Integrand size = 43, antiderivative size = 100 \[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {2 \sqrt {1-2 k+k^2-c^2 k^2} \arctan \left (\frac {\sqrt {1-2 k+k^2-c^2 k^2} x}{-1+c k x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{(1-k+c k) (-1+k+c k)} \]
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)
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 )}} \]
((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*S qrt[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]]*S qrt[-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*Sqrt[k]*Sqrt[4 + c^2*k]), ArcSin[x], k^2])/(2 + c^2*k - c*Sqrt[k]*Sqr t[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*S qrt[k]*Sqrt[4 + c^2*k]) + (8*c*Sqrt[k]*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*Ell ipticPi[(2*k)/(2 + c^2*k - c*Sqrt[k]*Sqrt[4 + c^2*k]), ArcSin[x], k^2])/(S qrt[4 + c^2*k]*(2 + c^2*k - c*Sqrt[k]*Sqrt[4 + c^2*k])) + (2*c^3*k^(3/2)*S qrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[(2*k)/(2 + c^2*k - c*Sqrt[k]*...
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.37 (sec) , antiderivative size = 1288, normalized size of antiderivative = 12.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2048, 7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {k x^2+1}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (c k x+k x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {k x^2+1}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (c k x+k x^2-1\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2-c k x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (c k x+k x^2-1\right )}+\frac {1}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\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}}\) |
-(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])*ArcT an[(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])^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]) ...
3.14.87.3.1 Defintions of rubi rules used
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] /; F reeQ[{a, b, c, d, e, n, p}, x]
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] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
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 | \(\text {Expression too large to display}\) | \(1699\) |
-(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)
Time = 0.54 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.44 \[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\left [\frac {\log \left (-\frac {{\left ({\left (2 \, c^{2} - 1\right )} k^{4} + 2 \, k^{3} - k^{2}\right )} x^{4} + 2 \, {\left (c k^{4} - 2 \, c k^{3} + c k^{2}\right )} x^{3} + {\left (2 \, c^{2} - 1\right )} k^{2} - {\left ({\left (c^{2} - 2\right )} k^{4} + 2 \, {\left (c^{2} + 3\right )} k^{3} + {\left (c^{2} - 8\right )} k^{2} + 6 \, k - 2\right )} x^{2} + 2 \, \sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left (c k^{2} x^{2} - c k + {\left (k^{2} - 2 \, k + 1\right )} x\right )} \sqrt {{\left (c^{2} - 1\right )} k^{2} + 2 \, k - 1} - 2 \, {\left (c k^{3} - 2 \, c k^{2} + c k\right )} x + 2 \, k - 1}{2 \, c k^{2} x^{3} + k^{2} x^{4} - 2 \, c k x + {\left (c^{2} k^{2} - 2 \, k\right )} x^{2} + 1}\right )}{2 \, \sqrt {{\left (c^{2} - 1\right )} k^{2} + 2 \, k - 1}}, -\frac {\sqrt {-{\left (c^{2} - 1\right )} k^{2} - 2 \, k + 1} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {-{\left (c^{2} - 1\right )} k^{2} - 2 \, k + 1}}{c k^{2} x^{2} - c k + {\left (k^{2} - 2 \, k + 1\right )} x}\right )}{{\left (c^{2} - 1\right )} k^{2} + 2 \, k - 1}\right ] \]
[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)*arctan(s qrt(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)]
\[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k x^{2} + 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (c k x + k x^{2} - 1\right )}\, dx \]
\[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {k x^{2} + 1}{{\left (c k x + k x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
\[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {k x^{2} + 1}{{\left (c k x + k x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
Timed out. \[ \int \frac {1+k x^2}{\left (-1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k\,x^2+1}{\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}\,\left (k\,x^2+c\,k\,x-1\right )} \,d x \]