3.13 \(\int \frac{1+x^4}{1+x^8} \, dx\)

Optimal. Leaf size=347 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2-\sqrt{2}}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2-\sqrt{2}}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2+\sqrt{2}}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2+\sqrt{2}}}-\frac{1}{4} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{4} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.564652, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2-\sqrt{2}}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2-\sqrt{2}}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2+\sqrt{2}}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2+\sqrt{2}}}-\frac{1}{4} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{4} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)/(1 + x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 39.2716, size = 270, normalized size = 0.78 \[ - \frac{\log{\left (x^{2} - x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{2} + 2}} + \frac{\log{\left (x^{2} + x \sqrt{- \sqrt{2} + 2} + 1 \right )}}{8 \sqrt{- \sqrt{2} + 2}} - \frac{\log{\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1 \right )}}{8 \sqrt{\sqrt{2} + 2}} + \frac{\log{\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1 \right )}}{8 \sqrt{\sqrt{2} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{2} + 2}}{\sqrt{- \sqrt{2} + 2}} \right )}}{4 \sqrt{- \sqrt{2} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{4 \sqrt{\sqrt{2} + 2}} + \frac{\operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}} \right )}}{4 \sqrt{\sqrt{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.334667, size = 258, normalized size = 0.74 \[ \frac{1}{8} \left (-\left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\left (\sin \left (\frac{\pi }{8}\right )-\cos \left (\frac{\pi }{8}\right )\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\left (\cos \left (\frac{\pi }{8}\right )-\sin \left (\frac{\pi }{8}\right )\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )+2 \left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )+2 \left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+2 \left (\cos \left (\frac{\pi }{8}\right )-\sin \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right )+2 \left (\cos \left (\frac{\pi }{8}\right )-\sin \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (x \sec \left (\frac{\pi }{8}\right )-\tan \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])]*(Cos[Pi/8] - Sin[Pi/8]) + 2*ArcTan[x*Sec[Pi
/8] - Tan[Pi/8]]*(Cos[Pi/8] - Sin[Pi/8]) + Log[1 + x^2 + 2*x*Cos[Pi/8]]*(Cos[Pi/
8] - Sin[Pi/8]) + Log[1 + x^2 - 2*x*Cos[Pi/8]]*(-Cos[Pi/8] + Sin[Pi/8]) + 2*ArcT
an[(x - Cos[Pi/8])*Csc[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) + 2*ArcTan[(x + Cos[Pi/8])
*Csc[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) - Log[1 + x^2 - 2*x*Sin[Pi/8]]*(Cos[Pi/8] +
Sin[Pi/8]) + Log[1 + x^2 + 2*x*Sin[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]))/8

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Maple [C]  time = 0.009, size = 27, normalized size = 0.1 \[{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{ \left ({{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/(x^8+1),x)

[Out]

1/8*sum((_R^4+1)/_R^7*ln(x-_R),_R=RootOf(_Z^8+1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4 + 1)/(x^8 + 1), x)

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Fricas [A]  time = 0.281364, size = 1343, normalized size = 3.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(2)*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))) - 1/8*sqr
t(2)*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))) + 1/8*sqrt(2)*sqr
t(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))) + 1/8*sqrt(2)*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))) - 1/8*(sqrt(sqrt(2) + 2) - 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))) - 1/8*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*arcta
n(sqrt(sqrt(2) + 2)/(2*x + 2*sqrt(x^2 - x*sqrt(-sqrt(2) + 2) + 1) - sqrt(-sqrt(2
) + 2))) - 1/8*(sqrt(sqrt(2) + 2) + 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))) - 1/8*(sqrt
(sqrt(2) + 2) + 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))) + 1/32*sqrt(2)*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) + 1/32*sqrt(2)*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) - 1/32*sqrt(2)*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) - 1/3
2*sqrt(2)*sqrt(-sqrt(2) + 2)*log(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqr
t(2)*x*sqrt(-sqrt(2) + 2) + 1) + 1/32*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*l
og(x^2 + x*sqrt(sqrt(2) + 2) + 1) - 1/32*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2)
)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1) + 1/32*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) +
2))*log(x^2 + x*sqrt(-sqrt(2) + 2) + 1) - 1/32*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(
2) + 2))*log(x^2 - x*sqrt(-sqrt(2) + 2) + 1)

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Sympy [A]  time = 4.44836, size = 19, normalized size = 0.05 \[ \operatorname{RootSum}{\left (1048576 t^{8} + 1, \left ( t \mapsto t \log{\left (4096 t^{5} + 4 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+1)/(x**8+1),x)

[Out]

RootSum(1048576*_t**8 + 1, Lambda(_t, _t*log(4096*_t**5 + 4*_t + x)))

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GIAC/XCAS [A]  time = 0.298775, size = 333, normalized size = 0.96 \[ \frac{1}{8} \, \sqrt{-2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{-2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{16} \, \sqrt{-2 \, \sqrt{2} + 4}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{-2 \, \sqrt{2} + 4}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 4}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 4}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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