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 = 16.24 (sec) , antiderivative size = 1374, normalized size of antiderivative = 13.74 \[ \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*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]]*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)*Sq rt[-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]*Ellipt icPi[(-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]*Sqrt[ -4 + c^2*k])) - (2*c^3*k^(3/2)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*Elliptic...
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.90 (sec) , antiderivative size = 1193, normalized size of antiderivative = 11.93, 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 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}-\frac {c k x+2}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (c k x+k x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\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 (c \sqrt {k}-\sqrt {c^2 k-4}\right ) \text {arctanh}\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 ) \text {arctanh}\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}} \operatorname {EllipticF}\left (2 \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}} \operatorname {EllipticF}\left (2 \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}} \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 (-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}} \operatorname {EllipticPi}\left (\frac {c^2 k}{4},2 \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}} \operatorname {EllipticPi}\left (\frac {c^2 k}{4},2 \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}}\) |
-(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])*ArcTa nh[(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]]*Sqr t[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*S qrt[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*Sqr t[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[S qrt[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]*Sqrt[-4 + c^2*k])*(1 + k*x^2)*Sqrt[(1 - (1 + k^2...
3.14.88.3.1 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 = 2.62 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {\ln \left (2\right )+\ln \left (\frac {\sqrt {\left (c k +k +1\right ) \left (c k -k -1\right )}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+\left (-c \,x^{2}-x \right ) k^{2}+\left (-c -2 x \right ) k -x}{1+x \left (c +x \right ) k}\right )}{\sqrt {-1+\left (c^{2}-1\right ) k^{2}-2 k}}\) | \(96\) |
| pseudoelliptic | \(-\frac {\ln \left (2\right )+\ln \left (\frac {\sqrt {\left (c k +k +1\right ) \left (c k -k -1\right )}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+\left (-c \,x^{2}-x \right ) k^{2}+\left (-c -2 x \right ) k -x}{1+x \left (c +x \right ) k}\right )}{\sqrt {-1+\left (c^{2}-1\right ) k^{2}-2 k}}\) | \(96\) |
| 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 = 0.53 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.42 \[ \int \frac {-1+k x^2}{\left (1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\left [\frac {\log \left (-\frac {{\left ({\left (2 \, c^{2} - 1\right )} k^{4} - 2 \, k^{3} - k^{2}\right )} x^{4} + 2 \, {\left (c k^{4} + 2 \, c k^{3} + c k^{2}\right )} x^{3} + {\left (2 \, c^{2} - 1\right )} k^{2} - {\left ({\left (c^{2} - 2\right )} k^{4} - 2 \, {\left (c^{2} + 3\right )} k^{3} + {\left (c^{2} - 8\right )} k^{2} - 6 \, k - 2\right )} x^{2} + 2 \, \sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left (c k^{2} x^{2} + c k + {\left (k^{2} + 2 \, k + 1\right )} x\right )} \sqrt {{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1} + 2 \, {\left (c k^{3} + 2 \, c k^{2} + c k\right )} x - 2 \, k - 1}{2 \, c k^{2} x^{3} + k^{2} x^{4} + 2 \, c k x + {\left (c^{2} k^{2} + 2 \, k\right )} x^{2} + 1}\right )}{2 \, \sqrt {{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1}}, -\frac {\sqrt {-{\left (c^{2} - 1\right )} k^{2} + 2 \, k + 1} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {-{\left (c^{2} - 1\right )} k^{2} + 2 \, k + 1}}{c k^{2} x^{2} + c k + {\left (k^{2} + 2 \, k + 1\right )} x}\right )}{{\left (c^{2} - 1\right )} k^{2} - 2 \, k - 1}\right ] \]
[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)]
\[ \int \frac {-1+k x^2}{\left (1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k x^{2} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (c k x + k x^{2} + 1\right )}\, dx \]
Exception generated. \[ \int \frac {-1+k x^2}{\left (1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c^2*k-4>0)', see `assume?` for m ore detail
\[ \int \frac {-1+k x^2}{\left (1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {k x^{2} - 1}{{\left (c k x + k x^{2} + 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
Timed out. \[ \int \frac {-1+k x^2}{\left (1+c k x+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k\,x^2-1}{\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}\,\left (k\,x^2+c\,k\,x+1\right )} \,d x \]