3.288 \(\int \frac{1}{x^6 \left (1+2 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=113 \[ -\frac{9}{20 x^5}+\frac{9 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{9 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{1}{4 x^5 \left (x^4+1\right )}+\frac{9}{4 x}-\frac{9 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{9 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

[Out]

-9/(20*x^5) + 9/(4*x) + 1/(4*x^5*(1 + x^4)) - (9*ArcTan[1 - Sqrt[2]*x])/(8*Sqrt[
2]) + (9*ArcTan[1 + Sqrt[2]*x])/(8*Sqrt[2]) + (9*Log[1 - Sqrt[2]*x + x^2])/(16*S
qrt[2]) - (9*Log[1 + Sqrt[2]*x + x^2])/(16*Sqrt[2])

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Rubi [A]  time = 0.115369, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562 \[ -\frac{9}{20 x^5}+\frac{9 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{9 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{1}{4 x^5 \left (x^4+1\right )}+\frac{9}{4 x}-\frac{9 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{9 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 + 2*x^4 + x^8)),x]

[Out]

-9/(20*x^5) + 9/(4*x) + 1/(4*x^5*(1 + x^4)) - (9*ArcTan[1 - Sqrt[2]*x])/(8*Sqrt[
2]) + (9*ArcTan[1 + Sqrt[2]*x])/(8*Sqrt[2]) + (9*Log[1 - Sqrt[2]*x + x^2])/(16*S
qrt[2]) - (9*Log[1 + Sqrt[2]*x + x^2])/(16*Sqrt[2])

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Rubi in Sympy [A]  time = 20.4032, size = 104, normalized size = 0.92 \[ \frac{9 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} - \frac{9 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} + \frac{9}{4 x} - \frac{9}{20 x^{5}} + \frac{1}{4 x^{5} \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**8+2*x**4+1),x)

[Out]

9*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 - 9*sqrt(2)*log(x**2 + sqrt(2)*x + 1)/32
+ 9*sqrt(2)*atan(sqrt(2)*x - 1)/16 + 9*sqrt(2)*atan(sqrt(2)*x + 1)/16 + 9/(4*x)
- 9/(20*x**5) + 1/(4*x**5*(x**4 + 1))

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Mathematica [A]  time = 0.132757, size = 103, normalized size = 0.91 \[ \frac{1}{160} \left (-\frac{32}{x^5}+45 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-45 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )+\frac{40 x^3}{x^4+1}+\frac{320}{x}-90 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+90 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 + 2*x^4 + x^8)),x]

[Out]

(-32/x^5 + 320/x + (40*x^3)/(1 + x^4) - 90*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 90*Sq
rt[2]*ArcTan[1 + Sqrt[2]*x] + 45*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - 45*Sqrt[2]*L
og[1 + Sqrt[2]*x + x^2])/160

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Maple [A]  time = 0.015, size = 80, normalized size = 0.7 \[{\frac{{x}^{3}}{4\,{x}^{4}+4}}+{\frac{9\,\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{16}}+{\frac{9\,\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}-\sqrt{2}x}{1+{x}^{2}+\sqrt{2}x}} \right ) }+{\frac{9\,\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{16}}-{\frac{1}{5\,{x}^{5}}}+2\,{x}^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^8+2*x^4+1),x)

[Out]

1/4*x^3/(x^4+1)+9/16*arctan(2^(1/2)*x-1)*2^(1/2)+9/32*2^(1/2)*ln((1+x^2-2^(1/2)*
x)/(1+x^2+2^(1/2)*x))+9/16*arctan(1+2^(1/2)*x)*2^(1/2)-1/5/x^5+2/x

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Maxima [A]  time = 0.873783, size = 128, normalized size = 1.13 \[ \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{9}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{9}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{45 \, x^{8} + 36 \, x^{4} - 4}{20 \,{\left (x^{9} + x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^6),x, algorithm="maxima")

[Out]

9/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 9/16*sqrt(2)*arctan(1/2*sqrt(
2)*(2*x - sqrt(2))) - 9/32*sqrt(2)*log(x^2 + sqrt(2)*x + 1) + 9/32*sqrt(2)*log(x
^2 - sqrt(2)*x + 1) + 1/20*(45*x^8 + 36*x^4 - 4)/(x^9 + x^5)

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Fricas [A]  time = 0.276045, size = 198, normalized size = 1.75 \[ \frac{360 \, x^{8} + 288 \, x^{4} - 180 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) - 180 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) - 45 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) + 45 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) - 32}{160 \,{\left (x^{9} + x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^6),x, algorithm="fricas")

[Out]

1/160*(360*x^8 + 288*x^4 - 180*sqrt(2)*(x^9 + x^5)*arctan(1/(sqrt(2)*x + sqrt(2)
*sqrt(x^2 + sqrt(2)*x + 1) + 1)) - 180*sqrt(2)*(x^9 + x^5)*arctan(1/(sqrt(2)*x +
sqrt(2)*sqrt(x^2 - sqrt(2)*x + 1) - 1)) - 45*sqrt(2)*(x^9 + x^5)*log(x^2 + sqrt
(2)*x + 1) + 45*sqrt(2)*(x^9 + x^5)*log(x^2 - sqrt(2)*x + 1) - 32)/(x^9 + x^5)

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Sympy [A]  time = 0.738045, size = 102, normalized size = 0.9 \[ \frac{9 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} - \frac{9 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} + \frac{45 x^{8} + 36 x^{4} - 4}{20 x^{9} + 20 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**8+2*x**4+1),x)

[Out]

9*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 - 9*sqrt(2)*log(x**2 + sqrt(2)*x + 1)/32
+ 9*sqrt(2)*atan(sqrt(2)*x - 1)/16 + 9*sqrt(2)*atan(sqrt(2)*x + 1)/16 + (45*x**8
+ 36*x**4 - 4)/(20*x**9 + 20*x**5)

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GIAC/XCAS [A]  time = 0.263094, size = 130, normalized size = 1.15 \[ \frac{x^{3}}{4 \,{\left (x^{4} + 1\right )}} + \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{9}{32} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) + \frac{9}{32} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) + \frac{10 \, x^{4} - 1}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^6),x, algorithm="giac")

[Out]

1/4*x^3/(x^4 + 1) + 9/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 9/16*sqrt
(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) - 9/32*sqrt(2)*ln(x^2 + sqrt(2)*x + 1) +
9/32*sqrt(2)*ln(x^2 - sqrt(2)*x + 1) + 1/5*(10*x^4 - 1)/x^5