3.1504 \(\int \frac{1}{x^4 \left (1+x^8\right )} \, dx\)

Optimal. Leaf size=354 \[ -\frac{1}{3 x^3}+\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}} \]

[Out]

-1/(3*x^3) - 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])]) + 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])]) - 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 = 0.540179, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ -\frac{1}{3 x^3}+\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 + x^8)),x]

[Out]

-1/(3*x^3) - 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])]) + 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])]) - 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 = 55.8956, size = 318, normalized size = 0.9 \[ \frac{\sqrt{2} \log{\left (x^{2} - x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{16 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \log{\left (x^{2} + x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{16 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \log{\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1 \right )}}{16 \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \log{\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1 \right )}}{16 \sqrt{\sqrt{2} + 2}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{8 \sqrt{- \sqrt{2} + 2}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{8 \sqrt{\sqrt{2} + 2}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{8 \sqrt{\sqrt{2} + 2}} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log(x**2 - x*sqrt(-sqrt(2) + 2) + 1)/(16*sqrt(-sqrt(2) + 2)) - sqrt(2)*l
og(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(sqr
t(2) + 2) + 1)/(16*sqrt(sqrt(2) + 2)) - sqrt(2)*atan((2*x - sqrt(sqrt(2) + 2))/s
qrt(-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(-sqr
t(2) + 2))/sqrt(sqrt(2) + 2))/(8*sqrt(sqrt(2) + 2)) - 1/(3*x**3)

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Mathematica [A]  time = 0.0112836, size = 216, normalized size = 0.61 \[ -\frac{1}{3 x^3}+\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(1 + x^8)),x]

[Out]

-1/(3*x^3) - (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 = 0.01, size = 28, normalized size = 0.1 \[ -{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}}-{\frac{1}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^8+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.250024, size = 1474, normalized size = 4.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*sqrt(2)*(12*sqrt(2)*x^3*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))) + 12*sqrt(2)*x^3
*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))) - 12*sqrt(2)*x^3*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))
) - 12*sqrt(2)*x^3*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))) - 3*sqrt(2)*x^3*sqrt(-sqrt(2)
+ 2)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) + 3*sqrt(2)*x^3*sqrt(-sqrt(2) + 2)*log(x
^2 - x*sqrt(sqrt(2) + 2) + 1) + 3*sqrt(2)*x^3*sqrt(sqrt(2) + 2)*log(x^2 + x*sqrt
(-sqrt(2) + 2) + 1) - 3*sqrt(2)*x^3*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(-sqrt(2)
+ 2) + 1) + 12*(x^3*sqrt(sqrt(2) + 2) - x^3*sqrt(-sqrt(2) + 2))*arctan((sqrt(sqr
t(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))) + 12*(x^3*sqrt(sqrt(2) + 2) - x^3*sqrt(-sqrt(2) + 2))*arc
tan((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))) + 12*(x^3*sqrt(sqrt(2) + 2) + x^3*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))) + 12*(x^3*sqrt(sqrt(2) + 2) +
x^3*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*sqr
t(-sqrt(2) + 2) + 1) - sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 3*(x^3*sqrt(sq
rt(2) + 2) - x^3*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) + 3*(x^3*sqrt(sqrt(2) + 2) + x^3*sqrt(-sq
rt(2) + 2))*log(x^2 + 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt
(2) + 2) + 1) - 3*(x^3*sqrt(sqrt(2) + 2) + x^3*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) + 3*(x^3*sq
rt(sqrt(2) + 2) - x^3*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) + 32*sqrt(2))/x^3

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Sympy [A]  time = 4.3808, size = 22, normalized size = 0.06 \[ \operatorname{RootSum}{\left (16777216 t^{8} + 1, \left ( t \mapsto t \log{\left (32768 t^{5} + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**8+1),x)

[Out]

RootSum(16777216*_t**8 + 1, Lambda(_t, _t*log(32768*_t**5 + x))) - 1/(3*x**3)

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GIAC/XCAS [A]  time = 0.258338, size = 329, normalized size = 0.93 \[ \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{-\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{\sqrt{2} + 2}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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