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

Optimal. Leaf size=83 \[ -\frac {1}{4} (a-b) \log \left (x^2-x+1\right )+\frac {1}{4} (a-b) \log \left (x^2+x+1\right )-\frac {(a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(a+b) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1169, 634, 618, 204, 628} \begin {gather*} -\frac {1}{4} (a-b) \log \left (x^2-x+1\right )+\frac {1}{4} (a-b) \log \left (x^2+x+1\right )-\frac {(a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(a+b) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(1 + x^2 + x^4),x]

[Out]

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

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 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 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]

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 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]

Rubi steps

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

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Mathematica [C]  time = 0.13, size = 97, normalized size = 1.17 \begin {gather*} \frac {\left (2 i a+\left (\sqrt {3}-i\right ) b\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x\right )}{\sqrt {6+6 i \sqrt {3}}}+\frac {\left (\left (\sqrt {3}+i\right ) b-2 i a\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x\right )}{\sqrt {6-6 i \sqrt {3}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x^2}{1+x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.04, size = 69, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} - x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 69, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} - x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 114, normalized size = 1.37 \begin {gather*} \frac {\sqrt {3}\, a \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\sqrt {3}\, a \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {a \ln \left (x^{2}-x +1\right )}{4}+\frac {a \ln \left (x^{2}+x +1\right )}{4}+\frac {\sqrt {3}\, b \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\sqrt {3}\, b \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {b \ln \left (x^{2}-x +1\right )}{4}-\frac {b \ln \left (x^{2}+x +1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/(x^4+x^2+1),x)

[Out]

1/4*ln(x^2+x+1)*a-1/4*ln(x^2+x+1)*b+1/6*3^(1/2)*arctan(1/3*(2*x+1)*3^(1/2))*a+1/6*3^(1/2)*arctan(1/3*(2*x+1)*3
^(1/2))*b-1/4*ln(x^2-x+1)*a+1/4*ln(x^2-x+1)*b+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*a+1/6*3^(1/2)*arctan(1/3
*(2*x-1)*3^(1/2))*b

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maxima [A]  time = 2.43, size = 69, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} - x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.50, size = 827, normalized size = 9.96

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/(x^2 + x^4 + 1),x)

[Out]

- atan(((x*(4*a*b - 4*a^2 + 2*b^2) + (12*a + 24*x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 -
a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)*1i + (x*(4*a
*b - 4*a^2 + 2*b^2) - (12*a - 24*x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*
a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)*1i)/((x*(4*a*b - 4*a^2 + 2*
b^2) + (12*a + 24*x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(
1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - (x*(4*a*b - 4*a^2 + 2*b^2) - (12*a - 24*
x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b
/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - 2*a*b^2 + 2*a^2*b + 2*b^3))*((a*1i)/2 - (b*1i)/2 + (3^(1/2
)*a)/6 + (3^(1/2)*b)/6) - atan(((x*(4*a*b - 4*a^2 + 2*b^2) + (12*a + 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^
(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2
)*b*1i)/12)*1i + (x*(4*a*b - 4*a^2 + 2*b^2) - (12*a - 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)
)*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)*1i)
/((x*(4*a*b - 4*a^2 + 2*b^2) + (12*a + 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 +
(3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - (x*(4*a*b - 4*a^
2 + 2*b^2) - (12*a - 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12
+ (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - 2*a*b^2 + 2*a^2*b + 2*b^3))*((b*1i
)/2 - (a*1i)/2 + (3^(1/2)*a)/6 + (3^(1/2)*b)/6)

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sympy [C]  time = 1.26, size = 740, normalized size = 8.92 \begin {gather*} \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} + \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} + \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} + \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a/4 + b/4 - sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12) + 6*a**2*b*(-a/4 + b/4
- sqrt(3)*I*(a + b)/12) - 12*a*b**2*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12) + 24*a*(-a/4 + b/4 - sqrt(3)*I*(a + b
)/12)**3 + 2*b**3*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12) - 48*b*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12)**3)/(a**4 - a
**3*b + a*b**3 - b**4)) + (-a/4 + b/4 + sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(-a/4 + b/4 + sqrt(3)*I*(a + b)/
12) + 6*a**2*b*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) - 12*a*b**2*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) + 24*a*(-a/
4 + b/4 + sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) - 48*b*(-a/4 + b/4 + sqrt(3)*I
*(a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4)) + (a/4 - b/4 - sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(a/4 -
b/4 - sqrt(3)*I*(a + b)/12) + 6*a**2*b*(a/4 - b/4 - sqrt(3)*I*(a + b)/12) - 12*a*b**2*(a/4 - b/4 - sqrt(3)*I*(
a + b)/12) + 24*a*(a/4 - b/4 - sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(a/4 - b/4 - sqrt(3)*I*(a + b)/12) - 48*b*(a/
4 - b/4 - sqrt(3)*I*(a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4)) + (a/4 - b/4 + sqrt(3)*I*(a + b)/12)*log(
x + (2*a**3*(a/4 - b/4 + sqrt(3)*I*(a + b)/12) + 6*a**2*b*(a/4 - b/4 + sqrt(3)*I*(a + b)/12) - 12*a*b**2*(a/4
- b/4 + sqrt(3)*I*(a + b)/12) + 24*a*(a/4 - b/4 + sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(a/4 - b/4 + sqrt(3)*I*(a
+ b)/12) - 48*b*(a/4 - b/4 + sqrt(3)*I*(a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4))

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