3.391 \(\int \frac{1-x^2}{a+b \left (1-x^2\right )^4} \, dx\)

Optimal. Leaf size=663 \[ \frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (-\sqrt{2} \sqrt [8]{b} x \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (\sqrt{2} \sqrt [8]{b} x \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}\right )}{4 \sqrt{-a} b^{3/8} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}-\sqrt{2} \sqrt [8]{b} x}{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}}}\right )}{4 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}+\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{2} \sqrt [8]{b} x}{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}}}\right )}{4 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 \sqrt{-a} b^{3/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}} \]

[Out]

-ArcTan[(b^(1/8)*x)/Sqrt[(-a)^(1/4) - b^(1/4)]]/(4*Sqrt[-a]*Sqrt[(-a)^(1/4) - b^
(1/4)]*b^(3/8)) - (Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] - b^(1/4)]*ArcTan[(Sqrt[Sqrt[Sq
rt[-a] + Sqrt[b]] + b^(1/4)] - Sqrt[2]*b^(1/8)*x)/Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]]
- b^(1/4)]])/(4*Sqrt[2]*Sqrt[-a]*Sqrt[Sqrt[-a] + Sqrt[b]]*b^(3/8)) + (Sqrt[Sqrt[
Sqrt[-a] + Sqrt[b]] - b^(1/4)]*ArcTan[(Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] + b^(1/4)]
+ Sqrt[2]*b^(1/8)*x)/Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] - b^(1/4)]])/(4*Sqrt[2]*Sqrt[
-a]*Sqrt[Sqrt[-a] + Sqrt[b]]*b^(3/8)) + ArcTanh[(b^(1/8)*x)/Sqrt[(-a)^(1/4) + b^
(1/4)]]/(4*Sqrt[-a]*Sqrt[(-a)^(1/4) + b^(1/4)]*b^(3/8)) + (Sqrt[Sqrt[Sqrt[-a] +
Sqrt[b]] + b^(1/4)]*Log[Sqrt[Sqrt[-a] + Sqrt[b]] - Sqrt[2]*Sqrt[Sqrt[Sqrt[-a] +
Sqrt[b]] + b^(1/4)]*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*Sqrt[-a]*Sqrt[Sqrt[-a]
+ Sqrt[b]]*b^(3/8)) - (Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] + b^(1/4)]*Log[Sqrt[Sqrt[-a
] + Sqrt[b]] + Sqrt[2]*Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] + b^(1/4)]*b^(1/8)*x + b^(1
/4)*x^2])/(8*Sqrt[2]*Sqrt[-a]*Sqrt[Sqrt[-a] + Sqrt[b]]*b^(3/8))

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Rubi [A]  time = 3.07506, antiderivative size = 663, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435 \[ \frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (-\sqrt{2} \sqrt [8]{b} x \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (\sqrt{2} \sqrt [8]{b} x \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}\right )}{4 \sqrt{-a} b^{3/8} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}-\sqrt{2} \sqrt [8]{b} x}{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}}}\right )}{4 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}+\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{2} \sqrt [8]{b} x}{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}}}\right )}{4 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 \sqrt{-a} b^{3/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)/(a + b*(1 - x^2)^4),x]

[Out]

