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

Optimal. Leaf size=370 \[ -\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.612242, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ -\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.1026, size = 318, normalized size = 0.86 \[ \frac{\sqrt{3} \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{24 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{24 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{24 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{24 \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{12 \sqrt{- \sqrt{3} + 2}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{12 \sqrt{\sqrt{3} + 2}} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(24*sqrt(-sqrt(3) + 2)) - sqrt(3)*l
og(x**2 + x*sqrt(-sqrt(3) + 2) + 1)/(24*sqrt(-sqrt(3) + 2)) - sqrt(3)*log(x**2 -
x*sqrt(sqrt(3) + 2) + 1)/(24*sqrt(sqrt(3) + 2)) + sqrt(3)*log(x**2 + x*sqrt(sqr
t(3) + 2) + 1)/(24*sqrt(sqrt(3) + 2)) - sqrt(3)*atan((2*x - sqrt(sqrt(3) + 2))/s
qrt(-sqrt(3) + 2))/(12*sqrt(-sqrt(3) + 2)) - sqrt(3)*atan((2*x + sqrt(sqrt(3) +
2))/sqrt(-sqrt(3) + 2))/(12*sqrt(-sqrt(3) + 2)) + sqrt(3)*atan((2*x - sqrt(-sqrt
(3) + 2))/sqrt(sqrt(3) + 2))/(12*sqrt(sqrt(3) + 2)) + sqrt(3)*atan((2*x + sqrt(-
sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(12*sqrt(sqrt(3) + 2)) - 1/(3*x**3)

_______________________________________________________________________________________

Mathematica [C]  time = 0.022147, size = 47, normalized size = 0.13 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right ]-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*x^3) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1)/(-1 + 2*#1^4) & ]/4

_______________________________________________________________________________________

Maple [C]  time = 0.014, size = 46, normalized size = 0.1 \[ -{\frac{1}{3\,{x}^{3}}}-{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/x^4/(x^8-x^4+1),x)

[Out]

-1/3/x^3-1/4*sum(_R^4/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{3 \, x^{3}} - \int \frac{x^{4}}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/x^3 - integrate(x^4/(x^8 - x^4 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.29883, size = 1197, normalized size = 3.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(4*sqrt(3)*x^3*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*arctan((sqrt(3)*sqrt(2)
+ 2*sqrt(2))/(2*sqrt(2*x^2 + 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2)*(sqrt
(3) + 2)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2*(sqrt(3)*sqrt(2)*x + 2*sqrt(2)*
x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + sqrt(2))) + 4*sqrt(3)*x^3*sqrt((sqrt(3)
- 2)/(4*sqrt(3) - 7))*arctan((sqrt(3)*sqrt(2) + 2*sqrt(2))/(2*sqrt(2*x^2 - 2*x*
sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2)*(sqrt(3) + 2)*sqrt((sqrt(3) + 2)/(4*sqr
t(3) + 7)) + 2*(sqrt(3)*sqrt(2)*x + 2*sqrt(2)*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) +
7)) - sqrt(2))) - 4*(7*sqrt(3)*x^3 + 12*x^3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)
)*arctan((sqrt(3)*sqrt(2) - 2*sqrt(2))/(2*sqrt(2*x^2 + 2*x*sqrt((sqrt(3) - 2)/(4
*sqrt(3) - 7)) + 2)*(sqrt(3) - 2)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2*(sqrt(
3)*sqrt(2)*x - 2*sqrt(2)*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) - sqrt(2))) - 4*
(7*sqrt(3)*x^3 + 12*x^3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*arctan((sqrt(3)*sqr
t(2) - 2*sqrt(2))/(2*sqrt(2*x^2 - 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2)*(
sqrt(3) - 2)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2*(sqrt(3)*sqrt(2)*x - 2*sqrt
(2)*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + sqrt(2))) + (2*sqrt(3)*x^3 + 3*x^3)
*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*log(2*x^2 + 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(
3) + 7)) + 2) - (2*sqrt(3)*x^3 + 3*x^3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*log(
2*x^2 - 2*x*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2) - (2*sqrt(3)*x^3 + 3*x^3)*s
qrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*log(2*x^2 + 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3)
- 7)) + 2) + (2*sqrt(3)*x^3 + 3*x^3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*log(2*
x^2 - 2*x*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + 2) + 8*(sqrt(3) + 2)*sqrt((sqrt(
3) + 2)/(4*sqrt(3) + 7))*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)))/((sqrt(3)*x^3 + 2*
x^3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)))

_______________________________________________________________________________________

Sympy [A]  time = 4.68959, size = 32, normalized size = 0.09 \[ - \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (- 18432 t^{5} + 4 t + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-18432*_t**5 + 4*_t +
x))) - 1/(3*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.294462, size = 348, normalized size = 0.94 \[ -\frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))
) - 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(
2))) - 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sq
rt(2))) - 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) -
sqrt(2))) - 1/48*(sqrt(6) - 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1)
+ 1/48*(sqrt(6) - 3*sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqr
t(6) + 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) + 1/48*(sqrt(6) + 3*sq
rt(2))*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/3/x^3