3.1505 \(\int \frac{1}{x^6 \left (1+x^8\right )} \, dx\)

Optimal. Leaf size=346 \[ -\frac{1}{5 x^5}-\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right ) \]

[Out]

-1/(5*x^5) - (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2
]]])/8 + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]])
/8 + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]])/8 -
(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]])/8 - Log
[1 - Sqrt[2 - Sqrt[2]]*x + x^2]/(8*Sqrt[2*(2 - Sqrt[2])]) + Log[1 + Sqrt[2 - Sqr
t[2]]*x + x^2]/(8*Sqrt[2*(2 - Sqrt[2])]) + Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2]/(8
*Sqrt[2*(2 + Sqrt[2])]) - Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2]/(8*Sqrt[2*(2 + Sqrt
[2])])

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Rubi [A]  time = 0.522588, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{1}{5 x^5}-\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 + x^8)),x]

[Out]

-1/(5*x^5) - (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2
]]])/8 + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]])
/8 + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]])/8 -
(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]])/8 - Log
[1 - Sqrt[2 - Sqrt[2]]*x + x^2]/(8*Sqrt[2*(2 - Sqrt[2])]) + Log[1 + Sqrt[2 - Sqr
t[2]]*x + x^2]/(8*Sqrt[2*(2 - Sqrt[2])]) + Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2]/(8
*Sqrt[2*(2 + Sqrt[2])]) - Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2]/(8*Sqrt[2*(2 + Sqrt
[2])])

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Rubi in Sympy [A]  time = 39.6639, size = 318, normalized size = 0.92 \[ - \frac{\sqrt{2} \log{\left (x^{2} - x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{16 \sqrt{- \sqrt{2} + 2}} + \frac{\sqrt{2} \log{\left (x^{2} + x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{16 \sqrt{- \sqrt{2} + 2}} + \frac{\sqrt{2} \log{\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1 \right )}}{16 \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \log{\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1 \right )}}{16 \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{8 \sqrt{- \sqrt{2} + 2}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{8 \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{8 \sqrt{\sqrt{2} + 2}} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**8+1),x)

[Out]

-sqrt(2)*log(x**2 - x*sqrt(-sqrt(2) + 2) + 1)/(16*sqrt(-sqrt(2) + 2)) + sqrt(2)*
log(x**2 + x*sqrt(-sqrt(2) + 2) + 1)/(16*sqrt(-sqrt(2) + 2)) + sqrt(2)*log(x**2
- x*sqrt(sqrt(2) + 2) + 1)/(16*sqrt(sqrt(2) + 2)) - sqrt(2)*log(x**2 + x*sqrt(sq
rt(2) + 2) + 1)/(16*sqrt(sqrt(2) + 2)) - sqrt(2)*atan((2*x - sqrt(sqrt(2) + 2))/
sqrt(-sqrt(2) + 2))/(8*sqrt(-sqrt(2) + 2)) - sqrt(2)*atan((2*x + sqrt(sqrt(2) +
2))/sqrt(-sqrt(2) + 2))/(8*sqrt(-sqrt(2) + 2)) + sqrt(2)*atan((2*x - sqrt(-sqrt(
2) + 2))/sqrt(sqrt(2) + 2))/(8*sqrt(sqrt(2) + 2)) + sqrt(2)*atan((2*x + sqrt(-sq
rt(2) + 2))/sqrt(sqrt(2) + 2))/(8*sqrt(sqrt(2) + 2)) - 1/(5*x**5)

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Mathematica [A]  time = 0.0116317, size = 216, normalized size = 0.62 \[ -\frac{1}{5 x^5}-\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 + x^8)),x]

[Out]

-1/(5*x^5) - (ArcTan[(x - Cos[Pi/8])*Csc[Pi/8]]*Cos[Pi/8])/4 - (ArcTan[(x + Cos[
Pi/8])*Csc[Pi/8]]*Cos[Pi/8])/4 - (Cos[Pi/8]*Log[1 + x^2 - 2*x*Sin[Pi/8]])/8 + (C
os[Pi/8]*Log[1 + x^2 + 2*x*Sin[Pi/8]])/8 + (ArcTan[Sec[Pi/8]*(x - Sin[Pi/8])]*Si
n[Pi/8])/4 + (ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])]*Sin[Pi/8])/4 + (Log[1 + x^2 - 2*
x*Cos[Pi/8]]*Sin[Pi/8])/8 - (Log[1 + x^2 + 2*x*Cos[Pi/8]]*Sin[Pi/8])/8

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Maple [C]  time = 0.008, size = 28, normalized size = 0.1 \[ -{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}}}}-{\frac{1}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^8+1),x)

[Out]

