3.1492 \(\int \frac{x}{1+x^8} \, dx\)

Optimal. Leaf size=93 \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} x^2\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x^2+1\right )}{4 \sqrt{2}}-\frac{\log \left (x^4-\sqrt{2} x^2+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^4+\sqrt{2} x^2+1\right )}{8 \sqrt{2}} \]

[Out]

-ArcTan[1 - Sqrt[2]*x^2]/(4*Sqrt[2]) + ArcTan[1 + Sqrt[2]*x^2]/(4*Sqrt[2]) - Log
[1 - Sqrt[2]*x^2 + x^4]/(8*Sqrt[2]) + Log[1 + Sqrt[2]*x^2 + x^4]/(8*Sqrt[2])

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Rubi [A]  time = 0.141226, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} x^2\right )}{4 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x^2+1\right )}{4 \sqrt{2}}-\frac{\log \left (x^4-\sqrt{2} x^2+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^4+\sqrt{2} x^2+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x^8),x]

[Out]

-ArcTan[1 - Sqrt[2]*x^2]/(4*Sqrt[2]) + ArcTan[1 + Sqrt[2]*x^2]/(4*Sqrt[2]) - Log
[1 - Sqrt[2]*x^2 + x^4]/(8*Sqrt[2]) + Log[1 + Sqrt[2]*x^2 + x^4]/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 15.5103, size = 80, normalized size = 0.86 \[ - \frac{\sqrt{2} \log{\left (x^{4} - \sqrt{2} x^{2} + 1 \right )}}{16} + \frac{\sqrt{2} \log{\left (x^{4} + \sqrt{2} x^{2} + 1 \right )}}{16} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} - 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**8+1),x)

[Out]

-sqrt(2)*log(x**4 - sqrt(2)*x**2 + 1)/16 + sqrt(2)*log(x**4 + sqrt(2)*x**2 + 1)/
16 + sqrt(2)*atan(sqrt(2)*x**2 - 1)/8 + sqrt(2)*atan(sqrt(2)*x**2 + 1)/8

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Mathematica [A]  time = 0.0553939, size = 149, normalized size = 1.6 \[ -\frac{-\log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )-2 \tan ^{-1}\left (x \sec \left (\frac{\pi }{8}\right )-\tan \left (\frac{\pi }{8}\right )\right )+2 \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+2 \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-x \csc \left (\frac{\pi }{8}\right )\right )+2 \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + x^8),x]

[Out]

-(2*ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]] + 2*ArcTan[Cot[Pi/8] - x*Csc[Pi/8]] + 2*Ar
cTan[Sec[Pi/8]*(x + Sin[Pi/8])] - 2*ArcTan[x*Sec[Pi/8] - Tan[Pi/8]] + Log[1 + x^
2 - 2*x*Cos[Pi/8]] + Log[1 + x^2 + 2*x*Cos[Pi/8]] - Log[1 + x^2 - 2*x*Sin[Pi/8]]
- Log[1 + x^2 + 2*x*Sin[Pi/8]])/(8*Sqrt[2])

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Maple [A]  time = 0.003, size = 66, normalized size = 0.7 \[{\frac{\arctan \left ( 1+{x}^{2}\sqrt{2} \right ) \sqrt{2}}{8}}+{\frac{\arctan \left ({x}^{2}\sqrt{2}-1 \right ) \sqrt{2}}{8}}+{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{4}+{x}^{2}\sqrt{2}}{1+{x}^{4}-{x}^{2}\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^8+1),x)

[Out]

1/8*arctan(1+x^2*2^(1/2))*2^(1/2)+1/8*arctan(x^2*2^(1/2)-1)*2^(1/2)+1/16*2^(1/2)
*ln((1+x^4+x^2*2^(1/2))/(1+x^4-x^2*2^(1/2)))

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Maxima [A]  time = 1.58577, size = 108, normalized size = 1.16 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} + \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} - \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{4} + \sqrt{2} x^{2} + 1\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{4} - \sqrt{2} x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2))) + 1/8*sqrt(2)*arctan(1/2*sqrt(
2)*(2*x^2 - sqrt(2))) + 1/16*sqrt(2)*log(x^4 + sqrt(2)*x^2 + 1) - 1/16*sqrt(2)*l
og(x^4 - sqrt(2)*x^2 + 1)

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Fricas [A]  time = 0.230934, size = 147, normalized size = 1.58 \[ -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x^{2} + \sqrt{2} \sqrt{x^{4} + \sqrt{2} x^{2} + 1} + 1}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x^{2} + \sqrt{2} \sqrt{x^{4} - \sqrt{2} x^{2} + 1} - 1}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{4} + \sqrt{2} x^{2} + 1\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{4} - \sqrt{2} x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*arctan(1/(sqrt(2)*x^2 + sqrt(2)*sqrt(x^4 + sqrt(2)*x^2 + 1) + 1)) -
1/4*sqrt(2)*arctan(1/(sqrt(2)*x^2 + sqrt(2)*sqrt(x^4 - sqrt(2)*x^2 + 1) - 1)) +
1/16*sqrt(2)*log(x^4 + sqrt(2)*x^2 + 1) - 1/16*sqrt(2)*log(x^4 - sqrt(2)*x^2 +
1)

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Sympy [A]  time = 0.460552, size = 80, normalized size = 0.86 \[ - \frac{\sqrt{2} \log{\left (x^{4} - \sqrt{2} x^{2} + 1 \right )}}{16} + \frac{\sqrt{2} \log{\left (x^{4} + \sqrt{2} x^{2} + 1 \right )}}{16} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} - 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**8+1),x)

[Out]

-sqrt(2)*log(x**4 - sqrt(2)*x**2 + 1)/16 + sqrt(2)*log(x**4 + sqrt(2)*x**2 + 1)/
16 + sqrt(2)*atan(sqrt(2)*x**2 - 1)/8 + sqrt(2)*atan(sqrt(2)*x**2 + 1)/8

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GIAC/XCAS [A]  time = 0.241424, size = 108, normalized size = 1.16 \[ \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} + \sqrt{2}\right )}\right ) + \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} - \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} + 1\right ) - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2))) + 1/8*sqrt(2)*arctan(1/2*sqrt(
2)*(2*x^2 - sqrt(2))) + 1/16*sqrt(2)*ln(x^4 + sqrt(2)*x^2 + 1) - 1/16*sqrt(2)*ln
(x^4 - sqrt(2)*x^2 + 1)