Integrand size = 41, antiderivative size = 189 \[ \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=-\frac {i \arctan \left (\frac {\sqrt {-2+k^2-2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{-1+k^2 x}\right )}{2 \sqrt {-1+k^2} \sqrt {-2+k^2-2 i \sqrt {-1+k^2}}}+\frac {i \arctan \left (\frac {\sqrt {-2+k^2+2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{-1+k^2 x}\right )}{2 \sqrt {-1+k^2} \sqrt {-2+k^2+2 i \sqrt {-1+k^2}}} \]
-1/2*I*arctan((-2+k^2-2*I*(k^2-1)^(1/2))^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1 /2)/(k^2*x-1))/(k^2-1)^(1/2)/(-2+k^2-2*I*(k^2-1)^(1/2))^(1/2)+1/2*I*arctan ((-2+k^2+2*I*(k^2-1)^(1/2))^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/(k^2*x-1) )/(k^2-1)^(1/2)/(-2+k^2+2*I*(k^2-1)^(1/2))^(1/2)
Time = 5.37 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=-\frac {i \left (\frac {\arctan \left (\frac {\sqrt {-2+k^2-2 i \sqrt {-1+k^2}} (-1+x) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{\sqrt {-2+k^2-2 i \sqrt {-1+k^2}}}-\frac {\arctan \left (\frac {\sqrt {-2+k^2+2 i \sqrt {-1+k^2}} (-1+x) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{\sqrt {-2+k^2+2 i \sqrt {-1+k^2}}}\right )}{2 \sqrt {-1+k^2}} \]
((-1/2*I)*(ArcTan[(Sqrt[-2 + k^2 - (2*I)*Sqrt[-1 + k^2]]*(-1 + x)*x)/Sqrt[ (-1 + x)*x*(-1 + k^2*x)]]/Sqrt[-2 + k^2 - (2*I)*Sqrt[-1 + k^2]] - ArcTan[( Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]]*(-1 + x)*x)/Sqrt[(-1 + x)*x*(-1 + k^ 2*x)]]/Sqrt[-2 + k^2 + (2*I)*Sqrt[-1 + k^2]]))/Sqrt[-1 + k^2]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.66 (sec) , antiderivative size = 910, normalized size of antiderivative = 4.81, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {2027, 2467, 25, 2035, 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.
| \(\displaystyle \int \frac {x^2-x}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (k^2 x^2-2 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {(x-1) x}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (k^2 x^2-2 x+1\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int -\frac {(1-x) \sqrt {x}}{\left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {(1-x) \sqrt {x}}{\left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {(1-x) x}{\left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {1-\left (2-k^2\right ) x}{k^2 \left (k^2 x^2-2 x+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {1}{k^2 \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (\frac {\left (-k^2+2 \sqrt {1-k^2}+2\right ) \text {arctanh}\left (\frac {\sqrt {1-k^2} \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 k^2 \left (-k^2+\sqrt {1-k^2}+1\right )}-\frac {\left (-k^2-2 \sqrt {1-k^2}+2\right ) \text {arctanh}\left (\frac {\sqrt {1-k^2} \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 k^2 \left (-k^2-\sqrt {1-k^2}+1\right )}+\frac {\left (-k^2+2 \sqrt {1-k^2}+2\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{4 k^{5/2} \left (k+\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\left (-k^2-2 \sqrt {1-k^2}+2\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{4 k^{5/2} \left (k-\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {(k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{2 k^{5/2} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}+\frac {\left (-k+\sqrt {1-k^2}+1\right ) \left (k^2-2 \left (\sqrt {1-k^2}+1\right )\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (\frac {k+1}{2 k},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{5/2} \left (\sqrt {1-k^2}+1\right ) \left (k+\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {\left (-k^2-2 \sqrt {1-k^2}+2\right ) \left (-k-\sqrt {1-k^2}+1\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticPi}\left (\frac {k+1}{2 k},2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{5/2} \left (1-\sqrt {1-k^2}\right ) \left (k-\sqrt {1-k^2}+1\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
