Integrand size = 47, antiderivative size = 38 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2 \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{x \left (-1+k^2 x\right )} \]
Time = 11.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.55 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2 (-1+x)}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}} \]
Result contains higher order function than in optimal. Order 4 vs. order 2 in optimal.
Time = 1.51 (sec) , antiderivative size = 297, normalized size of antiderivative = 7.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {2467, 25, 2035, 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.
| \(\displaystyle \int \frac {k^2 x^2-2 k^2 x+1}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (k^2 x-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int -\frac {x^2 k^2-2 x k^2+1}{x^{3/2} \left (1-k^2 x\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{x^{3/2} \left (1-k^2 x\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {x^2 k^2-2 x k^2+1}{x \left (1-k^2 x\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \int \frac {x^2 k^2-2 x k^2+1}{\sqrt {1-x} x \left (1-k^2 x\right )^{3/2}}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \int \left (\frac {x k^2}{\sqrt {1-x} \left (1-k^2 x\right )^{3/2}}-\frac {2 k^2}{\sqrt {1-x} \left (1-k^2 x\right )^{3/2}}+\frac {1}{\sqrt {1-x} x \left (1-k^2 x\right )^{3/2}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )-\frac {2 k^2 E\left (\arcsin \left (\sqrt {x}\right )|k^2\right )}{1-k^2}+\frac {k E\left (\arcsin \left (k \sqrt {x}\right )|\frac {1}{k^2}\right )}{1-k^2}-\frac {\left (1-2 k^2\right ) E\left (\arcsin \left (\sqrt {x}\right )|k^2\right )}{1-k^2}-\frac {k^2 \sqrt {1-x} \sqrt {x}}{\left (1-k^2\right ) \sqrt {1-k^2 x}}-\frac {k^2 \sqrt {1-x}}{\left (1-k^2\right ) \sqrt {x} \sqrt {1-k^2 x}}-\frac {\left (1-2 k^2\right ) \sqrt {1-x} \sqrt {1-k^2 x}}{\left (1-k^2\right ) \sqrt {x}}+\frac {2 k^4 \sqrt {1-x} \sqrt {x}}{\left (1-k^2\right ) \sqrt {1-k^2 x}}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
(-2*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 - k^2*x]*(-((k^2*Sqrt[1 - x])/((1 - k^2)*Sq rt[x]*Sqrt[1 - k^2*x])) - (k^2*Sqrt[1 - x]*Sqrt[x])/((1 - k^2)*Sqrt[1 - k^ 2*x]) + (2*k^4*Sqrt[1 - x]*Sqrt[x])/((1 - k^2)*Sqrt[1 - k^2*x]) - ((1 - 2* k^2)*Sqrt[1 - x]*Sqrt[1 - k^2*x])/((1 - k^2)*Sqrt[x]) - (2*k^2*EllipticE[A rcSin[Sqrt[x]], k^2])/(1 - k^2) - ((1 - 2*k^2)*EllipticE[ArcSin[Sqrt[x]], k^2])/(1 - k^2) + (k*EllipticE[ArcSin[k*Sqrt[x]], k^(-2)])/(1 - k^2) + Ell ipticF[ArcSin[Sqrt[x]], k^2]))/Sqrt[(1 - x)*x*(1 - k^2*x)]
3.5.75.3.1 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[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 0.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53
| method | result | size |
| gosper | \(-\frac {2 \left (x -1\right )}{\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}\) | \(20\) |
| trager | \(-\frac {2 \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}{x \left (k^{2} x -1\right )}\) | \(39\) |
| risch | \(\frac {2 \left (x -1\right ) \left (k^{2} x -1\right )}{\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}-\frac {2 x \left (x -1\right ) k^{2}}{\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}\) | \(51\) |
| elliptic | \(\frac {2 k^{2} x^{2}-2 k^{2} x -2 x +2}{\sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}-\frac {2 \left (k^{2} x^{2}-k^{2} x \right )}{\sqrt {\left (x -\frac {1}{k^{2}}\right ) \left (k^{2} x^{2}-k^{2} x \right )}}\) | \(84\) |
| default | \(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 k^{2} x^{2}-2 k^{2} x -2 x +2}{\sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}+\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \left (\left (\frac {1}{k^{2}}-1\right ) \operatorname {EllipticE}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )+\operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )\right )}{\sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\left (-k^{2}+1\right ) \left (\frac {2 k^{2} x^{2}-2 k^{2} x}{\left (k^{2}-1\right ) \sqrt {\left (x -\frac {1}{k^{2}}\right ) \left (k^{2} x^{2}-k^{2} x \right )}}-\frac {2 \left (-1+\frac {k^{2}}{k^{2}-1}\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {x -1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \left (\left (\frac {1}{k^{2}}-1\right ) \operatorname {EllipticE}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )+\operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}\right )\) | \(548\) |
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x}}{k^{2} x^{2} - x} \]
\[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=\int \frac {k^{2} x^{2} - 2 k^{2} x + 1}{x \sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k^{2} x - 1\right )}\, dx \]
\[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{{\left (k^{2} x - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} x} \,d x } \]
\[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, k^{2} x + 1}{{\left (k^{2} x - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} x} \,d x } \]
Time = 4.91 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {1-2 k^2 x+k^2 x^2}{x \sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x\right )} \, dx=-\frac {2\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}}{x\,\left (k^2\,x-1\right )} \]