3.49 \(\int \frac{1}{1+x^8} \, dx\)

Optimal. Leaf size=339 \[ -\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

[Out]

-ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqrt[2])]) -
ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqrt[2])]) +
ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqrt[2])]) +
ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqrt[2])]) -
(Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/16 + (Sqrt[2 - Sqrt[2]]*
Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2])/16 - (Sqrt[2 + Sqrt[2]]*Log[1 - Sqrt[2 + Sqr
t[2]]*x + x^2])/16 + (Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2])/16

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Rubi [A]  time = 0.680687, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857 \[ -\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^8)^(-1),x]

[Out]

-ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqrt[2])]) -
ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqrt[2])]) +
ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqrt[2])]) +
ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqrt[2])]) -
(Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/16 + (Sqrt[2 - Sqrt[2]]*
Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2])/16 - (Sqrt[2 + Sqrt[2]]*Log[1 - Sqrt[2 + Sqr
t[2]]*x + x^2])/16 + (Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2])/16

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Rubi in Sympy [A]  time = 25.6116, size = 529, normalized size = 1.56 \[ \frac{\sqrt{2} \left (- \frac{\sqrt{2}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \left (- \frac{\sqrt{2}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \log{\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1 \right )}}{8 \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \log{\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1 \right )}}{8 \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \left (- \frac{\left (1 + \sqrt{2}\right ) \sqrt{\sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{\sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \left (- \frac{\left (1 + \sqrt{2}\right ) \sqrt{\sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{\sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \left (\frac{\left (- \sqrt{2} + 1\right ) \sqrt{- \sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{- \sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \left (\frac{\left (- \sqrt{2} + 1\right ) \sqrt{- \sqrt{2} + 2}}{2} + \sqrt{2} \sqrt{- \sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**8+1),x)

[Out]

sqrt(2)*(-sqrt(2)/2 + 1/2)*log(x**2 - x*sqrt(-sqrt(2) + 2) + 1)/(8*sqrt(-sqrt(2)
+ 2)) - sqrt(2)*(-sqrt(2)/2 + 1/2)*log(x**2 + x*sqrt(-sqrt(2) + 2) + 1)/(8*sqrt
(-sqrt(2) + 2)) - sqrt(2)*(1/2 + sqrt(2)/2)*log(x**2 - x*sqrt(sqrt(2) + 2) + 1)/
(8*sqrt(sqrt(2) + 2)) + sqrt(2)*(1/2 + sqrt(2)/2)*log(x**2 + x*sqrt(sqrt(2) + 2)
+ 1)/(8*sqrt(sqrt(2) + 2)) + sqrt(2)*(-(1 + sqrt(2))*sqrt(sqrt(2) + 2)/2 + sqrt
(2)*sqrt(sqrt(2) + 2))*atan((2*x - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2))/(4*sqr
t(-sqrt(2) + 2)*sqrt(sqrt(2) + 2)) + sqrt(2)*(-(1 + sqrt(2))*sqrt(sqrt(2) + 2)/2
+ sqrt(2)*sqrt(sqrt(2) + 2))*atan((2*x + sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2))
/(4*sqrt(-sqrt(2) + 2)*sqrt(sqrt(2) + 2)) + sqrt(2)*((-sqrt(2) + 1)*sqrt(-sqrt(2
) + 2)/2 + sqrt(2)*sqrt(-sqrt(2) + 2))*atan((2*x - sqrt(-sqrt(2) + 2))/sqrt(sqrt
(2) + 2))/(4*sqrt(-sqrt(2) + 2)*sqrt(sqrt(2) + 2)) + sqrt(2)*((-sqrt(2) + 1)*sqr
t(-sqrt(2) + 2)/2 + sqrt(2)*sqrt(-sqrt(2) + 2))*atan((2*x + sqrt(-sqrt(2) + 2))/
sqrt(sqrt(2) + 2))/(4*sqrt(-sqrt(2) + 2)*sqrt(sqrt(2) + 2))

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Mathematica [A]  time = 0.00770615, size = 209, normalized size = 0.62 \[ -\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^8)^(-1),x]