-ArcTan[(b^(1/8)*x)/Sqrt[(-a)^(1/4) - b^(1/4)]]/(4*Sqrt[-a]*Sqrt[(-a)^(1/4) - b^
(1/4)]*b^(3/8)) - (Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] - b^(1/4)]*ArcTan[(Sqrt[Sqrt[Sq
rt[-a] + Sqrt[b]] + b^(1/4)] - Sqrt[2]*b^(1/8)*x)/Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]]
- b^(1/4)]])/(4*Sqrt[2]*Sqrt[-a]*Sqrt[Sqrt[-a] + Sqrt[b]]*b^(3/8)) + (Sqrt[Sqrt[
Sqrt[-a] + Sqrt[b]] - b^(1/4)]*ArcTan[(Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] + b^(1/4)]
+ Sqrt[2]*b^(1/8)*x)/Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] - b^(1/4)]])/(4*Sqrt[2]*Sqrt[
-a]*Sqrt[Sqrt[-a] + Sqrt[b]]*b^(3/8)) + ArcTanh[(b^(1/8)*x)/Sqrt[(-a)^(1/4) + b^
(1/4)]]/(4*Sqrt[-a]*Sqrt[(-a)^(1/4) + b^(1/4)]*b^(3/8)) + (Sqrt[Sqrt[Sqrt[-a] +
Sqrt[b]] + b^(1/4)]*Log[Sqrt[Sqrt[-a] + Sqrt[b]] - Sqrt[2]*Sqrt[Sqrt[Sqrt[-a] +
Sqrt[b]] + b^(1/4)]*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*Sqrt[-a]*Sqrt[Sqrt[-a]
+ Sqrt[b]]*b^(3/8)) - (Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] + b^(1/4)]*Log[Sqrt[Sqrt[-a
] + Sqrt[b]] + Sqrt[2]*Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] + b^(1/4)]*b^(1/8)*x + b^(1
/4)*x^2])/(8*Sqrt[2]*Sqrt[-a]*Sqrt[Sqrt[-a] + Sqrt[b]]*b^(3/8))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- \frac{- x^{2} + 1}{a + b \left (- x^{2} + 1\right )^{4}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)/(a+b*(-x**2+1)**4),x)

[Out]

Integral(-(-x**2 + 1)/(a + b*(-x**2 + 1)**4), x)

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Mathematica [C]  time = 0.0530493, size = 63, normalized size = 0.1 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+a+b\&,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^5-2 \text{$\#$1}^3+\text{$\#$1}}\&\right ]}{8 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^2)/(a + b*(1 - x^2)^4),x]

[Out]

-RootSum[a + b - 4*b*#1^2 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , Log[x - #1]/(#1 - 2
*#1^3 + #1^5) & ]/(8*b)

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Maple [C]  time = 0.095, size = 69, normalized size = 0.1 \[{\frac{1}{8\,b}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{8}-4\,b{{\it \_Z}}^{6}+6\,b{{\it \_Z}}^{4}-4\,b{{\it \_Z}}^{2}+a+b \right ) }{\frac{ \left ( -{{\it \_R}}^{2}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}-3\,{{\it \_R}}^{5}+3\,{{\it \_R}}^{3}-{\it \_R}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(a+b*(-x^2+1)^4),x)

[Out]

1/8/b*sum((-_R^2+1)/(_R^7-3*_R^5+3*_R^3-_R)*ln(x-_R),_R=RootOf(_Z^8*b-4*_Z^6*b+6
*_Z^4*b-4*_Z^2*b+a+b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2} - 1}{{\left (x^{2} - 1\right )}^{4} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((x^2 - 1)/((x^2 - 1)^4*b + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 11.2101, size = 133, normalized size = 0.2 \[ - \operatorname{RootSum}{\left (t^{8} \left (16777216 a^{5} b^{3} + 16777216 a^{4} b^{4}\right ) + 1048576 t^{6} a^{3} b^{3} + 24576 t^{4} a^{2} b^{2} + 256 t^{2} a b + 1, \left ( t \mapsto t \log{\left (- 6291456 t^{7} a^{4} b^{3} - 6291456 t^{7} a^{3} b^{4} + 65536 t^{5} a^{3} b^{2} - 327680 t^{5} a^{2} b^{3} - 512 t^{3} a^{2} b - 5632 t^{3} a b^{2} - 32 t b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)/(a+b*(-x**2+1)**4),x)

[Out]

-RootSum(_t**8*(16777216*a**5*b**3 + 16777216*a**4*b**4) + 1048576*_t**6*a**3*b*
*3 + 24576*_t**4*a**2*b**2 + 256*_t**2*a*b + 1, Lambda(_t, _t*log(-6291456*_t**7
*a**4*b**3 - 6291456*_t**7*a**3*b**4 + 65536*_t**5*a**3*b**2 - 327680*_t**5*a**2
*b**3 - 512*_t**3*a**2*b - 5632*_t**3*a*b**2 - 32*_t*b + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2} - 1}{{\left (x^{2} - 1\right )}^{4} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(x^2 - 1)/((x^2 - 1)^4*b + a), x)