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

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]*ArcT
an[(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))

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Rubi [A]  time = 0.162047, 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 \[ -\frac{\log \left (b^{2/3}-\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} x+x^2\right )}{4 \sqrt [3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{b}+2 x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 \sqrt [3]{b}} \]

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]*ArcT
an[(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))

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Rubi in Sympy [A]  time = 32.4405, size = 117, normalized size = 0.94 \[ - \frac{\log{\left (b^{\frac{2}{3}} - \sqrt [3]{b} x + x^{2} \right )}}{4 \sqrt [3]{b}} + \frac{\log{\left (b^{\frac{2}{3}} + \sqrt [3]{b} x + x^{2} \right )}}{4 \sqrt [3]{b}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{b}}{3} - \frac{2 x}{3}\right )}{\sqrt [3]{b}} \right )}}{2 \sqrt [3]{b}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{b}}{3} + \frac{2 x}{3}\right )}{\sqrt [3]{b}} \right )}}{2 \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*b**(2/3)+x**2)/(b**(4/3)+b**(2/3)*x**2+x**4),x)

[Out]

-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)) - sqrt(3)*atan(sqrt(3)*(b**(1/3)/3 - 2*x/3)/b**(1/3))/(2*b**(1/
3)) + sqrt(3)*atan(sqrt(3)*(b**(1/3)/3 + 2*x/3)/b**(1/3))/(2*b**(1/3))

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Mathematica [C]  time = 0.22584, 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 + Sq
rt[3]]*b^(1/3))] - Sqrt[I + Sqrt[3]]*(3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sqrt[-
I + Sqrt[3]]*b^(1/3))]))/(2*Sqrt[6]*b^(1/3))

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Maple [A]  time = 0.021, size = 89, normalized size = 0.7 \[{\frac{1}{4}\ln \left ({b}^{{\frac{2}{3}}}+\sqrt [3]{b}x+{x}^{2} \right ){\frac{1}{\sqrt [3]{b}}}}+{\frac{\sqrt{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ( \sqrt [3]{b}+2\,x \right ){\frac{1}{\sqrt [3]{b}}}} \right ){\frac{1}{\sqrt [3]{b}}}}-{\frac{1}{4}\ln \left ({b}^{{\frac{2}{3}}}-\sqrt [3]{b}x+{x}^{2} \right ){\frac{1}{\sqrt [3]{b}}}}+{\frac{\sqrt{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ( -\sqrt [3]{b}+2\,x \right ){\frac{1}{\sqrt [3]{b}}}} \right ){\frac{1}{\sqrt [3]{b}}}} \]

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

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Maxima [A]  time = 0.816896, size = 170, normalized size = 1.37 \[ -\frac{i \, \sqrt{3} \log \left (\frac{2 \, x - i \, \sqrt{3} b^{\frac{1}{3}} + b^{\frac{1}{3}}}{2 \, x + i \, \sqrt{3} b^{\frac{1}{3}} + b^{\frac{1}{3}}}\right )}{4 \, b^{\frac{1}{3}}} - \frac{i \, \sqrt{3} \log \left (\frac{2 \, x - i \, \sqrt{3} b^{\frac{1}{3}} - b^{\frac{1}{3}}}{2 \, x + i \, \sqrt{3} b^{\frac{1}{3}} - b^{\frac{1}{3}}}\right )}{4 \, b^{\frac{1}{3}}} + \frac{\log \left (x^{2} + b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} - \frac{\log \left (x^{2} - b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.323741, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{3} b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{2 \, b^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (2 \, b x + b^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} + 2 \, b^{\frac{2}{3}} x - b}{b^{\frac{1}{3}} x^{2} + b^{\frac{2}{3}} x + b}\right ) + \sqrt{3} b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{2 \, b^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (2 \, b x - b^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - 2 \, b^{\frac{2}{3}} x - b}{b^{\frac{1}{3}} x^{2} - b^{\frac{2}{3}} x + b}\right ) + b^{\frac{2}{3}} \log \left (b^{\frac{1}{3}} x^{2} + b^{\frac{2}{3}} x + b\right ) - b^{\frac{2}{3}} \log \left (b^{\frac{1}{3}} x^{2} - b^{\frac{2}{3}} x + b\right )}{4 \, b}, \frac{2 \, \sqrt{3} b^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}}{3 \, b^{\frac{2}{3}}}\right ) - 2 \, \sqrt{3} b^{\frac{2}{3}} \arctan \left (-\frac{\sqrt{3}{\left (2 \, b^{\frac{1}{3}} x - b^{\frac{2}{3}}\right )}}{3 \, b^{\frac{2}{3}}}\right ) + b^{\frac{2}{3}} \log \left (b^{\frac{1}{3}} x^{2} + b^{\frac{2}{3}} x + b\right ) - b^{\frac{2}{3}} \log \left (b^{\frac{1}{3}} x^{2} - b^{\frac{2}{3}} x + b\right )}{4 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.41341, size = 143, normalized size = 1.15 \[ \frac{\left (- \frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (- \frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) + x \right )} + \left (- \frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (- \frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) + x \right )} + \left (\frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (\frac{1}{4} - \frac{\sqrt{3} i}{4}\right ) + x \right )} + \left (\frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) \log{\left (2 \sqrt [3]{b} \left (\frac{1}{4} + \frac{\sqrt{3} i}{4}\right ) + x \right )}}{\sqrt [3]{b}} \]

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 + sqrt(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)

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GIAC/XCAS [A]  time = 0.281628, size = 124, normalized size = 1. \[ \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + b^{\frac{1}{3}}\right )}}{3 \,{\left | b \right |}^{\frac{1}{3}}}\right )}{2 \,{\left | b \right |}^{\frac{1}{3}}} + \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, x - b^{\frac{1}{3}}\right )}}{3 \,{\left | b \right |}^{\frac{1}{3}}}\right )}{2 \,{\left | b \right |}^{\frac{1}{3}}} + \frac{{\rm ln}\left (x^{2} + b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} - \frac{{\rm ln}\left (x^{2} - b^{\frac{1}{3}} x + b^{\frac{2}{3}}\right )}{4 \, b^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*b^(2/3))/(x^4 + b^(2/3)*x^2 + b^(4/3)),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*ln(x
^2 + b^(1/3)*x + b^(2/3))/b^(1/3) - 1/4*ln(x^2 - b^(1/3)*x + b^(2/3))/b^(1/3)