\(\int \frac {1+k x^2}{(-1+c k x+k x^2) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\) [1387]

Optimal result

Integrand size = 43, antiderivative size = 100 \[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {2 \sqrt {1-2 k+k^2-c^2 k^2} \arctan \left (\frac {\sqrt {1-2 k+k^2-c^2 k^2} x}{-1+c k x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{(1-k+c k) (-1+k+c k)} \]

[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)

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.75 (sec) , antiderivative size = 1288, normalized size of antiderivative = 12.88, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {1976, 6860, 1117, 1738, 1230, 1720, 1261, 738, 210} \[ \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 (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}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right ) \left (c \sqrt {k}-\sqrt {k c^2+4}\right )^2}{4 \sqrt {k} \left (k c^2-\sqrt {k} \sqrt {k c^2+4} c+4\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {c \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}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (k c^2+4\right ),2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right ) \left (c \sqrt {k}-\sqrt {k c^2+4}\right )^2}{8 \left (k^{3/2} c^3-k \sqrt {k c^2+4} c^2+4 \sqrt {k} c-2 \sqrt {k c^2+4}\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\arctan \left (\frac {-2 \left (2 \left (k^2+1\right )-\left (c k-\sqrt {k} \sqrt {k c^2+4}\right )^2\right ) x^2 k^2+8 k^2-\left (k^2+1\right ) \left (c k-\sqrt {k} \sqrt {k c^2+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 {k c^2+4} c+2} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right ) \left (c \sqrt {k}-\sqrt {k c^2+4}\right )}{2 \sqrt {2} \sqrt {\left (1-c^2\right ) k^2-2 k+1} \sqrt {k c^2-\sqrt {k} \sqrt {k c^2+4} c+2}}-\frac {\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 (\sqrt {k} c+\sqrt {k c^2+4}\right ) \arctan \left (\frac {-2 \left (2 \left (k^2+1\right )-\left (c k+\sqrt {k c^2+4} \sqrt {k}\right )^2\right ) x^2 k^2+8 k^2-\left (k^2+1\right ) \left (c k+\sqrt {k c^2+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 {k c^2+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 {k c^2+4} c+2}}+\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}} \operatorname {EllipticF}\left (2 \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 (\sqrt {k} c+\sqrt {k c^2+4}\right )^2 \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}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{4 \sqrt {k} \left (k c^2+\sqrt {k} \sqrt {k c^2+4} c+4\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {c \left (\sqrt {k} c+\sqrt {k c^2+4}\right )^2 \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}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (k c^2+4\right ),2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{8 \left (k^{3/2} c^3+k \sqrt {k c^2+4} c^2+4 \sqrt {k} c+2 \sqrt {k c^2+4}\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}} \]

[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])*ArcTan[(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]]) - ((c*Sqrt[k] + Sqrt[4 + c^2*k])*ArcTan[(8*k^2 - (1 + k^2)*(c*k + Sqr
t[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])^2*(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*Sqrt[k]*(4 + c^2*k - c*Sqrt[k]*Sqrt[4 + c^2*k])*
Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((c*Sqrt[k] + Sqrt[4 + c^2*k])^2*(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*Sqrt[k]*(4 + c^2*k + c*Sqrt[k]*Sqrt[
4 + c^2*k])*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) + (c*(c*Sqrt[k] - Sqrt[4 + c^2*k])^2*(1 + k*x^2)*Sqrt[(1 - (1 +
k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*EllipticPi[(4 + c^2*k)/4, 2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(8*(4*c*S
qrt[k] + c^3*k^(3/2) - 2*Sqrt[4 + c^2*k] - c^2*k*Sqrt[4 + c^2*k])*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) + (c*(c*S
qrt[k] + Sqrt[4 + c^2*k])^2*(1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*EllipticPi[(4 + c^2*
k)/4, 2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(8*(4*c*Sqrt[k] + c^3*k^(3/2) + 2*Sqrt[4 + c^2*k] + c^2*k*Sqrt[4
+ c^2*k])*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])

