\(\int \frac {-1+a k x+k x^2}{(1+k x^2) \sqrt {(1-x^2) (1-k^2 x^2)}} \, dx\) [2398]
Optimal result
Integrand size = 43, antiderivative size = 193 \[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {\arctan \left (\frac {\left (-1-2 \sqrt {k}-k\right ) x}{-1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}+\frac {\arctan \left (\frac {\left (-1+2 \sqrt {k}-k\right ) x}{-1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}-\frac {a \sqrt {k} \text {arctanh}\left (\frac {\left (2 \sqrt {k}+2 k^{3/2}\right ) x^2}{1+2 k x^2+k^2 x^4+\left (-1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 (1+k)}
\]
[Out]
arctan((-1-2*k^(1/2)-k)*x/(-1+k*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(1+k)+arctan((-1+2*k^(1/2)-k)*x/(-1+k*x^2
+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(1+k)-a*k^(1/2)*arctanh((2*k^(1/2)+2*k^(3/2))*x^2/(1+2*k*x^2+k^2*x^4+(k*x^2-
1)*(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/(2+2*k)
Rubi [A] (verified)
Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51, number of
steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {1976, 1701, 1712, 209, 12,
1261, 738, 212} \[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=-\frac {a \sqrt {k} \text {arctanh}\left (\frac {-k (k+1) x^2+k+1}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 (k+1)}-\frac {\arctan \left (\frac {(k+1) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{k+1}
\]
[In]
Int[(-1 + a*k*x + k*x^2)/((1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]
[Out]
-(ArcTan[((1 + k)*x)/Sqrt[1 + (-1 - k^2)*x^2 + k^2*x^4]]/(1 + k)) - (a*Sqrt[k]*ArcTanh[(1 + k - k*(1 + k)*x^2)
/(2*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])])/(2*(1 + k))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
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 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 1701
Int[(Pr_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{r = Expo
n[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2*k]*x^(2*k), {k, 0, r/2}]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] + Int[x*
Sum[Coeff[Pr, x, 2*k + 1]*x^(2*k), {k, 0, (r - 1)/2}]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b,
c, d, e, p, q}, x] && PolyQ[Pr, x] && !PolyQ[Pr, x^2]
Rule 1712
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], 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] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
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]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \frac {a k x}{\left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\int \frac {-1+k x^2}{\left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = (a k) \int \frac {x}{\left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\text {Subst}\left (\int \frac {1}{1-\left (-1-2 k-k^2\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {(1+k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}+\frac {1}{2} (a k) \text {Subst}\left (\int \frac {1}{(1+k x) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {\arctan \left (\frac {(1+k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}-(a k) \text {Subst}\left (\int \frac {1}{8 k^2-4 k \left (-1-k^2\right )-x^2} \, dx,x,\frac {1+2 k+k^2-k (1+k)^2 x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {(1+k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}-\frac {a \sqrt {k} \text {arctanh}\left (\frac {1+k-k (1+k) x^2}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{2 (1+k)} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.82
\[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {a \sqrt {k} \sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \text {arctanh}\left (\frac {\sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-1+x^2}}\right )+(1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticF}\left (\arcsin (x),k^2\right )-2 (1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (-k,\arcsin (x),k^2\right )}{(1+k) \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}}
