3.59 \(\int \frac{1-x^4}{x^3 \left (1-x^4+x^8\right )} \, dx\)

Optimal. Leaf size=89 \[ -\frac{1}{2 x^2}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )-\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]

[Out]

-1/(2*x^2) + ArcTan[Sqrt[3] - 2*x^2]/4 - ArcTan[Sqrt[3] + 2*x^2]/4 - Log[1 - Sqr
t[3]*x^2 + x^4]/(8*Sqrt[3]) + Log[1 + Sqrt[3]*x^2 + x^4]/(8*Sqrt[3])

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Rubi [A]  time = 0.188991, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{1}{2 x^2}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )-\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^4)/(x^3*(1 - x^4 + x^8)),x]

[Out]

-1/(2*x^2) + ArcTan[Sqrt[3] - 2*x^2]/4 - ArcTan[Sqrt[3] + 2*x^2]/4 - Log[1 - Sqr
t[3]*x^2 + x^4]/(8*Sqrt[3]) + Log[1 + Sqrt[3]*x^2 + x^4]/(8*Sqrt[3])

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Rubi in Sympy [A]  time = 40.0168, size = 76, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} - \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+1)/x**3/(x**8-x**4+1),x)

[Out]

-sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/24 + sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/
24 - atan(2*x**2 - sqrt(3))/4 - atan(2*x**2 + sqrt(3))/4 - 1/(2*x**2)

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Mathematica [C]  time = 0.0242982, size = 49, normalized size = 0.55 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right ]-\frac{1}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^4)/(x^3*(1 - x^4 + x^8)),x]

[Out]

-1/(2*x^2) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^2)/(-1 + 2*#1^4) & ]/4

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Maple [A]  time = 0.007, size = 70, normalized size = 0.8 \[ -{\frac{1}{2\,{x}^{2}}}-{\frac{\arctan \left ( 2\,{x}^{2}-\sqrt{3} \right ) }{4}}-{\frac{\arctan \left ( 2\,{x}^{2}+\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/x^3/(x^8-x^4+1),x)

[Out]

-1/2/x^2-1/4*arctan(2*x^2-3^(1/2))-1/4*arctan(2*x^2+3^(1/2))-1/24*ln(1+x^4-x^2*3
^(1/2))*3^(1/2)+1/24*ln(1+x^4+x^2*3^(1/2))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2 \, x^{2}} - \int \frac{x^{5}}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/((x^8 - x^4 + 1)*x^3),x, algorithm="maxima")

[Out]

-1/2/x^2 - integrate(x^5/(x^8 - x^4 + 1), x)

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Fricas [A]  time = 0.267569, size = 188, normalized size = 2.11 \[ \frac{\sqrt{3}{\left (4 \, \sqrt{3} x^{2} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} + \sqrt{3} x^{2} + 1} + 3}\right ) + 4 \, \sqrt{3} x^{2} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} - \sqrt{3} x^{2} + 1} - 3}\right ) + x^{2} \log \left (x^{4} + \sqrt{3} x^{2} + 1\right ) - x^{2} \log \left (x^{4} - \sqrt{3} x^{2} + 1\right ) - 4 \, \sqrt{3}\right )}}{24 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/((x^8 - x^4 + 1)*x^3),x, algorithm="fricas")

[Out]

1/24*sqrt(3)*(4*sqrt(3)*x^2*arctan(sqrt(3)/(2*sqrt(3)*x^2 + 2*sqrt(3)*sqrt(x^4 +
sqrt(3)*x^2 + 1) + 3)) + 4*sqrt(3)*x^2*arctan(sqrt(3)/(2*sqrt(3)*x^2 + 2*sqrt(3
)*sqrt(x^4 - sqrt(3)*x^2 + 1) - 3)) + x^2*log(x^4 + sqrt(3)*x^2 + 1) - x^2*log(x
^4 - sqrt(3)*x^2 + 1) - 4*sqrt(3))/x^2

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Sympy [A]  time = 0.698227, size = 76, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} - \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+1)/x**3/(x**8-x**4+1),x)

[Out]

-sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/24 + sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/
24 - atan(2*x**2 - sqrt(3))/4 - atan(2*x**2 + sqrt(3))/4 - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.348744, size = 348, normalized size = 3.91 \[ -\frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/((x^8 - x^4 + 1)*x^3),x, algorithm="giac")

[Out]

-1/48*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))
) - 1/48*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(
2))) - 1/48*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sq
rt(2))) - 1/48*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) -
sqrt(2))) - 1/96*(sqrt(6) - 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1)
+ 1/96*(sqrt(6) - 3*sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/96*(sqr
t(6) + 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) + 1/96*(sqrt(6) + 3*sq
rt(2))*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/2/x^2