x^2 + 2*a - b)*log(x^2 + x + 1) + 9*((2*a - b)*x^4 + (2*a - b)*x^2 + 2*a - b)*log(x^2 - x +
1))/(x^4 + x^2 + 1)

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Sympy [C]  time = 1.41187, size = 876, normalized size = 7.36 \begin{align*} \text{result too large to display} \end{align*}

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 + 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 + 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))

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Giac [A]  time = 1.13813, size = 147, normalized size = 1.24 \begin{align*} \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{a x^{3} - 2 \, b x^{3} - a x - b x}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \end{align*}

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