Integrand size = 54, antiderivative size = 40 \[ \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx=-\frac {\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}{x \left (-1+k^2 x^2\right )} \]
Time = 8.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx=\frac {1-x^2}{x \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
Result contains higher order function than in optimal. Order 4 vs. order 2 in optimal.
Time = 1.31 (sec) , antiderivative size = 467, normalized size of antiderivative = 11.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2048, 1395, 7293, 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.
\(\displaystyle \int \frac {k^2 x^4-2 k^2 x^2+1}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (k^2 x^2-1\right )} \, dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int \frac {k^2 x^4-2 k^2 x^2+1}{x^2 \left (k^2 x^2-1\right ) \sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx\) |
\(\Big \downarrow \) 1395 |
\(\displaystyle \frac {\sqrt {x^2-1} \sqrt {k^2 x^2-1} \int \frac {k^2 x^4-2 k^2 x^2+1}{x^2 \sqrt {x^2-1} \left (k^2 x^2-1\right )^{3/2}}dx}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\sqrt {x^2-1} \sqrt {k^2 x^2-1} \int \left (\frac {x^2 k^2}{\sqrt {x^2-1} \left (k^2 x^2-1\right )^{3/2}}-\frac {2 k^2}{\sqrt {x^2-1} \left (k^2 x^2-1\right )^{3/2}}+\frac {1}{x^2 \sqrt {x^2-1} \left (k^2 x^2-1\right )^{3/2}}\right )dx}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {x^2-1} \sqrt {k^2 x^2-1} \left (-\frac {2 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticF}\left (\arcsin (x),k^2\right )}{\sqrt {x^2-1} \sqrt {k^2 x^2-1}}-\frac {2 k^2 \sqrt {1-x^2} \sqrt {k^2 x^2-1} E\left (\arcsin (x)\left |k^2\right .\right )}{\left (1-k^2\right ) \sqrt {x^2-1} \sqrt {1-k^2 x^2}}-\frac {\left (1-2 k^2\right ) k \sqrt {x^2-1} \sqrt {1-k^2 x^2} E\left (\arcsin (k x)\left |\frac {1}{k^2}\right .\right )}{\left (1-k^2\right ) \sqrt {1-x^2} \sqrt {k^2 x^2-1}}+\frac {k \sqrt {x^2-1} \sqrt {1-k^2 x^2} E\left (\arcsin (k x)\left |\frac {1}{k^2}\right .\right )}{\left (1-k^2\right ) \sqrt {1-x^2} \sqrt {k^2 x^2-1}}-\frac {k^2 x \sqrt {x^2-1}}{\left (1-k^2\right ) \sqrt {k^2 x^2-1}}-\frac {k^2 \sqrt {x^2-1}}{\left (1-k^2\right ) x \sqrt {k^2 x^2-1}}+\frac {\left (1-2 k^2\right ) \sqrt {x^2-1} \sqrt {k^2 x^2-1}}{\left (1-k^2\right ) x}+\frac {2 k^4 x \sqrt {x^2-1}}{\left (1-k^2\right ) \sqrt {k^2 x^2-1}}\right )}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\) |
(Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*(-((k^2*Sqrt[-1 + x^2])/((1 - k^2)*x*Sq rt[-1 + k^2*x^2])) - (k^2*x*Sqrt[-1 + x^2])/((1 - k^2)*Sqrt[-1 + k^2*x^2]) + (2*k^4*x*Sqrt[-1 + x^2])/((1 - k^2)*Sqrt[-1 + k^2*x^2]) + ((1 - 2*k^2)* Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2])/((1 - k^2)*x) - (2*k^2*Sqrt[1 - x^2]*Sq rt[-1 + k^2*x^2]*EllipticE[ArcSin[x], k^2])/((1 - k^2)*Sqrt[-1 + x^2]*Sqrt [1 - k^2*x^2]) + (k*Sqrt[-1 + x^2]*Sqrt[1 - k^2*x^2]*EllipticE[ArcSin[k*x] , k^(-2)])/((1 - k^2)*Sqrt[1 - x^2]*Sqrt[-1 + k^2*x^2]) - (k*(1 - 2*k^2)*S qrt[-1 + x^2]*Sqrt[1 - k^2*x^2]*EllipticE[ArcSin[k*x], k^(-2)])/((1 - k^2) *Sqrt[1 - x^2]*Sqrt[-1 + k^2*x^2]) - (2*k^2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2 ]*EllipticF[ArcSin[x], k^2])/(Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2])))/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]