-1/8*sum(1/_R^5*ln(x-_R),_R=RootOf(_Z^8+1))-1/5/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^6),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254584, size = 3472, normalized size = 10.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^6),x, algorithm="fricas")

[Out]

-1/320*sqrt(2)*(20*sqrt(2)*x^5*sqrt(sqrt(2) + 2)*arctan(-((sqrt(2) + 2)^(3/2) +
3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2))/(3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(
2) + 2)^(3/2) + 8*x + sqrt((sqrt(2) + 2)^3 - 3*(sqrt(2) + 2)^2*(sqrt(2) - 2) + 3
*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2) - 2)^3 + 48*x*(sqrt(2) + 2)*sqrt(-sqrt
(2) + 2) - 16*x*(-sqrt(2) + 2)^(3/2) + 64*x^2))) + 20*sqrt(2)*x^5*sqrt(sqrt(2) +
2)*arctan(((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2))/(3*(sqrt(2)
+ 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2) - 8*x - sqrt((sqrt(2) + 2)^3 - 3
*(sqrt(2) + 2)^2*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2) - 2)
^3 - 48*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 16*x*(-sqrt(2) + 2)^(3/2) + 64*x^2)
)) - 20*sqrt(2)*x^5*sqrt(-sqrt(2) + 2)*arctan(-(3*(sqrt(2) + 2)*sqrt(-sqrt(2) +
2) - (-sqrt(2) + 2)^(3/2))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) -
2) + 8*x + sqrt((sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2) -
2)^3 + 16*x*(sqrt(2) + 2)^(3/2) + 64*x^2 - 3*((sqrt(2) + 2)^2 - 16*x*sqrt(sqrt(2
) + 2))*(sqrt(2) - 2)))) - 20*sqrt(2)*x^5*sqrt(-sqrt(2) + 2)*arctan((3*(sqrt(2)
+ 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sq
rt(2) + 2)*(sqrt(2) - 2) - 8*x - sqrt((sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2)
- 2)^2 - (sqrt(2) - 2)^3 - 16*x*(sqrt(2) + 2)^(3/2) + 64*x^2 - 3*((sqrt(2) + 2)
^2 + 16*x*sqrt(sqrt(2) + 2))*(sqrt(2) - 2)))) + 5*sqrt(2)*x^5*sqrt(-sqrt(2) + 2)
*log(1/64*(sqrt(2) + 2)^3 - 3/64*(sqrt(2) + 2)^2*(sqrt(2) - 2) + 3/64*(sqrt(2) +
2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 3/4*x*(sqrt(2) + 2)*sqrt(-sqrt(2) +
2) - 1/4*x*(-sqrt(2) + 2)^(3/2) + x^2) - 5*sqrt(2)*x^5*sqrt(-sqrt(2) + 2)*log(1
/64*(sqrt(2) + 2)^3 - 3/64*(sqrt(2) + 2)^2*(sqrt(2) - 2) + 3/64*(sqrt(2) + 2)*(s
qrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 - 3/4*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) +
1/4*x*(-sqrt(2) + 2)^(3/2) + x^2) - 5*sqrt(2)*x^5*sqrt(sqrt(2) + 2)*log(1/64*(sq
rt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 1/4*x
*(sqrt(2) + 2)^(3/2) + x^2 - 3/64*((sqrt(2) + 2)^2 - 16*x*sqrt(sqrt(2) + 2))*(sq
rt(2) - 2)) + 5*sqrt(2)*x^5*sqrt(sqrt(2) + 2)*log(1/64*(sqrt(2) + 2)^3 + 3/64*(s
qrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 - 1/4*x*(sqrt(2) + 2)^(3/2) +
x^2 - 3/64*((sqrt(2) + 2)^2 + 16*x*sqrt(sqrt(2) + 2))*(sqrt(2) - 2)) - 20*(x^5*
sqrt(sqrt(2) + 2) - x^5*sqrt(-sqrt(2) + 2))*arctan(-((sqrt(2) + 2)^(3/2) + 3*sqr
t(sqrt(2) + 2)*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) +
2)^(3/2))/(8*sqrt(2)*x + (sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2)
- 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + (-sqrt(2) + 2)^(3/2) + sqrt(2)*sqrt(8*sq
rt(2)*x*(sqrt(2) + 2)^(3/2) + (sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2
- (sqrt(2) - 2)^3 - 24*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 8*sqrt(2)*x*
(-sqrt(2) + 2)^(3/2) + 64*x^2 + 3*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2) + 2)
^2)*(sqrt(2) - 2)))) - 20*(x^5*sqrt(sqrt(2) + 2) - x^5*sqrt(-sqrt(2) + 2))*arcta
n(-((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*sq
rt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2))/(8*sqrt(2)*x - (sqrt(2) + 2)^(3/2) - 3*
sqrt(sqrt(2) + 2)*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2)
+ 2)^(3/2) + 8*sqrt(2)*sqrt(-1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2)
+ 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 3/8*sqrt(2)
*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 -
3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2) + 2)^2)*(sqrt(2) - 2)))) + 20*(x
^5*sqrt(sqrt(2) + 2) + x^5*sqrt(-sqrt(2) + 2))*arctan(((sqrt(2) + 2)^(3/2) + 3*s
qrt(sqrt(2) + 2)*(sqrt(2) - 2) - 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + (-sqrt(2)
+ 2)^(3/2))/(8*sqrt(2)*x + (sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) -
2) + 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2) + sqrt(2)*sqrt(8*
sqrt(2)*x*(sqrt(2) + 2)^(3/2) + (sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^
2 - (sqrt(2) - 2)^3 + 24*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - 8*sqrt(2)*
x*(-sqrt(2) + 2)^(3/2) + 64*x^2 + 3*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2) +
2)^2)*(sqrt(2) - 2)))) + 20*(x^5*sqrt(sqrt(2) + 2) + x^5*sqrt(-sqrt(2) + 2))*arc
tan(((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 3*(sqrt(2) + 2)*s
qrt(-sqrt(2) + 2) + (-sqrt(2) + 2)^(3/2))/(8*sqrt(2)*x - (sqrt(2) + 2)^(3/2) - 3
*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + (-sqrt(2
) + 2)^(3/2) + 8*sqrt(2)*sqrt(-1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2)
+ 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 - 3/8*sqrt(2
)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2
- 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2) + 2)^2)*(sqrt(2) - 2)))) + 5*(x
^5*sqrt(sqrt(2) + 2) - x^5*sqrt(-sqrt(2) + 2))*log(1/8*sqrt(2)*x*(sqrt(2) + 2)^(
3/2) + 1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2)
- 2)^3 + 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - 1/8*sqrt(2)*x*(-sqrt(
2) + 2)^(3/2) + x^2 + 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2) + 2)^2)*(sq
rt(2) - 2)) + 5*(x^5*sqrt(sqrt(2) + 2) + x^5*sqrt(-sqrt(2) + 2))*log(1/8*sqrt(2)
*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)
^2 - 1/64*(sqrt(2) - 2)^3 - 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 1/8
*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 + 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) - (s
qrt(2) + 2)^2)*(sqrt(2) - 2)) - 5*(x^5*sqrt(sqrt(2) + 2) + x^5*sqrt(-sqrt(2) + 2
))*log(-1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2)
+ 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt(
-sqrt(2) + 2) - 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 - 3/64*(8*sqrt(2)*x*sqr
t(sqrt(2) + 2) + (sqrt(2) + 2)^2)*(sqrt(2) - 2)) - 5*(x^5*sqrt(sqrt(2) + 2) - x^
5*sqrt(-sqrt(2) + 2))*log(-1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2) + 2
)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 - 3/8*sqrt(2)*x*
(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 - 3/
64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2) + 2)^2)*(sqrt(2) - 2)) + 32*sqrt(2)
)/x^5

