3.4.92 \(\int \frac {1-x^2}{a+b (-1+x^2)^4} \, dx\) [392]

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

[Out]

-1/4*arctan(b^(1/8)*x/((-a)^(1/4)-b^(1/4))^(1/2))/b^(3/8)/(-a)^(1/2)/((-a)^(1/4)-b^(1/4))^(1/2)+1/4*arctanh(b^
(1/8)*x/((-a)^(1/4)+b^(1/4))^(1/2))/b^(3/8)/(-a)^(1/2)/((-a)^(1/4)+b^(1/4))^(1/2)-1/8*arctan((-b^(1/8)*x*2^(1/
2)+(b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2))^(1/2))/(-b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2))^(1/2))*(-b^(1/4)+((-a)^(1
/2)+b^(1/2))^(1/2))^(1/2)/b^(3/8)*2^(1/2)/(-a)^(1/2)/((-a)^(1/2)+b^(1/2))^(1/2)+1/8*arctan((b^(1/8)*x*2^(1/2)+
(b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2))^(1/2))/(-b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2))^(1/2))*(-b^(1/4)+((-a)^(1/2)
+b^(1/2))^(1/2))^(1/2)/b^(3/8)*2^(1/2)/(-a)^(1/2)/((-a)^(1/2)+b^(1/2))^(1/2)+1/16*ln(b^(1/4)*x^2+((-a)^(1/2)+b
^(1/2))^(1/2)-b^(1/8)*x*2^(1/2)*(b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2))^(1/2))*(b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2
))^(1/2)/b^(3/8)*2^(1/2)/(-a)^(1/2)/((-a)^(1/2)+b^(1/2))^(1/2)-1/16*ln(b^(1/4)*x^2+((-a)^(1/2)+b^(1/2))^(1/2)+
b^(1/8)*x*2^(1/2)*(b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2))^(1/2))*(b^(1/4)+((-a)^(1/2)+b^(1/2))^(1/2))^(1/2)/b^(3/
8)*2^(1/2)/(-a)^(1/2)/((-a)^(1/2)+b^(1/2))^(1/2)

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Rubi [A]
time = 0.71, antiderivative size = 663, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6872, 2015, 1180, 211, 214, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\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}} \text {ArcTan}\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}} \text {ArcTan}\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}} \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 {\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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(a + b*(-1 + x^2)^4),x]

[Out]

-1/4*ArcTan[(b^(1/8)*x)/Sqrt[(-a)^(1/4) - b^(1/4)]]/(Sqrt[-a]*Sqrt[(-a)^(1/4) - b^(1/4)]*b^(3/8)) - (Sqrt[Sqrt
[Sqrt[-a] + Sqrt[b]] - b^(1/4)]*ArcTan[(Sqrt[Sqrt[Sqrt[-a] + Sqrt[b]] + b^(1/4)] - Sqrt[2]*b^(1/8)*x)/Sqrt[Sqr
t[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))

Rule 210

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

Rule 211

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

Rule 214

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

Rule 632

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 642

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]

Rule 648

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 1180

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 1183

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 2015

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 6872

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]

Rubi steps

\begin {align*} \int \frac {1-x^2}{a+b \left (-1+x^2\right )^4} \, dx &=-\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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.02, size = 63, normalized size = 0.10 \begin {gather*} -\frac {\text {RootSum}\left [a+b-4 b \text {$\#$1}^2+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{\text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*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) & ]/b

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.03, size = 69, normalized size = 0.10

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{8}-4 b \,\textit {\_Z}^{6}+6 \textit {\_Z}^{4} b -4 \textit {\_Z}^{2} b +a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+3 \textit {\_R}^{3}-\textit {\_R}}}{8 b}\)	\(69\)
risch	\(\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{8}-4 b \,\textit {\_Z}^{6}+6 \textit {\_Z}^{4} b -4 \textit {\_Z}^{2} b +a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+3 \textit {\_R}^{3}-\textit {\_R}}}{8 b}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)/(a+b*(x^2-1)^4),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.91, size = 322185, normalized size = 485.95 \begin {gather*} \text {too large to display} \end {gather*}

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]