3.6.22.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
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]
Time = 5.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(-\frac {\left (x -1\right ) \left (1+x \right )}{\sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}\, x}\) | \(29\) |
elliptic | \(\frac {-x^{2}+1}{x \sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}\, k^{2}}\) | \(35\) |
trager | \(-\frac {\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}{x \left (k^{2} x^{2}-1\right )}\) | \(41\) |
risch | \(\frac {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}{x \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}}-\frac {\left (1+x \right ) \left (x -1\right ) k^{2} x}{\sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}}\) | \(66\) |
default | \(\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}{x}-\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x , k\right )-\operatorname {EllipticE}\left (x , k\right )\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\left (-\frac {k^{2}}{2}+\frac {1}{2}\right ) \left (\frac {k^{2} x^{3}-k^{2} x +k \,x^{2}-k}{\left (k^{2}-1\right ) \sqrt {\left (x -\frac {1}{k}\right ) \left (k^{2} x^{3}-k^{2} x +k \,x^{2}-k \right )}}+\frac {\left (-\frac {k^{2}-2}{2 \left (k^{2}-1\right )}+\frac {k^{2}}{2 k^{2}-2}\right ) \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x , k\right )-\operatorname {EllipticE}\left (x , k\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )+\left (\frac {k^{2}}{2}-\frac {1}{2}\right ) \left (-\frac {k^{2} x^{3}-k^{2} x -k \,x^{2}+k}{\left (k^{2}-1\right ) \sqrt {\left (x +\frac {1}{k}\right ) \left (k^{2} x^{3}-k^{2} x -k \,x^{2}+k \right )}}+\frac {\left (\frac {k^{2}-2}{2 k^{2}-2}-\frac {k^{2}}{2 \left (k^{2}-1\right )}\right ) \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x , k\right )-\operatorname {EllipticE}\left (x , k\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )\) | \(551\) |
int((k^2*x^4-2*k^2*x^2+1)/x^2/((-x^2+1)*(-k^2*x^2+1))^(1/2)/(k^2*x^2-1),x, method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx=-\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1}}{k^{2} x^{3} - x} \]
integrate((k^2*x^4-2*k^2*x^2+1)/x^2/((-x^2+1)*(-k^2*x^2+1))^(1/2)/(k^2*x^2 -1),x, algorithm="fricas")
\[ \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx=\int \frac {k^{2} x^{4} - 2 k^{2} x^{2} + 1}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x - 1\right ) \left (k x + 1\right )}\, dx \]
Integral((k**2*x**4 - 2*k**2*x**2 + 1)/(x**2*sqrt((x - 1)*(x + 1)*(k*x - 1 )*(k*x + 1))*(k*x - 1)*(k*x + 1)), x)
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx=-\frac {x^{2} - 1}{\sqrt {k x + 1} \sqrt {k x - 1} \sqrt {x + 1} \sqrt {x - 1} x} \]
integrate((k^2*x^4-2*k^2*x^2+1)/x^2/((-x^2+1)*(-k^2*x^2+1))^(1/2)/(k^2*x^2 -1),x, algorithm="maxima")
\[ \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx=\int { \frac {k^{2} x^{4} - 2 \, k^{2} x^{2} + 1}{{\left (k^{2} x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}} x^{2}} \,d x } \]
integrate((k^2*x^4-2*k^2*x^2+1)/x^2/((-x^2+1)*(-k^2*x^2+1))^(1/2)/(k^2*x^2 -1),x, algorithm="giac")
Time = 4.94 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx=-\frac {\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}}{x\,\left (k^2\,x^2-1\right )} \]