3.20 \(\int \frac{1-x^4}{1+b x^4+x^8} \, dx\)

Optimal. Leaf size=511 \[ \frac{\sqrt{2-\sqrt{2-b}} \log \left (-\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left (\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{\sqrt{2-b}+2} \log \left (-\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}+\frac{\sqrt{\sqrt{2-b}+2} \log \left (\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}} \]

[Out]

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

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Rubi [A]  time = 0.875449, antiderivative size = 511, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\sqrt{2-\sqrt{2-b}} \log \left (-\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{2-\sqrt{2-b}} \log \left (\sqrt{2-\sqrt{2-b}} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{\sqrt{2-b}+2} \log \left (-\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}+\frac{\sqrt{\sqrt{2-b}+2} \log \left (\sqrt{\sqrt{2-b}+2} x+x^2+1\right )}{8 \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}-2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}-2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}}+\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2-b}}+2 x}{\sqrt{\sqrt{2-b}+2}}\right )}{4 \sqrt{2-\sqrt{2-b}} \sqrt{2-b}}-\frac{\sqrt{b+2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{2-b}+2}+2 x}{\sqrt{2-\sqrt{2-b}}}\right )}{4 \sqrt{\sqrt{2-b}+2} \sqrt{2-b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 119.994, size = 359, normalized size = 0.7 \[ \frac{\sqrt{- \sqrt{- b + 2} + 2} \log{\left (x^{2} - x \sqrt{- \sqrt{- b + 2} + 2} + 1 \right )}}{8 \sqrt{- b + 2}} - \frac{\sqrt{- \sqrt{- b + 2} + 2} \log{\left (x^{2} + x \sqrt{- \sqrt{- b + 2} + 2} + 1 \right )}}{8 \sqrt{- b + 2}} - \frac{\sqrt{- \sqrt{- b + 2} + 2} \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{- b + 2} + 2}}{\sqrt{- \sqrt{- b + 2} + 2}} \right )}}{4 \sqrt{- b + 2}} - \frac{\sqrt{- \sqrt{- b + 2} + 2} \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{- b + 2} + 2}}{\sqrt{- \sqrt{- b + 2} + 2}} \right )}}{4 \sqrt{- b + 2}} - \frac{\sqrt{\sqrt{- b + 2} + 2} \log{\left (x^{2} - x \sqrt{\sqrt{- b + 2} + 2} + 1 \right )}}{8 \sqrt{- b + 2}} + \frac{\sqrt{\sqrt{- b + 2} + 2} \log{\left (x^{2} + x \sqrt{\sqrt{- b + 2} + 2} + 1 \right )}}{8 \sqrt{- b + 2}} + \frac{\sqrt{\sqrt{- b + 2} + 2} \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{- b + 2} + 2}}{\sqrt{\sqrt{- b + 2} + 2}} \right )}}{4 \sqrt{- b + 2}} + \frac{\sqrt{\sqrt{- b + 2} + 2} \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{- b + 2} + 2}}{\sqrt{\sqrt{- b + 2} + 2}} \right )}}{4 \sqrt{- b + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0358951, size = 57, normalized size = 0.11 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+\text{$\#$1}^4 b+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+\text{$\#$1}^3 b}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

-RootSum[1 + b*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(b*#1^3 + 2*#1^
7) & ]/4

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Maple [C]  time = 0.004, size = 44, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+{{\it \_R}}^{3}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8+b*x^4+1),x)

[Out]

1/4*sum((-_R^4+1)/(2*_R^7+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8+_Z^4*b+1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((x^4 - 1)/(x^8 + b*x^4 + 1), x)

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Fricas [A]  time = 0.291944, size = 1458, normalized size = 2.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(sqrt(1/2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) +
b)/(b^2 - 4*b + 4)))*arctan(1/2*((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*
b - 8)) - b + 2)*sqrt(sqrt(1/2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2
+ 12*b - 8)) + b)/(b^2 - 4*b + 4)))/(x + sqrt(x^2 + 1/2*sqrt(1/2)*(b^2 - (b^3 -
6*b^2 + 12*b - 8)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - 2*b)*sqrt(-((b^2 - 4
*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4))))) + sqrt(s
qrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2
- 4*b + 4)))*arctan(1/2*((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8))
+ b - 2)*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b -
8)) - b)/(b^2 - 4*b + 4)))/(x + sqrt(x^2 + 1/2*sqrt(1/2)*(b^2 + (b^3 - 6*b^2 +
12*b - 8)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - 2*b)*sqrt(((b^2 - 4*b + 4)*sq
rt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4))))) + 1/4*sqrt(sqrt(1/
2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b
+ 4)))*log(1/2*((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b + 2)
*sqrt(sqrt(1/2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) +
b)/(b^2 - 4*b + 4))) + x) - 1/4*sqrt(sqrt(1/2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b +
2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4)))*log(-1/2*((b^2 - 4*b + 4)*sq
rt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b + 2)*sqrt(sqrt(1/2)*sqrt(-((b^2 - 4*b +
4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4))) + x) - 1/4*sqr
t(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b
^2 - 4*b + 4)))*log(1/2*((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8))
+ b - 2)*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b -
8)) - b)/(b^2 - 4*b + 4))) + x) + 1/4*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt
((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4)))*log(-1/2*((b^2 - 4*b +
4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b - 2)*sqrt(sqrt(1/2)*sqrt(((b^2 -
4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4))) + x)

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Sympy [A]  time = 7.47082, size = 76, normalized size = 0.15 \[ - \operatorname{RootSum}{\left (t^{8} \left (65536 b^{4} - 524288 b^{3} + 1572864 b^{2} - 2097152 b + 1048576\right ) + t^{4} \left (256 b^{3} - 1024 b^{2} + 1024 b\right ) + 1, \left ( t \mapsto t \log{\left (1024 t^{5} b^{2} - 4096 t^{5} b + 4096 t^{5} + 4 t b - 4 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(_t**8*(65536*b**4 - 524288*b**3 + 1572864*b**2 - 2097152*b + 1048576) +
_t**4*(256*b**3 - 1024*b**2 + 1024*b) + 1, Lambda(_t, _t*log(1024*_t**5*b**2 -
4096*_t**5*b + 4096*_t**5 + 4*_t*b - 4*_t + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{4} - 1}{x^{8} + b x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(x^4 - 1)/(x^8 + b*x^4 + 1), x)