3.14.88 \(\int \frac {-1+k x^2}{(1+c k x+k x^2) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\) [1388]

Optimal. Leaf size=100 \[ \frac {2 \sqrt {1+2 k+k^2-c^2 k^2} \text {ArcTan}\left (\frac {\sqrt {1+2 k+k^2-c^2 k^2} x}{1+c k x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{(-1-k+c k) (1+k+c k)} \]

[Out]

2*(-c^2*k^2+k^2+2*k+1)^(1/2)*arctan((-c^2*k^2+k^2+2*k+1)^(1/2)*x/(1+c*k*x+k*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)
))/(c*k-k-1)/(c*k+k+1)

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.77, antiderivative size = 1193, normalized size of antiderivative = 11.93, number of steps used = 20, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {1976, 6860, 1117, 1738, 1230, 1720, 1261, 738, 212} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\left (1-c^2\right ) k^2+2 k+1} x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{\sqrt {\left (1-c^2\right ) k^2+2 k+1}}+\frac {\left (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \tanh ^{-1}\left (\frac {-4 \left (\left (1-c^2\right ) k^2+c \sqrt {c^2 k-4} k^{3/2}+2 k+1\right ) x^2 k^2+8 k^2-\left (k^2+1\right ) \left (c k-\sqrt {k} \sqrt {c^2 k-4}\right )^2}{4 \sqrt {2} k^{3/2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2+\sqrt {k} \sqrt {c^2 k-4} c+2} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 \sqrt {2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2+\sqrt {k} \sqrt {c^2 k-4} c+2}}+\frac {\left (\sqrt {k} c+\sqrt {c^2 k-4}\right ) \tanh ^{-1}\left (\frac {-2 \left (2 \left (k^2+1\right )-\left (c k+\sqrt {c^2 k-4} \sqrt {k}\right )^2\right ) x^2 k^2+8 k^2-\left (k^2+1\right ) \left (c k+\sqrt {c^2 k-4} \sqrt {k}\right )^2}{4 \sqrt {2} k^{3/2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2-\sqrt {k} \sqrt {c^2 k-4} c+2} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 \sqrt {2} \sqrt {\left (1-c^2\right ) k^2+2 k+1} \sqrt {-k c^2-\sqrt {k} \sqrt {c^2 k-4} c+2}}-\frac {\left (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (\sqrt {k} c+\sqrt {c^2 k-4}\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (-k c^2-\sqrt {k} \sqrt {c^2 k-4} c+4\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \Pi \left (\frac {c^2 k}{4};2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \left (\sqrt {k} c+\sqrt {c^2 k-4}\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (-k c^2+\sqrt {k} \sqrt {c^2 k-4} c+4\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \Pi \left (\frac {c^2 k}{4};2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{4 c k \left (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-1 + k*x^2)/((1 + c*k*x + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]

[Out]

-(ArcTan[(Sqrt[1 + 2*k + (1 - c^2)*k^2]*x)/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]]/Sqrt[1 + 2*k + (1 - c^2)*k^2]) +
((c*Sqrt[k] - Sqrt[-4 + c^2*k])*ArcTanh[(8*k^2 - (1 + k^2)*(c*k - Sqrt[k]*Sqrt[-4 + c^2*k])^2 - 4*k^2*(1 + 2*
k + (1 - c^2)*k^2 + c*k^(3/2)*Sqrt[-4 + c^2*k])*x^2)/(4*Sqrt[2]*k^(3/2)*Sqrt[1 + 2*k + (1 - c^2)*k^2]*Sqrt[2 -
c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])])/(2*Sqrt[2]*Sqrt[1 + 2*k + (1 - c^2)*
k^2]*Sqrt[2 - c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]]) + ((c*Sqrt[k] + Sqrt[-4 + c^2*k])*ArcTanh[(8*k^2 - (1 + k^2
)*(c*k + Sqrt[k]*Sqrt[-4 + c^2*k])^2 - 2*k^2*(2*(1 + k^2) - (c*k + Sqrt[k]*Sqrt[-4 + c^2*k])^2)*x^2)/(4*Sqrt[2
]*k^(3/2)*Sqrt[1 + 2*k + (1 - c^2)*k^2]*Sqrt[2 - c^2*k - c*Sqrt[k]*Sqrt[-4 + c^2*k]]*Sqrt[1 - (1 + k^2)*x^2 +
k^2*x^4])])/(2*Sqrt[2]*Sqrt[1 + 2*k + (1 - c^2)*k^2]*Sqrt[2 - c^2*k - c*Sqrt[k]*Sqrt[-4 + c^2*k]]) + ((1 + k*x
^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*EllipticF[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(2*Sqrt
[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((c*Sqrt[k] - Sqrt[-4 + c^2*k])*(1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 +
k^2*x^4)/(1 + k*x^2)^2]*EllipticF[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(4*c*k*Sqrt[1 - (1 + k^2)*x^2 + k^2*
x^4]) - ((c*Sqrt[k] + Sqrt[-4 + c^2*k])*(1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*Elliptic
F[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(4*c*k*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((4 - c^2*k - c*Sqrt[k]*S
qrt[-4 + c^2*k])*(1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*EllipticPi[(c^2*k)/4, 2*ArcTan[
Sqrt[k]*x], (1 + k)^2/(4*k)])/(4*c*k*(c*Sqrt[k] + Sqrt[-4 + c^2*k])*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((4 -
c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k])*(1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*EllipticPi[
(c^2*k)/4, 2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(4*c*k*(c*Sqrt[k] - Sqrt[-4 + c^2*k])*Sqrt[1 - (1 + k^2)*x^2
+ k^2*x^4])