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 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*} \text {integral}& = \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (\frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2-c k x}{\left (-1+c k x+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\int \frac {2-c k x}{\left (-1+c k x+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\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+\left (-1-k^2\right ) x^2+k^2 x^4}}+\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+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\left (\sqrt {k} \left (c \sqrt {k}-\sqrt {4+c^2 k}\right )\right ) \int \frac {1}{\left (c k-\sqrt {k} \sqrt {4+c^2 k}+2 k x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\left (\sqrt {k} \left (c \sqrt {k}+\sqrt {4+c^2 k}\right )\right ) \int \frac {1}{\left (c k+\sqrt {k} \sqrt {4+c^2 k}+2 k x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\left (2 k^{3/2} \left (c \sqrt {k}-\sqrt {4+c^2 k}\right )\right ) \int \frac {x}{\left (\left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\left (k \left (c \sqrt {k}-\sqrt {4+c^2 k}\right )^2\right ) \int \frac {1}{\left (\left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\left (2 k^{3/2} \left (c \sqrt {k}+\sqrt {4+c^2 k}\right )\right ) \int \frac {x}{\left (\left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\left (k \left (c \sqrt {k}+\sqrt {4+c^2 k}\right )^2\right ) \int \frac {1}{\left (\left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\left (k^{3/2} \left (c \sqrt {k}-\sqrt {4+c^2 k}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )+\left (k^{3/2} \left (c \sqrt {k}+\sqrt {4+c^2 k}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )-\frac {\left (c \sqrt {k}-\sqrt {4+c^2 k}\right )^2 \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{2 \left (4+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}\right )}-\frac {\left (2 k \left (c \sqrt {k}-\sqrt {4+c^2 k}\right )^2\right ) \int \frac {1+k x^2}{\left (\left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{4+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}}-\frac {\left (c \sqrt {k}+\sqrt {4+c^2 k}\right )^2 \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{2 \left (4+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}\right )}-\frac {\left (2 k \left (c \sqrt {k}+\sqrt {4+c^2 k}\right )^2\right ) \int \frac {1+k x^2}{\left (\left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2-4 k^2 x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{4+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}} \\ & = -\frac {\arctan \left (\frac {\sqrt {1-2 k+\left (1-c^2\right ) k^2} x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{\sqrt {1-2 k+\left (1-c^2\right ) k^2}}+\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (c \sqrt {k}-\sqrt {4+c^2 k}\right )^2 \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \sqrt {k} \left (4+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (c \sqrt {k}+\sqrt {4+c^2 k}\right )^2 \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \sqrt {k} \left (4+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {c \left (c \sqrt {k}-\sqrt {4+c^2 k}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (4+c^2 k\right ),2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \left (4+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {c \left (c \sqrt {k}+\sqrt {4+c^2 k}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (4+c^2 k\right ),2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \left (4+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\left (2 k^{3/2} \left (c \sqrt {k}-\sqrt {4+c^2 k}\right )\right ) \text {Subst}\left (\int \frac {1}{64 k^4+16 k^2 \left (-1-k^2\right ) \left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2+4 k^2 \left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^4-x^2} \, dx,x,\frac {-8 k^2-\left (-1-k^2\right ) \left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2-2 k^2 \left (2 \left (-1-k^2\right )+\left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2\right ) x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )-\left (2 k^{3/2} \left (c \sqrt {k}+\sqrt {4+c^2 k}\right )\right ) \text {Subst}\left (\int \frac {1}{64 k^4+16 k^2 \left (-1-k^2\right ) \left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2+4 k^2 \left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^4-x^2} \, dx,x,\frac {-8 k^2-\left (-1-k^2\right ) \left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2-2 k^2 \left (2 \left (-1-k^2\right )+\left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2\right ) x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {\sqrt {1-2 k+\left (1-c^2\right ) k^2} x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{\sqrt {1-2 k+\left (1-c^2\right ) k^2}}-\frac {\left (c \sqrt {k}-\sqrt {4+c^2 k}\right ) \arctan \left (\frac {8 k^2-\left (1+k^2\right ) \left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2-2 k^2 \left (2 \left (1+k^2\right )-\left (c k-\sqrt {k} \sqrt {4+c^2 k}\right )^2\right ) x^2}{4 \sqrt {2} k^{3/2} \sqrt {1-2 k+\left (1-c^2\right ) k^2} \sqrt {2+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{2 \sqrt {2} \sqrt {1-2 k+\left (1-c^2\right ) k^2} \sqrt {2+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}}}-\frac {\left (c \sqrt {k}+\sqrt {4+c^2 k}\right ) \arctan \left (\frac {8 k^2-\left (1+k^2\right ) \left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2-2 k^2 \left (2 \left (1+k^2\right )-\left (c k+\sqrt {k} \sqrt {4+c^2 k}\right )^2\right ) x^2}{4 \sqrt {2} k^{3/2} \sqrt {1-2 k+\left (1-c^2\right ) k^2} \sqrt {2+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{2 \sqrt {2} \sqrt {1-2 k+\left (1-c^2\right ) k^2} \sqrt {2+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}}}+\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (c \sqrt {k}-\sqrt {4+c^2 k}\right )^2 \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \sqrt {k} \left (4+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (c \sqrt {k}+\sqrt {4+c^2 k}\right )^2 \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \sqrt {k} \left (4+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {c \left (c \sqrt {k}-\sqrt {4+c^2 k}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (4+c^2 k\right ),2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \left (4+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {c \left (c \sqrt {k}+\sqrt {4+c^2 k}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (4+c^2 k\right ),2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{4 \left (4+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}\right ) \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 16.38 (sec) , antiderivative size = 1373, normalized size of antiderivative = 13.73 \[ \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 {\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 ) \text {arctanh}\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 ) \text {arctanh}\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} \operatorname {EllipticF}\left (\arcsin (x),k^2\right )-\frac {4 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k-c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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} \operatorname {EllipticPi}\left (\frac {2 k}{2+c^2 k+c \sqrt {k} \sqrt {4+c^2 k}},\arcsin (x),k^2\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 )}} \]

