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

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[S
qrt[-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)) + (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*S
qrt[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 = 1.10775, 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, Rules used = {6740, 1990, 1166, 205, 208, 1169, 634, 618, 204, 628} \[ \frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (-\sqrt{2} \sqrt [8]{b} x \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (\sqrt{2} \sqrt [8]{b} x \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}\right )}{4 \sqrt{-a} b^{3/8} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}-\sqrt{2} \sqrt [8]{b} x}{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}}}\right )}{4 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}+\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}+\sqrt{2} \sqrt [8]{b} x}{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt [4]{b}}}\right )}{4 \sqrt{2} \sqrt{-a} b^{3/8} \sqrt{\sqrt{-a}+\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 \sqrt{-a} b^{3/8} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}} \]

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[S
qrt[-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)) + (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*S
qrt[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))

Rule 6740

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]

Rule 1990

Int[(u_)^(q_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*ExpandToSum[v, x]^p, x] /; FreeQ[{p, q}, x] &&
BinomialQ[u, x] && TrinomialQ[v, x] &&  !(BinomialMatchQ[u, x] && TrinomialMatchQ[v, x])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1-x^2}{a+b \left (1-x^2\right )^4} \, dx &=\int \left (-\frac{\sqrt{b} \left (1-x^2\right )}{2 \sqrt{-a} \left (\sqrt{-a} \sqrt{b}-b \left (1-x^2\right )^2\right )}-\frac{\sqrt{b} \left (1-x^2\right )}{2 \sqrt{-a} \left (\sqrt{-a} \sqrt{b}+b \left (1-x^2\right )^2\right )}\right ) \, dx\\ &=-\frac{\sqrt{b} \int \frac{1-x^2}{\sqrt{-a} \sqrt{b}-b \left (1-x^2\right )^2} \, dx}{2 \sqrt{-a}}-\frac{\sqrt{b} \int \frac{1-x^2}{\sqrt{-a} \sqrt{b}+b \left (1-x^2\right )^2} \, dx}{2 \sqrt{-a}}\\ &=-\frac{\sqrt{b} \int \frac{1-x^2}{\left (\sqrt{-a}-\sqrt{b}\right ) \sqrt{b}+2 b x^2-b x^4} \, dx}{2 \sqrt{-a}}-\frac{\sqrt{b} \int \frac{1-x^2}{\left (\sqrt{-a}+\sqrt{b}\right ) \sqrt{b}-2 b x^2+b x^4} \, dx}{2 \sqrt{-a}}\\ &=-\frac{\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}}{\sqrt [8]{b}}-\left (1+\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}\right ) x}{\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} x}{\sqrt [8]{b}}+x^2} \, dx}{4 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \sqrt{\sqrt{-a}+\sqrt{b}} \sqrt [8]{b}}-\frac{\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}}{\sqrt [8]{b}}+\left (1+\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}\right ) x}{\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} x}{\sqrt [8]{b}}+x^2} \, dx}{4 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \sqrt{\sqrt{-a}+\sqrt{b}} \sqrt [8]{b}}+\frac{\sqrt{b} \int \frac{1}{-\sqrt [4]{-a} b^{3/4}+b-b x^2} \, dx}{4 \sqrt{-a}}+\frac{\sqrt{b} \int \frac{1}{\sqrt [4]{-a} b^{3/4}+b-b x^2} \, dx}{4 \sqrt{-a}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}\right )}{4 \sqrt{-a} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}} b^{3/8}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 \sqrt{-a} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}} b^{3/8}}+\frac{\left (1-\frac{\sqrt [4]{b}}{\sqrt{\sqrt{-a}+\sqrt{b}}}\right ) \int \frac{1}{\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt{-a} \sqrt{b}}+\frac{\left (1-\frac{\sqrt [4]{b}}{\sqrt{\sqrt{-a}+\sqrt{b}}}\right ) \int \frac{1}{\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt{-a} \sqrt{b}}+\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \int \frac{-\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}}{\sqrt [8]{b}}+2 x}{\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \int \frac{\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}}{\sqrt [8]{b}}+2 x}{\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} x}{\sqrt [8]{b}}+x^2} \, dx}{8 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}\right )}{4 \sqrt{-a} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}} b^{3/8}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 \sqrt{-a} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}} b^{3/8}}+\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}-\frac{\left (1-\frac{\sqrt [4]{b}}{\sqrt{\sqrt{-a}+\sqrt{b}}}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}\right )-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}}{\sqrt [8]{b}}+2 x\right )}{4 \sqrt{-a} \sqrt{b}}-\frac{\left (1-\frac{\sqrt [4]{b}}{\sqrt{\sqrt{-a}+\sqrt{b}}}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\frac{\sqrt{\sqrt{-a}+\sqrt{b}}}{\sqrt [4]{b}}\right )-x^2} \, dx,x,\frac{\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}}}{\sqrt [8]{b}}+2 x\right )}{4 \sqrt{-a} \sqrt{b}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}-\sqrt [4]{b}}}\right )}{4 \sqrt{-a} \sqrt{\sqrt [4]{-a}-\sqrt [4]{b}} b^{3/8}}-\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} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}+\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} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt{\sqrt [4]{-a}+\sqrt [4]{b}}}\right )}{4 \sqrt{-a} \sqrt{\sqrt [4]{-a}+\sqrt [4]{b}} b^{3/8}}+\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (\sqrt{\sqrt{-a}+\sqrt{b}}-\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}-\frac{\sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \log \left (\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt{2} \sqrt{\sqrt{\sqrt{-a}+\sqrt{b}}+\sqrt [4]{b}} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} \sqrt{-a} \sqrt{\sqrt{-a}+\sqrt{b}} b^{3/8}}\\ \end{align*}

Mathematica [C]  time = 0.0384053, 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 = 0.072, size = 69, normalized size = 0.1 \begin{align*}{\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}}}} \end{align*}

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. \begin{align*} -\int \frac{x^{2} - 1}{{\left (x^{2} - 1\right )}^{4} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [A]  time = 2.98831, size = 133, normalized size = 0.2 \begin{align*} - \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 )} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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