Rule 212

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

Rule 738

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

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1230

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

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1720

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*
e*Rt[-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1738

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x
^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; Free
Q[{a, b, c, d, e}, x]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-1+k x^2}{\left (1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {-1+k x^2}{\sqrt {1-x^2} \left (1+c k x+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (\frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {2+c k x}{\sqrt {1-x^2} \left (1+c k x+k x^2\right ) \sqrt {1-k^2 x^2}}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {2+c k x}{\sqrt {1-x^2} \left (1+c k x+k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (\frac {c k-\sqrt {k} \sqrt {-4+c^2 k}}{\left (c k-\sqrt {k} \sqrt {-4+c^2 k}+2 k x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}+\frac {c k+\sqrt {k} \sqrt {-4+c^2 k}}{\left (c k+\sqrt {k} \sqrt {-4+c^2 k}+2 k x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt {k} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (c k-\sqrt {k} \sqrt {-4+c^2 k}+2 k x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt {k} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (c k+\sqrt {k} \sqrt {-4+c^2 k}+2 k x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^{3/2} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (k \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^{3/2} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (k \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (k^{3/2} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (k^{3/2} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-4 k^2 x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^{3/2} \left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{4 k^2-\left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2-\left (4 k^2-k^2 \left (c k-\sqrt {k} \sqrt {-4+c^2 k}\right )^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^{3/2} \left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{4 k^2-\left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2-\left (4 k^2-k^2 \left (c k+\sqrt {k} \sqrt {-4+c^2 k}\right )^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {2+2 k-c^2 k^2-c k^{3/2} \sqrt {-4+c^2 k}} \sqrt {1-x^2}}{\sqrt {2+\left (2-c^2\right ) k-c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {1-k^2 x^2}}\right )}{\sqrt {2+\left (2-c^2\right ) k-c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {2+2 k-c^2 k^2-c k^{3/2} \sqrt {-4+c^2 k}} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {2+2 k-c^2 k^2+c k^{3/2} \sqrt {-4+c^2 k}} \sqrt {1-x^2}}{\sqrt {2+\left (2-c^2\right ) k+c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {1-k^2 x^2}}\right )}{\sqrt {2+\left (2-c^2\right ) k+c \sqrt {k} \sqrt {-4+c^2 k}} \sqrt {2+2 k-c^2 k^2+c k^{3/2} \sqrt {-4+c^2 k}} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}-\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {4 k}{\left (c \sqrt {k}+\sqrt {-4+c^2 k}\right )^2};\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 14.23, size = 1374, normalized size = 13.74 \begin {gather*} \frac {\frac {\sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \left (-\frac {\left (-4+\left (-4+c^2\right ) k+c^2 k^2+c \sqrt {k} \sqrt {-4+c^2 k}-c k^{3/2} \sqrt {-4+c^2 k}\right ) \tanh ^{-1}\left (\frac {\sqrt {-2-4 k+2 \left (-1+c^2\right ) k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3+c \sqrt {k} \sqrt {-4+c^2 k}-c k^{5/2} \sqrt {-4+c^2 k}} \sqrt {-1+x^2}}\right )}{\sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3+c \sqrt {k} \sqrt {-4+c^2 k}-c k^{5/2} \sqrt {-4+c^2 k}}}+\frac {\left (-4+\left (-4+c^2\right ) k+c^2 k^2-c \sqrt {k} \sqrt {-4+c^2 k}+c k^{3/2} \sqrt {-4+c^2 k}\right ) \tanh ^{-1}\left (\frac {\sqrt {-2-4 k+2 \left (-1+c^2\right ) k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3-c \sqrt {k} \sqrt {-4+c^2 k}+c k^{5/2} \sqrt {-4+c^2 k}} \sqrt {-1+x^2}}\right )}{\sqrt {-2+\left (-4+c^2\right ) k-2 k^2+c^2 k^3-c \sqrt {k} \sqrt {-4+c^2 k}+c k^{5/2} \sqrt {-4+c^2 k}}}\right )}{\sqrt {-2+\frac {c^2 k}{2}} \sqrt {-1-2 k+\left (-1+c^2\right ) k^2}}+2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\text {ArcSin}(x)\left |k^2\right .\right )-\frac {4 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}+\frac {2 c^2 k \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}+\frac {8 c \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}-\frac {2 c^3 k^{3/2} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\frac {2 k}{2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (2-c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}+\frac {4 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}-\frac {2 c^2 k \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}}+\frac {8 c \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}-\frac {2 c^3 k^{3/2} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {2 k}{-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}};\text {ArcSin}(x)\left |k^2\right .\right )}{\sqrt {-4+c^2 k} \left (-2+c^2 k+c \sqrt {k} \sqrt {-4+c^2 k}\right )}}{2 \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-1 + k*x^2)/((1 + c*k*x + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]

