Integrand size = 43, antiderivative size = 193 \[ \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {\arctan \left (\frac {\left (-1-2 \sqrt {k}-k\right ) x}{-1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}+\frac {\arctan \left (\frac {\left (-1+2 \sqrt {k}-k\right ) x}{-1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1+k}-\frac {a \sqrt {k} \text {arctanh}\left (\frac {\left (2 \sqrt {k}+2 k^{3/2}\right ) x^2}{1+2 k x^2+k^2 x^4+\left (-1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 (1+k)} \]
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)
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 \[ \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {a \sqrt {k} \sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \text {arctanh}\left (\frac {\sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-1+x^2}}\right )+(1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticF}\left (\arcsin (x),k^2\right )-2 (1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (-k,\arcsin (x),k^2\right )}{(1+k) \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
(a*Sqrt[k]*Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*ArcTanh[Sqrt[-1 + k^2*x^2]/(S qrt[k]*Sqrt[-1 + x^2])] + (1 + k)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*Elliptic F[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)])
Time = 0.59 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.50, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2048, 2254, 27, 1576, 1154, 219, 2212, 216}
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.
| \(\displaystyle \int \frac {a k x+k x^2-1}{\left (k x^2+1\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {a k x+k x^2-1}{\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx\) |
| \(\Big \downarrow \) 2254 |
| \(\displaystyle \int \frac {a k x}{\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx+\int \frac {k x^2-1}{\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a k \int \frac {x}{\left (k x^2+1\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}dx+\int \frac {k x^2-1}{\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{2} a k \int \frac {1}{\left (k x^2+1\right ) \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}dx^2+\int \frac {k x^2-1}{\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \int \frac {k x^2-1}{\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx-a k \int \frac {1}{4 k (k+1)^2-x^4}d\frac {(k+1)^2 \left (1-k x^2\right )}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {k x^2-1}{\left (k x^2+1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx-\frac {a \sqrt {k} \text {arctanh}\left (\frac {(k+1) \left (1-k x^2\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 (k+1)}\) |
| \(\Big \downarrow \) 2212 |
| \(\displaystyle -\int \frac {1}{\frac {(k+1)^2 x^2}{k^2 x^4-\left (k^2+1\right ) x^2+1}+1}d\frac {x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {a \sqrt {k} \text {arctanh}\left (\frac {(k+1) \left (1-k x^2\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 (k+1)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {a \sqrt {k} \text {arctanh}\left (\frac {(k+1) \left (1-k x^2\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 (k+1)}-\frac {\arctan \left (\frac {(k+1) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{k+1}\) |
-(ArcTan[((1 + k)*x)/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]]/(1 + k)) - (a*Sqrt [k]*ArcTanh[((1 + k)*(1 - k*x^2))/(2*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2* x^4])])/(2*(1 + k))
3.24.98.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[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]
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[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Simp[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] & & EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
Int[(Pr_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_), x_Symbol] :> Module[{r = Expon[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[C oeff[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] && !Poly Q[Pr, x^2]
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 | \(-\frac {\left (a k +2 \sqrt {-k}\right ) \ln \left (\frac {-\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 x^{2} \left (-k \right )^{\frac {3}{2}}+2 \sqrt {-k}+\left (k^{2}-2 k +1\right ) x}{k \,x^{2}-2 \sqrt {-k}\, x -1}\right )+\left (-a k +2 \sqrt {-k}\right ) \ln \left (\frac {-\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 x^{2} \left (-k \right )^{\frac {3}{2}}-2 \sqrt {-k}+\left (k^{2}-2 k +1\right ) x}{k \,x^{2}+2 \sqrt {-k}\, x -1}\right )+4 \ln \left (2\right ) \sqrt {-k}}{4 \sqrt {-k}\, \sqrt {-\left (1+k \right )^{2}}}\) | \(202\) |
-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))
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 \[ \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\text {Too large to display} \]
-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*...
\[ \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {a k x + k x^{2} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x^{2} + 1\right )}\, dx \]
\[ \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {a k x + k x^{2} - 1}{{\left (k x^{2} + 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
\[ \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {a k x + k x^{2} - 1}{{\left (k x^{2} + 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
Timed out. \[ \int \frac {-1+a k x+k x^2}{\left (1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k\,x^2+a\,k\,x-1}{\left (k\,x^2+1\right )\,\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}} \,d x \]