\]
[In]
Integrate[(-1 + a*k*x + k*x^2)/((1 + k*x^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]),x]
[Out]
(a*Sqrt[k]*Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*ArcTanh[Sqrt[-1 + k^2*x^2]/(Sqrt[k]*Sqrt[-1 + x^2])] + (1 + k)*Sq
rt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[x], k^2] - 2*(1 + k)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi
[-k, ArcSin[x], k^2])/((1 + k)*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)])
Maple [A] (verified)
Time = 2.46 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.83
| | |
| method | result | size |
| | |
| elliptic |
\(-\frac {a \ln \left (\frac {\frac {2 k^{2}+4 k +2}{k}+\left (-k^{2}-2 k -1\right ) \left (x^{2}+\frac {1}{k}\right )+2 \sqrt {\frac {k^{2}+2 k +1}{k}}\, \sqrt {k^{2} \left (x^{2}+\frac {1}{k}\right )^{2}+\left (-k^{2}-2 k -1\right ) \left (x^{2}+\frac {1}{k}\right )+\frac {k^{2}+2 k +1}{k}}}{x^{2}+\frac {1}{k}}\right )}{2 \sqrt {\frac {k^{2}+2 k +1}{k}}}+\frac {\arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (1+k \right )}\right )}{1+k}\) |
\(160\) |
| default |
\(-\frac {\left (a k +2 \sqrt {-k}\right ) \ln \left (\frac {-\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 x^{2} \left (-k \right )^{\frac {3}{2}}+2 \sqrt {-k}+\left (k^{2}-2 k +1\right ) x}{k \,x^{2}-2 \sqrt {-k}\, x -1}\right )+\left (-a k +2 \sqrt {-k}\right ) \ln \left (\frac {-\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 x^{2} \left (-k \right )^{\frac {3}{2}}-2 \sqrt {-k}+\left (k^{2}-2 k +1\right ) x}{k \,x^{2}+2 \sqrt {-k}\, x -1}\right )+4 \ln \left (2\right ) \sqrt {-k}}{4 \sqrt {-k}\, \sqrt {-\left (1+k \right )^{2}}}\) |
\(202\) |
| pseudoelliptic |
\(-\frac {\left (a k +2 \sqrt {-k}\right ) \ln \left (\frac {-\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 x^{2} \left (-k \right )^{\frac {3}{2}}+2 \sqrt {-k}+\left (k^{2}-2 k +1\right ) x}{k \,x^{2}-2 \sqrt {-k}\, x -1}\right )+\left (-a k +2 \sqrt {-k}\right ) \ln \left (\frac {-\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 x^{2} \left (-k \right )^{\frac {3}{2}}-2 \sqrt {-k}+\left (k^{2}-2 k +1\right ) x}{k \,x^{2}+2 \sqrt {-k}\, x -1}\right )+4 \ln \left (2\right ) \sqrt {-k}}{4 \sqrt {-k}\, \sqrt {-\left (1+k \right )^{2}}}\) |
\(202\) |
| | |
|
|
|
|
[In]
int((a*k*x+k*x^2-1)/(k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/2*a/((k^2+2*k+1)/k)^(1/2)*ln((2*(k^2+2*k+1)/k+(-k^2-2*k-1)*(x^2+1/k)+2*((k^2+2*k+1)/k)^(1/2)*(k^2*(x^2+1/k)
^2+(-k^2-2*k-1)*(x^2+1/k)+(k^2+2*k+1)/k)^(1/2))/(x^2+1/k))+1/(1+k)*arctan(((-x^2+1)*(-k^2*x^2+1))^(1/2)/x/(1+k
))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1809 vs. \(2 (167) = 334\).
Time = 1.13 (sec) , antiderivative size = 1809, normalized size of antiderivative = 9.37
\[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\text {Too large to display}
\]
[In]
integrate((a*k*x+k*x^2-1)/(k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="fricas")
[Out]
-1/8*sqrt((a^2*k - 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1))*log(2*
(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(a^3*k - (a^3*k^2 + 4*a*k)*x^2 + 2*(a^2*k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k
+ 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*x + 4*a) + (2*a^2*k^3*x^4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 +
2*a^2*k - 2*(a^2*k^3 + a^2*k)*x^2 - 2*(a*k^2 + 2*a*k + a)*x + (4*(k^4 + 2*k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 +
6*a*k^3 + 4*a*k^2 + a*k)*x^3 - 4*(k^4 + 2*k^3 + 2*k^2 + 2*k + 1)*x^2 + 4*k^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4
*a*k + a)*x + 8*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1)))*sqrt((a^2*k - 4*sqrt(-a^2*k/(k^4 + 4*k^3