[Out]

((Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*(-(((-4 + (-4 + c^2)*k + c^2*k^2 + c*Sqrt[k]*Sqrt[-4 + c^2*k] - c*k^(3/2)*
Sqrt[-4 + c^2*k])*ArcTanh[(Sqrt[-2 - 4*k + 2*(-1 + c^2)*k^2]*Sqrt[-1 + k^2*x^2])/(Sqrt[k]*Sqrt[-2 + (-4 + c^2)
*k - 2*k^2 + c^2*k^3 + c*Sqrt[k]*Sqrt[-4 + c^2*k] - c*k^(5/2)*Sqrt[-4 + c^2*k]]*Sqrt[-1 + x^2])])/Sqrt[-2 + (-
4 + c^2)*k - 2*k^2 + c^2*k^3 + c*Sqrt[k]*Sqrt[-4 + c^2*k] - c*k^(5/2)*Sqrt[-4 + c^2*k]]) + ((-4 + (-4 + c^2)*k
+ c^2*k^2 - c*Sqrt[k]*Sqrt[-4 + c^2*k] + c*k^(3/2)*Sqrt[-4 + c^2*k])*ArcTanh[(Sqrt[-2 - 4*k + 2*(-1 + c^2)*k^
2]*Sqrt[-1 + k^2*x^2])/(Sqrt[k]*Sqrt[-2 + (-4 + c^2)*k - 2*k^2 + c^2*k^3 - c*Sqrt[k]*Sqrt[-4 + c^2*k] + c*k^(5
/2)*Sqrt[-4 + c^2*k]]*Sqrt[-1 + x^2])])/Sqrt[-2 + (-4 + c^2)*k - 2*k^2 + c^2*k^3 - c*Sqrt[k]*Sqrt[-4 + c^2*k]
+ c*k^(5/2)*Sqrt[-4 + c^2*k]]))/(Sqrt[-2 + (c^2*k)/2]*Sqrt[-1 - 2*k + (-1 + c^2)*k^2]) + 2*Sqrt[1 - x^2]*Sqrt[
1 - k^2*x^2]*EllipticF[ArcSin[x], k^2] - (4*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k)/(2 - c^2*k + c*S
qrt[k]*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/(2 - c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]) + (2*c^2*k*Sqrt[1 - x^2]*S
qrt[1 - k^2*x^2]*EllipticPi[(-2*k)/(2 - c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/(2 - c^2*k + c*S
qrt[k]*Sqrt[-4 + c^2*k]) + (8*c*Sqrt[k]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k)/(2 - c^2*k + c*Sqrt[
k]*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/(Sqrt[-4 + c^2*k]*(2 - c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k])) - (2*c^3*k^
(3/2)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(-2*k)/(2 - c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]), ArcSin[x], k
^2])/(Sqrt[-4 + c^2*k]*(2 - c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k])) + (4*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*Elliptic
Pi[(2*k)/(-2 + c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/(-2 + c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k])
- (2*c^2*k*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(2*k)/(-2 + c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]), ArcSin
[x], k^2])/(-2 + c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]) + (8*c*Sqrt[k]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi
[(2*k)/(-2 + c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k]), ArcSin[x], k^2])/(Sqrt[-4 + c^2*k]*(-2 + c^2*k + c*Sqrt[k]*S
qrt[-4 + c^2*k])) - (2*c^3*k^(3/2)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(2*k)/(-2 + c^2*k + c*Sqrt[k]*Sq
rt[-4 + c^2*k]), ArcSin[x], k^2])/(Sqrt[-4 + c^2*k]*(-2 + c^2*k + c*Sqrt[k]*Sqrt[-4 + c^2*k])))/(2*Sqrt[(-1 +
x^2)*(-1 + k^2*x^2)])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.33, size = 11374, normalized size = 113.74