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Sympy [A]  time = 4.45463, size = 22, normalized size = 0.06 \[ \operatorname{RootSum}{\left (16777216 t^{8} + 1, \left ( t \mapsto t \log{\left (512 t^{3} + x \right )} \right )\right )} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**8+1),x)

[Out]

RootSum(16777216*_t**8 + 1, Lambda(_t, _t*log(512*_t**3 + x))) - 1/(5*x**5)

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GIAC/XCAS [A]  time = 0.255283, size = 329, normalized size = 0.95 \[ \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^6),x, algorithm="giac")

[Out]

1/8*sqrt(-sqrt(2) + 2)*arctan((2*x + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 1/
8*sqrt(-sqrt(2) + 2)*arctan((2*x - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) - 1/8*
sqrt(sqrt(2) + 2)*arctan((2*x + sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 1/8*sqr
t(sqrt(2) + 2)*arctan((2*x - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 1/16*sqrt(
-sqrt(2) + 2)*ln(x^2 + x*sqrt(sqrt(2) + 2) + 1) + 1/16*sqrt(-sqrt(2) + 2)*ln(x^2
- x*sqrt(sqrt(2) + 2) + 1) + 1/16*sqrt(sqrt(2) + 2)*ln(x^2 + x*sqrt(-sqrt(2) +
2) + 1) - 1/16*sqrt(sqrt(2) + 2)*ln(x^2 - x*sqrt(-sqrt(2) + 2) + 1) - 1/5/x^5