3.108 \(\int \frac{2 b^{2/3}+x^2}{b^{4/3}+b^{2/3} x^2+x^4} \, dx\)

Optimal. Leaf size=124 \[ -\frac{\log \left (b^{2/3}-\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}+2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0774642, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1169, 634, 617, 204, 628} \[ -\frac{\log \left (b^{2/3}-\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}+2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Int[(2*b^(2/3) + x^2)/(b^(4/3) + b^(2/3)*x^2 + x^4),x]

[Out]

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

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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{2 b^{2/3}+x^2}{b^{4/3}+b^{2/3} x^2+x^4} \, dx &=\frac{\int \frac{2 b-b^{2/3} x}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx}{2 b}+\frac{\int \frac{2 b+b^{2/3} x}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx}{2 b}\\ &=\frac{3}{4} \int \frac{1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx+\frac{3}{4} \int \frac{1}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx-\frac{\int \frac{-\sqrt [3]{b}+2 x}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx}{4 \sqrt [3]{b}}+\frac{\int \frac{\sqrt [3]{b}+2 x}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx}{4 \sqrt [3]{b}}\\ &=-\frac{\log \left (b^{2/3}-\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{b}}\right )}{2 \sqrt [3]{b}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 x}{\sqrt [3]{b}}\right )}{2 \sqrt [3]{b}}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}+2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}}-\frac{\log \left (b^{2/3}-\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}\\ \end{align*}

Mathematica [C]  time = 0.134815, size = 115, normalized size = 0.93 \[ \frac{\sqrt [4]{-1} \left (\sqrt{\sqrt{3}-i} \left (\sqrt{3}-3 i\right ) \tan ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt [3]{b}}\right )-\sqrt{\sqrt{3}+i} \left (\sqrt{3}+3 i\right ) \tanh ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt [3]{b}}\right )\right )}{2 \sqrt{6} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*b^(2/3) + x^2)/(b^(4/3) + b^(2/3)*x^2 + x^4),x]

[Out]

((-1)^(1/4)*(Sqrt[-I + Sqrt[3]]*(-3*I + Sqrt[3])*ArcTan[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*b^(1/3))] - Sqrt[I + Sq
rt[3]]*(3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*b^(1/3))]))/(2*Sqrt[6]*b^(1/3))

________________________________________________________________________________________

Maple [A]  time = 0.055, size = 89, normalized size = 0.7 \begin{align*} -{\frac{1}{4}\ln \left ({b}^{{\frac{2}{3}}}-\sqrt [3]{b}x+{x}^{2} \right ){\frac{1}{\sqrt [3]{b}}}}+{\frac{\sqrt{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ( -\sqrt [3]{b}+2\,x \right ){\frac{1}{\sqrt [3]{b}}}} \right ){\frac{1}{\sqrt [3]{b}}}}+{\frac{1}{4}\ln \left ({b}^{{\frac{2}{3}}}+\sqrt [3]{b}x+{x}^{2} \right ){\frac{1}{\sqrt [3]{b}}}}+{\frac{\sqrt{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ( \sqrt [3]{b}+2\,x \right ){\frac{1}{\sqrt [3]{b}}}} \right ){\frac{1}{\sqrt [3]{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*b^(2/3)+x^2)/(b^(4/3)+b^(2/3)*x^2+x^4),x)

[Out]

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

________________________________________________________________________________________

Maxima [C]  time = 1.5106, size = 170, normalized size = 1.37 \begin{align*} -\frac{i \, \sqrt{3} \log \left (\frac{2 \, x - i \, \sqrt{3} b^{\frac{1}{3}} + b^{\frac{1}{3}}}{2 \, x + i \, \sqrt{3} b^{\frac{1}{3}} + b^{\frac{1}{3}}}\right )}{4 \, b^{\frac{1}{3}}} - \frac{i \, \sqrt{3} \log \left (\frac{2 \, x - i \, \sqrt{3} b^{\frac{1}{3}} - b^{\frac{1}{3}}}{2 \, x + i \, \sqrt{3} b^{\frac{1}{3}} - b^{\frac{1}{3}}}\right )}{4 \, b^{\frac{1}{3}}} + \frac{\log \left (x^{2} + b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} - \frac{\log \left (x^{2} - b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b^(2/3)+x^2)/(b^(4/3)+b^(2/3)*x^2+x^4),x, algorithm="maxima")

[Out]

-1/4*I*sqrt(3)*log((2*x - I*sqrt(3)*b^(1/3) + b^(1/3))/(2*x + I*sqrt(3)*b^(1/3) + b^(1/3)))/b^(1/3) - 1/4*I*sq
rt(3)*log((2*x - I*sqrt(3)*b^(1/3) - b^(1/3))/(2*x + I*sqrt(3)*b^(1/3) - b^(1/3)))/b^(1/3) + 1/4*log(x^2 + b^(
1/3)*x + b^(2/3))/b^(1/3) - 1/4*log(x^2 - b^(1/3)*x + b^(2/3))/b^(1/3)