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(4734\)
default	\(\text {Expression too large to display}\)	\(11374\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((k*x^2-1)/(c*k*x+k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((k*x^2-1)/(c*k*x+k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2*k-4>0)', see `assume?` for
more detail

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Fricas [A]
time = 0.67, size = 342, normalized size = 3.42 \begin {gather*} \left [\frac {\log \left (-\frac {{\left ({\left (2 \, c^{2} - 1\right )} k^{4} - 2 \, k^{3} - k^{2}\right )} x^{4} + 2 \, {\left (c k^{4} + 2 \, c k^{3} + c k^{2}\right )} x^{3} + {\left (2 \, c^{2} - 1\right )} k^{2} - {\left ({\left (c^{2} - 2\right )} k^{4} - 2 \, {\left (c^{2} + 3\right )} k^{3} + {\left (c^{2} - 8\right )} k^{2} - 6 \, k - 2\right )} x^{2} + 2 \, \sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left (c k^{2} x^{2} + c k + {\left (k^{2} + 2 \, k + 1\right )} x\right )} \sqrt {{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1} + 2 \, {\left (c k^{3} + 2 \, c k^{2} + c k\right )} x - 2 \, k - 1}{2 \, c k^{2} x^{3} + k^{2} x^{4} + 2 \, c k x + {\left (c^{2} k^{2} + 2 \, k\right )} x^{2} + 1}\right )}{2 \, \sqrt {{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1}}, -\frac {\sqrt {-{\left (c^{2} - 1\right )} k^{2} + 2 \, k + 1} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {-{\left (c^{2} - 1\right )} k^{2} + 2 \, k + 1}}{c k^{2} x^{2} + c k + {\left (k^{2} + 2 \, k + 1\right )} x}\right )}{{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((k*x^2-1)/(c*k*x+k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="fricas")

[Out]

[1/2*log(-(((2*c^2 - 1)*k^4 - 2*k^3 - k^2)*x^4 + 2*(c*k^4 + 2*c*k^3 + c*k^2)*x^3 + (2*c^2 - 1)*k^2 - ((c^2 - 2
)*k^4 - 2*(c^2 + 3)*k^3 + (c^2 - 8)*k^2 - 6*k - 2)*x^2 + 2*sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(c*k^2*x^2 + c*k
+ (k^2 + 2*k + 1)*x)*sqrt((c^2 - 1)*k^2 - 2*k - 1) + 2*(c*k^3 + 2*c*k^2 + c*k)*x - 2*k - 1)/(2*c*k^2*x^3 + k^2
*x^4 + 2*c*k*x + (c^2*k^2 + 2*k)*x^2 + 1))/sqrt((c^2 - 1)*k^2 - 2*k - 1), -sqrt(-(c^2 - 1)*k^2 + 2*k + 1)*arct
an(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*sqrt(-(c^2 - 1)*k^2 + 2*k + 1)/(c*k^2*x^2 + c*k + (k^2 + 2*k + 1)*x))/((c
^2 - 1)*k^2 - 2*k - 1)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k x^{2} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (c k x + k x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((k*x**2-1)/(c*k*x+k*x**2+1)/((-x**2+1)*(-k**2*x**2+1))**(1/2),x)

[Out]

Integral((k*x**2 - 1)/(sqrt((x - 1)*(x + 1)*(k*x - 1)*(k*x + 1))*(c*k*x + k*x**2 + 1)), 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((k*x^2-1)/(c*k*x+k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="giac")

[Out]

integrate((k*x^2 - 1)/((c*k*x + k*x^2 + 1)*sqrt((k^2*x^2 - 1)*(x^2 - 1))), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {k\,x^2-1}{\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}\,\left (k\,x^2+c\,k\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((k*x^2 - 1)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/2)*(k*x^2 + c*k*x + 1)),x)

[Out]

int((k*x^2 - 1)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/2)*(k*x^2 + c*k*x + 1)), x)

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