3.1503 \(\int \frac{1}{x^2 \left (1+x^8\right )} \, dx\)

Optimal. Leaf size=344 \[ -\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

[Out]

-x^(-1) + ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqr
t[2])]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqr
t[2])]) - ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqr
t[2])]) - ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqr
t[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 - Sqr
t[2 + Sqrt[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 = 0.606968, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ -\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(1 + x^8)),x]

[Out]

-x^(-1) + ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqr
t[2])]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqr
t[2])]) - ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqr
t[2])]) - ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2*(2 + Sqr
t[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 - Sqr
t[2 + Sqrt[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 = 76.0884, size = 512, normalized size = 1.49 \[ \frac{\sqrt{2} \left (- \frac{\sqrt{2}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \left (- \frac{\sqrt{2}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \log{\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1 \right )}}{8 \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \log{\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1 \right )}}{8 \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \left (- \sqrt{\sqrt{2} + 2} + \frac{\left (1 + \sqrt{2}\right ) \sqrt{\sqrt{2} + 2}}{2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \left (- \sqrt{\sqrt{2} + 2} + \frac{\left (1 + \sqrt{2}\right ) \sqrt{\sqrt{2} + 2}}{2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \left (- \frac{\left (- \sqrt{2} + 1\right ) \sqrt{- \sqrt{2} + 2}}{2} + \sqrt{- \sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \left (- \frac{\left (- \sqrt{2} + 1\right ) \sqrt{- \sqrt{2} + 2}}{2} + \sqrt{- \sqrt{2} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2} \sqrt{\sqrt{2} + 2}} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(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)*(-sqrt(sqrt(2) + 2) + (1 + sqrt(2))*sqrt(s
qrt(2) + 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(sqrt(2) + 2) + (1 + sqrt(2))*sqrt(sq
rt(2) + 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
(-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 + sq
rt(-sqrt(2) + 2))*atan((2*x + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2))/(4*sqrt(-sq
rt(2) + 2)*sqrt(sqrt(2) + 2)) - 1/x

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Mathematica [A]  time = 0.0114426, size = 214, normalized size = 0.62 \[ -\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{x}-\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(1 + x^8)),x]

[Out]

-x^(-1) - (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[P
i/8])/4 - (ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]]*Sin[Pi/8])/4 - (Log[1 + x^2 - 2*x*S
in[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 = 28, normalized size = 0.1 \[ -{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{\it \_R}}}}-{x}^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^8+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{x} - \int \frac{x^{6}}{x^{8} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^2),x, algorithm="maxima")

[Out]

-1/x - integrate(x^6/(x^8 + 1), x)

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Fricas [A]  time = 0.254252, size = 3402, normalized size = 9.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^2),x, algorithm="fricas")

[Out]

-1/64*sqrt(2)*(4*sqrt(2)*x*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*x + sqrt((sqrt(2) + 2)^3 - 3*(sqrt(2) + 2)^2*(sqrt(2) - 2) + 3*(s
qrt(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))) + 4*sqrt(2)*x*sqrt(-sqrt(2) + 2)*a
rctan(((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*(sqr
t(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))) +
4*sqrt(2)*x*sqrt(sqrt(2) + 2)*arctan(-(3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sq
rt(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)))) + 4*sqrt(2)*x*sqrt(sqrt(2) + 2)*arctan((3*(sqrt(2) + 2)*sqrt(-sq
rt(2) + 2) - (-sqrt(2) + 2)^(3/2))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(s
qrt(2) - 2) - 8*x - sqrt((sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sq
rt(2) - 2)^3 - 16*x*(sqrt(2) + 2)^(3/2) + 64*x^2 - 3*((sqrt(2) + 2)^2 + 16*x*sqr
t(sqrt(2) + 2))*(sqrt(2) - 2)))) - sqrt(2)*x*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) + sqrt(2)*x*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*(s
qrt(2) - 2)^3 - 3/4*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 1/4*x*(-sqrt(2) + 2)^(3
/2) + x^2) - sqrt(2)*x*sqrt(-sqrt(2) + 2)*log(1/64*(sqrt(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))*(sqrt(2) - 2)) + sqrt(2)*x*sq
rt(-sqrt(2) + 2)*log(1/64*(sqrt(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))*(sqrt(2) - 2)) + 4*(x*sqrt(sqrt(2) + 2) + x*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*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) + (sqr
t(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)))) + 4*(x*sqrt(s
qrt(2) + 2) + x*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*sqrt(2)*x - (sqrt(2) + 2)^(3/2) - 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) + 3*(sq
rt(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)))) + 4*(x*sqrt(sqrt(2) + 2) - x*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*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)))) + 4*(x*sqrt(sqrt(2) +
2) - x*sqrt(-sqrt(2) + 2))*arctan(((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(s
qrt(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) + 8*sqrt(2)*sqrt(-1/8*sqrt(2)*x*(sqr
t(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)))) - (x*sqrt(sqrt(2) + 2) + x*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)) + (x*sqrt(sqrt(2) + 2) - x*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)) - (x*sqrt(sqrt(2) + 2) - x*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)) + (x*sqrt(sqrt(2)
+ 2) + x*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*sqr
t(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

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Sympy [A]  time = 4.29776, size = 19, normalized size = 0.06 \[ \operatorname{RootSum}{\left (16777216 t^{8} + 1, \left ( t \mapsto t \log{\left (- 2097152 t^{7} + x \right )} \right )\right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(x**8+1),x)

[Out]

RootSum(16777216*_t**8 + 1, Lambda(_t, _t*log(-2097152*_t**7 + x))) - 1/x

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GIAC/XCAS [A]  time = 0.263129, size = 329, normalized size = 0.96 \[ -\frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^2),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*s
qrt(-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/x