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

Optimal. Leaf size=106 \[ -\frac{7}{12 x^3}+\frac{7 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{7 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{1}{4 x^3 \left (x^4+1\right )}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.104168, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562 \[ -\frac{7}{12 x^3}+\frac{7 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{7 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{1}{4 x^3 \left (x^4+1\right )}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.6087, size = 99, normalized size = 0.93 \[ \frac{7 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} - \frac{7 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} - \frac{7}{12 x^{3}} + \frac{1}{4 x^{3} \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.132524, size = 96, normalized size = 0.91 \[ \frac{1}{96} \left (-\frac{24 x}{x^4+1}-\frac{32}{x^3}+21 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-21 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )+42 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )-42 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-32/x^3 - (24*x)/(1 + x^4) + 42*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] - 42*Sqrt[2]*ArcT
an[1 + Sqrt[2]*x] + 21*Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - 21*Sqrt[2]*Log[1 + Sqr
t[2]*x + x^2])/96

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Maple [A]  time = 0.012, size = 73, normalized size = 0.7 \[ -{\frac{x}{4\,{x}^{4}+4}}-{\frac{7\,\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{16}}-{\frac{7\,\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{16}}-{\frac{7\,\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) }-{\frac{1}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^8+2*x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.852348, size = 122, normalized size = 1.15 \[ -\frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{7}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{7}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{7 \, x^{4} + 4}{12 \,{\left (x^{7} + x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) - 7/16*sqrt(2)*arctan(1/2*sqrt
(2)*(2*x - sqrt(2))) - 7/32*sqrt(2)*log(x^2 + sqrt(2)*x + 1) + 7/32*sqrt(2)*log(
x^2 - sqrt(2)*x + 1) - 1/12*(7*x^4 + 4)/(x^7 + x^3)

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Fricas [A]  time = 0.274103, size = 192, normalized size = 1.81 \[ -\frac{56 \, x^{4} - 84 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) - 84 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) + 21 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) - 21 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) + 32}{96 \,{\left (x^{7} + x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*(56*x^4 - 84*sqrt(2)*(x^7 + x^3)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt(x^2 +
sqrt(2)*x + 1) + 1)) - 84*sqrt(2)*(x^7 + x^3)*arctan(1/(sqrt(2)*x + sqrt(2)*sqrt
(x^2 - sqrt(2)*x + 1) - 1)) + 21*sqrt(2)*(x^7 + x^3)*log(x^2 + sqrt(2)*x + 1) -
21*sqrt(2)*(x^7 + x^3)*log(x^2 - sqrt(2)*x + 1) + 32)/(x^7 + x^3)

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Sympy [A]  time = 0.642443, size = 97, normalized size = 0.92 \[ - \frac{7 x^{4} + 4}{12 x^{7} + 12 x^{3}} + \frac{7 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} - \frac{7 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274178, size = 117, normalized size = 1.1 \[ -\frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{7}{32} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) + \frac{7}{32} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) - \frac{x}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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