________________________________________________________________________________________

Fricas [A]  time = 1.84745, size = 768, normalized size = 6.19 \begin{align*} \left [\frac{\sqrt{3} b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{2 \, x^{3} + \sqrt{3}{\left (2 \, b^{\frac{2}{3}} x^{2} + b x - b^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - 3 \, b^{\frac{2}{3}} x - b}{x^{3} + b}\right ) + \sqrt{3} b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{2 \, x^{3} + \sqrt{3}{\left (2 \, b^{\frac{2}{3}} x^{2} - b x - b^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - 3 \, b^{\frac{2}{3}} x + b}{x^{3} - b}\right ) + b^{\frac{2}{3}} \log \left (x^{2} + b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right ) - b^{\frac{2}{3}} \log \left (x^{2} - b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b}, \frac{2 \, \sqrt{3} b^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + b^{\frac{1}{3}}\right )}}{3 \, b^{\frac{1}{3}}}\right ) - 2 \, \sqrt{3} b^{\frac{2}{3}} \arctan \left (-\frac{\sqrt{3}{\left (2 \, x - b^{\frac{1}{3}}\right )}}{3 \, b^{\frac{1}{3}}}\right ) + b^{\frac{2}{3}} \log \left (x^{2} + b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right ) - b^{\frac{2}{3}} \log \left (x^{2} - b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b^(2/3)+x^2)/(b^(4/3)+b^(2/3)*x^2+x^4),x, algorithm="fricas")

[Out]

[1/4*(sqrt(3)*b*sqrt(-1/b^(2/3))*log((2*x^3 + sqrt(3)*(2*b^(2/3)*x^2 + b*x - b^(4/3))*sqrt(-1/b^(2/3)) - 3*b^(
2/3)*x - b)/(x^3 + b)) + sqrt(3)*b*sqrt(-1/b^(2/3))*log((2*x^3 + sqrt(3)*(2*b^(2/3)*x^2 - b*x - b^(4/3))*sqrt(
-1/b^(2/3)) - 3*b^(2/3)*x + b)/(x^3 - b)) + b^(2/3)*log(x^2 + b^(1/3)*x + b^(2/3)) - b^(2/3)*log(x^2 - b^(1/3)
*x + b^(2/3)))/b, 1/4*(2*sqrt(3)*b^(2/3)*arctan(1/3*sqrt(3)*(2*x + b^(1/3))/b^(1/3)) - 2*sqrt(3)*b^(2/3)*arcta
n(-1/3*sqrt(3)*(2*x - b^(1/3))/b^(1/3)) + b^(2/3)*log(x^2 + b^(1/3)*x + b^(2/3)) - b^(2/3)*log(x^2 - b^(1/3)*x
+ b^(2/3)))/b]

________________________________________________________________________________________

Sympy [C]  time = 0.398252, size = 143, normalized size = 1.15 \begin{align*} \frac{\left (- \frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (- \frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) + x \right )} + \left (- \frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (- \frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) + x \right )} + \left (\frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (\frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) + x \right )} + \left (\frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (\frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) + x \right )}}{\sqrt [3]{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b**(2/3)+x**2)/(b**(4/3)+b**(2/3)*x**2+x**4),x)

[Out]

((-1/4 - sqrt(3)*I/4)*log(2*b**(1/3)*(-1/4 - sqrt(3)*I/4) + x) + (-1/4 + sqrt(3)*I/4)*log(2*b**(1/3)*(-1/4 + s
qrt(3)*I/4) + x) + (1/4 - sqrt(3)*I/4)*log(2*b**(1/3)*(1/4 - sqrt(3)*I/4) + x) + (1/4 + sqrt(3)*I/4)*log(2*b**
(1/3)*(1/4 + sqrt(3)*I/4) + x))/b**(1/3)

________________________________________________________________________________________

Giac [A]  time = 1.15824, size = 124, normalized size = 1. \begin{align*} \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + b^{\frac{1}{3}}\right )}}{3 \,{\left | b \right |}^{\frac{1}{3}}}\right )}{2 \,{\left | b \right |}^{\frac{1}{3}}} + \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, x - b^{\frac{1}{3}}\right )}}{3 \,{\left | b \right |}^{\frac{1}{3}}}\right )}{2 \,{\left | b \right |}^{\frac{1}{3}}} + \frac{\log \left (x^{2} + b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} - \frac{\log \left (x^{2} - b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b^(2/3)+x^2)/(b^(4/3)+b^(2/3)*x^2+x^4),x, algorithm="giac")

[Out]

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