+ 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1)))/(k^2*x^4 + 2*k*x^2 + 1)) + 1/8*sqrt((a^2*k - 4*sqrt
(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1))*log(2*(sqrt(k^2*x^4 - (k^2 + 1)
*x^2 + 1)*(a^3*k - (a^3*k^2 + 4*a*k)*x^2 + 2*(a^2*k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k + 4)*sqrt(-a^2*k/(k^4 +
4*k^3 + 6*k^2 + 4*k + 1))*x + 4*a) - (2*a^2*k^3*x^4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 + 2*a^2*k - 2*(a^2*k^3 + a
^2*k)*x^2 - 2*(a*k^2 + 2*a*k + a)*x + (4*(k^4 + 2*k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 + 6*a*k^3 + 4*a*k^2 + a*k)
*x^3 - 4*(k^4 + 2*k^3 + 2*k^2 + 2*k + 1)*x^2 + 4*k^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4*a*k + a)*x + 8*k + 4)*sq
rt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1)))*sqrt((a^2*k - 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2
+ 2*k + 1) - 4)/(k^2 + 2*k + 1)))/(k^2*x^4 + 2*k*x^2 + 1)) - 1/8*sqrt((a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*
k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1))*log(2*(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(a^3*k - (a^3*k
^2 + 4*a*k)*x^2 - 2*(a^2*k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))
*x + 4*a) + (2*a^2*k^3*x^4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 + 2*a^2*k - 2*(a^2*k^3 + a^2*k)*x^2 - 2*(a*k^2 + 2*
a*k + a)*x - (4*(k^4 + 2*k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 + 6*a*k^3 + 4*a*k^2 + a*k)*x^3 - 4*(k^4 + 2*k^3 + 2
*k^2 + 2*k + 1)*x^2 + 4*k^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4*a*k + a)*x + 8*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 +
6*k^2 + 4*k + 1)))*sqrt((a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*
k + 1)))/(k^2*x^4 + 2*k*x^2 + 1)) + 1/8*sqrt((a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*
k + 1) - 4)/(k^2 + 2*k + 1))*log(2*(sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)*(a^3*k - (a^3*k^2 + 4*a*k)*x^2 - 2*(a^2*
k^3 + 2*(a^2 + 2)*k^2 + (a^2 + 8)*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*x + 4*a) - (2*a^2*k^3*x^
4 + 2*(a*k^3 + 2*a*k^2 + a*k)*x^3 + 2*a^2*k - 2*(a^2*k^3 + a^2*k)*x^2 - 2*(a*k^2 + 2*a*k + a)*x - (4*(k^4 + 2*
k^3 + k^2)*x^4 - (a*k^5 + 4*a*k^4 + 6*a*k^3 + 4*a*k^2 + a*k)*x^3 - 4*(k^4 + 2*k^3 + 2*k^2 + 2*k + 1)*x^2 + 4*k
^2 + (a*k^4 + 4*a*k^3 + 6*a*k^2 + 4*a*k + a)*x + 8*k + 4)*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1)))*sqrt((
a^2*k + 4*sqrt(-a^2*k/(k^4 + 4*k^3 + 6*k^2 + 4*k + 1))*(k^2 + 2*k + 1) - 4)/(k^2 + 2*k + 1)))/(k^2*x^4 + 2*k*x
^2 + 1))
Sympy [F]
\[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {a k x + k x^{2} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x^{2} + 1\right )}\, dx
\]
[In]
integrate((a*k*x+k*x**2-1)/(k*x**2+1)/((-x**2+1)*(-k**2*x**2+1))**(1/2),x)
[Out]
Integral((a*k*x + k*x**2 - 1)/(sqrt((x - 1)*(x + 1)*(k*x - 1)*(k*x + 1))*(k*x**2 + 1)), x)
Maxima [F]
\[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {a k x + k x^{2} - 1}{{\left (k x^{2} + 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x }
\]
[In]
integrate((a*k*x+k*x^2-1)/(k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="maxima")
[Out]
integrate((a*k*x + k*x^2 - 1)/((k*x^2 + 1)*sqrt((k^2*x^2 - 1)*(x^2 - 1))), x)
Giac [F]
\[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {a k x + k x^{2} - 1}{{\left (k x^{2} + 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x }
\]
[In]
integrate((a*k*x+k*x^2-1)/(k*x^2+1)/((-x^2+1)*(-k^2*x^2+1))^(1/2),x, algorithm="giac")
[Out]
integrate((a*k*x + k*x^2 - 1)/((k*x^2 + 1)*sqrt((k^2*x^2 - 1)*(x^2 - 1))), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k\,x^2+a\,k\,x-1}{\left (k\,x^2+1\right )\,\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}} \,d x
\]
[In]
int((k*x^2 + a*k*x - 1)/((k*x^2 + 1)*((x^2 - 1)*(k^2*x^2 - 1))^(1/2)),x)
[Out]
int((k*x^2 + a*k*x - 1)/((k*x^2 + 1)*((x^2 - 1)*(k^2*x^2 - 1))^(1/2)), x)