[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]*Elliptic
F[ArcSin[x], k^2] - (4*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]) - (2*c^2*k*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*Ellipti
cPi[(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]*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]*EllipticPi[(2*k)/(2 + c^2*k + c*Sqrt[k]*Sqr
t[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]*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])))/(2*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)])

Maple [A] (verified)

Time = 3.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {\ln \left (2\right )+\ln \left (\frac {-\sqrt {\left (c k +k -1\right ) \left (c k -k +1\right )}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+\left (c \,x^{2}+x \right ) k^{2}+\left (-c -2 x \right ) k +x}{-1+x \left (c +x \right ) k}\right )}{\sqrt {-1+\left (c^{2}-1\right ) k^{2}+2 k}}\)	\(92\)
pseudoelliptic	\(-\frac {\ln \left (2\right )+\ln \left (\frac {-\sqrt {\left (c k +k -1\right ) \left (c k -k +1\right )}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+\left (c \,x^{2}+x \right ) k^{2}+\left (-c -2 x \right ) k +x}{-1+x \left (c +x \right ) k}\right )}{\sqrt {-1+\left (c^{2}-1\right ) k^{2}+2 k}}\)	\(92\)
elliptic	\(\text {Expression too large to display}\)	\(1699\)

[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]

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

Fricas [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.44 \[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\left [\frac {\log \left (-\frac {{\left ({\left (2 \, c^{2} - 1\right )} k^{4} + 2 \, k^{3} - k^{2}\right )} x^{4} + 2 \, {\left (c k^{4} - 2 \, c k^{3} + c k^{2}\right )} x^{3} + {\left (2 \, c^{2} - 1\right )} k^{2} - {\left ({\left (c^{2} - 2\right )} k^{4} + 2 \, {\left (c^{2} + 3\right )} k^{3} + {\left (c^{2} - 8\right )} k^{2} + 6 \, k - 2\right )} x^{2} + 2 \, \sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left (c k^{2} x^{2} - c k + {\left (k^{2} - 2 \, k + 1\right )} x\right )} \sqrt {{\left (c^{2} - 1\right )} k^{2} + 2 \, k - 1} - 2 \, {\left (c k^{3} - 2 \, c k^{2} + c k\right )} x + 2 \, k - 1}{2 \, c k^{2} x^{3} + k^{2} x^{4} - 2 \, c k x + {\left (c^{2} k^{2} - 2 \, k\right )} x^{2} + 1}\right )}{2 \, \sqrt {{\left (c^{2} - 1\right )} k^{2} + 2 \, k - 1}}, -\frac {\sqrt {-{\left (c^{2} - 1\right )} k^{2} - 2 \, k + 1} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {-{\left (c^{2} - 1\right )} k^{2} - 2 \, k + 1}}{c k^{2} x^{2} - c k + {\left (k^{2} - 2 \, k + 1\right )} x}\right )}{{\left (c^{2} - 1\right )} k^{2} + 2 \, k - 1}\right ] \]

[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)]

Sympy [F]

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

[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)

Maxima [F]

\[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {k x^{2} + 1}{{\left (c k x + k x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]

[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]

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

Giac [F]

\[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {k x^{2} + 1}{{\left (c k x + k x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k\,x^2+1}{\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}\,\left (k\,x^2+c\,k\,x-1\right )} \,d x \]

[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)