[Out]

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

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Maple [C]  time = 0.01, size = 22, normalized size = 0.1 \[{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^8 + 1), x)

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Fricas [A]  time = 0.232609, size = 1345, normalized size = 3.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*sqrt(2)*(4*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*arctan((sqrt(sqrt(2) +
2) + sqrt(-sqrt(2) + 2))/(2*sqrt(2)*x + 2*sqrt(2)*sqrt(x^2 + 1/2*sqrt(2)*x*sqrt
(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + sqrt(sqrt(2) + 2) - sqrt
(-sqrt(2) + 2))) + 4*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*arctan((sqrt(sqrt(
2) + 2) + sqrt(-sqrt(2) + 2))/(2*sqrt(2)*x + 2*sqrt(2)*sqrt(x^2 - 1/2*sqrt(2)*x*
sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - sqrt(sqrt(2) + 2) +
sqrt(-sqrt(2) + 2))) - 4*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*arctan(-(sqrt(
sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(2*sqrt(2)*x + 2*sqrt(2)*sqrt(x^2 + 1/2*sqrt(
2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + sqrt(sqrt(2) +
2) + sqrt(-sqrt(2) + 2))) - 4*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*arctan(-(
sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(2*sqrt(2)*x + 2*sqrt(2)*sqrt(x^2 - 1/2*
sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - sqrt(sqrt(
2) + 2) - sqrt(-sqrt(2) + 2))) + 4*sqrt(2)*sqrt(sqrt(2) + 2)*arctan(sqrt(sqrt(2)
+ 2)/(2*x + 2*sqrt(x^2 + x*sqrt(-sqrt(2) + 2) + 1) + sqrt(-sqrt(2) + 2))) + 4*s
qrt(2)*sqrt(sqrt(2) + 2)*arctan(sqrt(sqrt(2) + 2)/(2*x + 2*sqrt(x^2 - x*sqrt(-sq
rt(2) + 2) + 1) - sqrt(-sqrt(2) + 2))) + 4*sqrt(2)*sqrt(-sqrt(2) + 2)*arctan(sqr
t(-sqrt(2) + 2)/(2*x + 2*sqrt(x^2 + x*sqrt(sqrt(2) + 2) + 1) + sqrt(sqrt(2) + 2)
)) + 4*sqrt(2)*sqrt(-sqrt(2) + 2)*arctan(sqrt(-sqrt(2) + 2)/(2*x + 2*sqrt(x^2 -
x*sqrt(sqrt(2) + 2) + 1) - sqrt(sqrt(2) + 2))) - (sqrt(sqrt(2) + 2) + sqrt(-sqrt
(2) + 2))*log(x^2 + 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2
) + 2) + 1) - (sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*log(x^2 + 1/2*sqrt(2)*x*s
qrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + (sqrt(sqrt(2) + 2) -
sqrt(-sqrt(2) + 2))*log(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sq
rt(-sqrt(2) + 2) + 1) + (sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*log(x^2 - 1/2*s
qrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - sqrt(2)*sqr
t(sqrt(2) + 2)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) + sqrt(2)*sqrt(sqrt(2) + 2)*lo
g(x^2 - x*sqrt(sqrt(2) + 2) + 1) - sqrt(2)*sqrt(-sqrt(2) + 2)*log(x^2 + x*sqrt(-
sqrt(2) + 2) + 1) + sqrt(2)*sqrt(-sqrt(2) + 2)*log(x^2 - x*sqrt(-sqrt(2) + 2) +
1))

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Sympy [A]  time = 1.93723, size = 14, normalized size = 0.04 \[ \operatorname{RootSum}{\left (16777216 t^{8} + 1, \left ( t \mapsto t \log{\left (8 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**8+1),x)

[Out]

RootSum(16777216*_t**8 + 1, Lambda(_t, _t*log(8*_t + x)))

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GIAC/XCAS [A]  time = 0.20441, size = 323, normalized size = 0.95 \[ \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^8 + 1),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*sq
rt(-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)) + 1/16*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)