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

Optimal. Leaf size=113 \[ -\frac{11}{28 x^7}+\frac{11}{12 x^3}-\frac{11 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{11 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{1}{4 x^7 \left (x^4+1\right )}-\frac{11 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{11 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

[Out]

-11/(28*x^7) + 11/(12*x^3) + 1/(4*x^7*(1 + x^4)) - (11*ArcTan[1 - Sqrt[2]*x])/(8
*Sqrt[2]) + (11*ArcTan[1 + Sqrt[2]*x])/(8*Sqrt[2]) - (11*Log[1 - Sqrt[2]*x + x^2
])/(16*Sqrt[2]) + (11*Log[1 + Sqrt[2]*x + x^2])/(16*Sqrt[2])

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Rubi [A]  time = 0.112744, 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 \[ -\frac{11}{28 x^7}+\frac{11}{12 x^3}-\frac{11 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{11 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{1}{4 x^7 \left (x^4+1\right )}-\frac{11 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{11 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-11/(28*x^7) + 11/(12*x^3) + 1/(4*x^7*(1 + x^4)) - (11*ArcTan[1 - Sqrt[2]*x])/(8
*Sqrt[2]) + (11*ArcTan[1 + Sqrt[2]*x])/(8*Sqrt[2]) - (11*Log[1 - Sqrt[2]*x + x^2
])/(16*Sqrt[2]) + (11*Log[1 + Sqrt[2]*x + x^2])/(16*Sqrt[2])

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Rubi in Sympy [A]  time = 18.9727, size = 105, normalized size = 0.93 \[ - \frac{11 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{11 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} + \frac{11}{12 x^{3}} - \frac{11}{28 x^{7}} + \frac{1}{4 x^{7} \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + 11*sqrt(2)*log(x**2 + sqrt(2)*x + 1)/
32 + 11*sqrt(2)*atan(sqrt(2)*x - 1)/16 + 11*sqrt(2)*atan(sqrt(2)*x + 1)/16 + 11/
(12*x**3) - 11/(28*x**7) + 1/(4*x**7*(x**4 + 1))

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Mathematica [A]  time = 0.140038, size = 101, normalized size = 0.89 \[ \frac{1}{672} \left (-\frac{96}{x^7}+\frac{168 x}{x^4+1}+\frac{448}{x^3}-231 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+231 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-462 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+462 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-96/x^7 + 448/x^3 + (168*x)/(1 + x^4) - 462*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 462
*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] - 231*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] + 231*Sqrt
[2]*Log[1 + Sqrt[2]*x + x^2])/672

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Maple [A]  time = 0.015, size = 78, normalized size = 0.7 \[{\frac{x}{4\,{x}^{4}+4}}+{\frac{11\,\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{16}}+{\frac{11\,\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) }+{\frac{11\,\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{16}}-{\frac{1}{7\,{x}^{7}}}+{\frac{2}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(x^8+2*x^4+1),x)

[Out]

1/4*x/(x^4+1)+11/16*arctan(2^(1/2)*x-1)*2^(1/2)+11/32*2^(1/2)*ln((1+x^2+2^(1/2)*
x)/(1+x^2-2^(1/2)*x))+11/16*arctan(1+2^(1/2)*x)*2^(1/2)-1/7/x^7+2/3/x^3

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Maxima [A]  time = 0.854048, size = 128, normalized size = 1.13 \[ \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{11}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{11}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{77 \, x^{8} + 44 \, x^{4} - 12}{84 \,{\left (x^{11} + x^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 11/16*sqrt(2)*arctan(1/2*sqr
t(2)*(2*x - sqrt(2))) + 11/32*sqrt(2)*log(x^2 + sqrt(2)*x + 1) - 11/32*sqrt(2)*l
og(x^2 - sqrt(2)*x + 1) + 1/84*(77*x^8 + 44*x^4 - 12)/(x^11 + x^7)

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Fricas [A]  time = 0.260469, size = 198, normalized size = 1.75 \[ \frac{616 \, x^{8} + 352 \, x^{4} - 924 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) - 924 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) + 231 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) - 231 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) - 96}{672 \,{\left (x^{11} + x^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/672*(616*x^8 + 352*x^4 - 924*sqrt(2)*(x^11 + x^7)*arctan(1/(sqrt(2)*x + sqrt(2
)*sqrt(x^2 + sqrt(2)*x + 1) + 1)) - 924*sqrt(2)*(x^11 + x^7)*arctan(1/(sqrt(2)*x
+ sqrt(2)*sqrt(x^2 - sqrt(2)*x + 1) - 1)) + 231*sqrt(2)*(x^11 + x^7)*log(x^2 +
sqrt(2)*x + 1) - 231*sqrt(2)*(x^11 + x^7)*log(x^2 - sqrt(2)*x + 1) - 96)/(x^11 +
x^7)

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Sympy [A]  time = 0.777807, size = 102, normalized size = 0.9 \[ - \frac{11 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{11 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} + \frac{77 x^{8} + 44 x^{4} - 12}{84 x^{11} + 84 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + 11*sqrt(2)*log(x**2 + sqrt(2)*x + 1)/
32 + 11*sqrt(2)*atan(sqrt(2)*x - 1)/16 + 11*sqrt(2)*atan(sqrt(2)*x + 1)/16 + (77
*x**8 + 44*x**4 - 12)/(84*x**11 + 84*x**7)

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GIAC/XCAS [A]  time = 0.309526, size = 127, normalized size = 1.12 \[ \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{11}{32} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{11}{32} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) + \frac{x}{4 \,{\left (x^{4} + 1\right )}} + \frac{14 \, x^{4} - 3}{21 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 11/16*sqrt(2)*arctan(1/2*sqr
t(2)*(2*x - sqrt(2))) + 11/32*sqrt(2)*ln(x^2 + sqrt(2)*x + 1) - 11/32*sqrt(2)*ln
(x^2 - sqrt(2)*x + 1) + 1/4*x/(x^4 + 1) + 1/21*(14*x^4 - 3)/x^7