1/8*sqrt(1/2)*sqrt(1/6)*sqrt((6*sqrt(1/2)*sqrt(1/6)*(a^2 + a*b)*sqrt(-((a^4*b + 2*a^3*b^2 + a^2*b^3)*(2*(1/2)^
(2/3)*(-I*sqrt(3) + 1)*((a^2*b*sqrt(-((2*a*sqrt(-1/(a*b)) + 1)*b - a)/(a^4*b^2*sqrt(-1/(a*b)) + 2*a^3*b^3*sqrt
(-1/(a*b)) + a^2*b^4*sqrt(-1/(a*b))))*sqrt(-1/(a*b)) + a*b^2*sqrt(-((2*a*sqrt(-1/(a*b)) + 1)*b - a)/(a^4*b^2*s
qrt(-1/(a*b)) + 2*a^3*b^3*sqrt(-1/(a*b)) + a^2*b^4*sqrt(-1/(a*b))))*sqrt(-1/(a*b)) - 3*b*sqrt(-1/(a*b)) - 1)^2
/(a^2*b*sqrt(-1/(a*b)) + a*b^2*sqrt(-1/(a*b)))^2 + 3*(2*(b^2*sqrt(-((2*a*sqrt(-1/(a*b)) + 1)*b - a)/(a^4*b^2*s
qrt(-1/(a*b)) + 2*a^3*b^3*sqrt(-1/(a*b)) + a^2*b^4*sqrt(-1/(a*b))))*sqrt(-1/(a*b)) + b*sqrt(-((2*a*sqrt(-1/(a*
b)) + 1)*b - a)/(a^4*b^2*sqrt(-1/(a*b)) + 2*a^3*b^3*sqrt(-1/(a*b)) + a^2*b^4*sqrt(-1/(a*b)))))*a - 3*b*sqrt(-1
/(a*b)) - 1)/(a^3*b^2*sqrt(-1/(a*b)) + a^2*b^3*sqrt(-1/(a*b))))/(27*(a^5*b^2*(-((2*a*sqrt(-1/(a*b)) + 1)*b - a
)/(a^4*b^2*sqrt(-1/(a*b)) + 2*a^3*b^3*sqrt(-1/(a*b)) + a^2*b^4*sqrt(-1/(a*b))))^(3/2)*sqrt(-1/(a*b)) + 2*a^4*b
^3*(-((2*a ...

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Sympy [A]
time = 2.15, size = 133, normalized size = 0.20 \begin {gather*} - \operatorname {RootSum} {\left (t^{8} \cdot \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.00, size = 328, normalized size = 0.49 \begin {gather*} \sum _{k=1}^8\ln \left (a\,b^5\,\left ({\mathrm {root}\left (16777216\,a^5\,b^3\,z^8+16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6+24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2+1,z,k\right )}^2\,a\,b\,64+1\right )\,\left ({\mathrm {root}\left (16777216\,a^5\,b^3\,z^8+16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6+24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2+1,z,k\right )}^4\,a^2\,b^2\,4096+{\mathrm {root}\left (16777216\,a^5\,b^3\,z^8+16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6+24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2+1,z,k\right )}^2\,a\,b\,128-{\mathrm {root}\left (16777216\,a^5\,b^3\,z^8+16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6+24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2+1,z,k\right )}^5\,a^3\,b^2\,x\,32768+1\right )\right )\,\mathrm {root}\left (16777216\,a^5\,b^3\,z^8+16777216\,a^4\,b^4\,z^8+1048576\,a^3\,b^3\,z^6+24576\,a^2\,b^2\,z^4+256\,a\,b\,z^2+1,z,k\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)/(a + b*(x^2 - 1)^4),x)

[Out]

symsum(log(a*b^5*(64*root(16777216*a^5*b^3*z^8 + 16777216*a^4*b^4*z^8 + 1048576*a^3*b^3*z^6 + 24576*a^2*b^2*z^
4 + 256*a*b*z^2 + 1, z, k)^2*a*b + 1)*(4096*root(16777216*a^5*b^3*z^8 + 16777216*a^4*b^4*z^8 + 1048576*a^3*b^3
*z^6 + 24576*a^2*b^2*z^4 + 256*a*b*z^2 + 1, z, k)^4*a^2*b^2 + 128*root(16777216*a^5*b^3*z^8 + 16777216*a^4*b^4
*z^8 + 1048576*a^3*b^3*z^6 + 24576*a^2*b^2*z^4 + 256*a*b*z^2 + 1, z, k)^2*a*b - 32768*root(16777216*a^5*b^3*z^
8 + 16777216*a^4*b^4*z^8 + 1048576*a^3*b^3*z^6 + 24576*a^2*b^2*z^4 + 256*a*b*z^2 + 1, z, k)^5*a^3*b^2*x + 1))*
root(16777216*a^5*b^3*z^8 + 16777216*a^4*b^4*z^8 + 1048576*a^3*b^3*z^6 + 24576*a^2*b^2*z^4 + 256*a*b*z^2 + 1,
z, k), k, 1, 8)

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