(-2*Sqrt[x]*Sqrt[1 - (1 + k^2)*x + k^2*x^2]*(-1/4*((2 - k^2 - 2*Sqrt[1 - k ^2])*ArcTanh[(Sqrt[1 - k^2]*Sqrt[x])/Sqrt[1 - (1 + k^2)*x + k^2*x^2]])/(k^ 2*(1 - k^2 - Sqrt[1 - k^2])) + ((2 - k^2 + 2*Sqrt[1 - k^2])*ArcTanh[(Sqrt[ 1 - k^2]*Sqrt[x])/Sqrt[1 - (1 + k^2)*x + k^2*x^2]])/(4*k^2*(1 - k^2 + Sqrt [1 - k^2])) - ((1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*Ell ipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(2*k^(5/2)*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + ((2 - k^2 - 2*Sqrt[1 - k^2])*(1 + k*x)*Sqrt[(1 - ( 1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(4*k^(5/2)*(1 + k - Sqrt[1 - k^2])*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + ((2 - k^2 + 2*Sqrt[1 - k^2])*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k )])/(4*k^(5/2)*(1 + k + Sqrt[1 - k^2])*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) - ((2 - k^2 - 2*Sqrt[1 - k^2])*(1 - k - Sqrt[1 - k^2])*(1 + k*x)*Sqrt[(1 - ( 1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticPi[(1 + k)/(2*k), 2*ArcTan[Sqrt [k]*Sqrt[x]], (1 + k)^2/(4*k)])/(8*k^(5/2)*(1 - Sqrt[1 - k^2])*(1 + k - Sq rt[1 - k^2])*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + ((1 - k + Sqrt[1 - k^2])*( k^2 - 2*(1 + Sqrt[1 - k^2]))*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticPi[(1 + k)/(2*k), 2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/ (4*k)])/(8*k^(5/2)*(1 + Sqrt[1 - k^2])*(1 + k + Sqrt[1 - k^2])*Sqrt[1 - (1 + k^2)*x + k^2*x^2])))/Sqrt[(1 - x)*x*(1 - k^2*x)]
3.24.67.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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]
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]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.59 (sec) , antiderivative size = 1121, normalized size of antiderivative = 5.93
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1121\) |
| elliptic | \(\text {Expression too large to display}\) | \(1132\) |
-2/k^4*(-(x-1/k^2)*k^2)^(1/2)*((-1+x)/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(k^2* x^3-k^2*x^2-x^2+x)^(1/2)*EllipticF((-(x-1/k^2)*k^2)^(1/2),(1/k^2/(1/k^2-1) )^(1/2))+1/k^2*(-2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^ (1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k ^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))+2/ (-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2) /(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/ k^2-(1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))/k^2-1/(-k^2+1)^(1/2)* (-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3 -k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(- k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))+2/(-k^2+1)^(1/2)*(-k^2*x+1)^(1 /2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+ x)^(1/2)*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(1+(-k^2+1)^(1/2)) /k^2),(1/k^2/(1/k^2-1))^(1/2))/k^2-2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2- 1)+1/(1/k^2-1)*x)^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*Ellipt icPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+(-1+(-k^2+1)^(1/2))/k^2),(1/k^2/( 1/k^2-1))^(1/2))+2/(-k^2+1)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^ (1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*EllipticPi((-(x-1/k^2)*k ^2)^(1/2),1/k^2/(1/k^2+(-1+(-k^2+1)^(1/2))/k^2),(1/k^2/(1/k^2-1))^(1/2))/k ^2+1/(-k^2+1)^(1/2)*(-k^2*x+1)^(1/2)*(-1/(1/k^2-1)+1/(1/k^2-1)*x)^(1/2)...
Time = 0.33 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.47 \[ \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\left [\frac {{\left (k^{2} - 1\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right ) - \sqrt {-k^{2} + 1} \log \left (\frac {k^{4} x^{4} - 4 \, {\left (2 \, k^{4} - k^{2}\right )} x^{3} + 2 \, {\left (4 \, k^{4} + k^{2} - 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {-k^{2} + 1} - 4 \, {\left (2 \, k^{2} - 1\right )} x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}, \frac {{\left (k^{2} - 1\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right ) + 2 \, \sqrt {k^{2} - 1} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {k^{2} - 1}}{2 \, {\left ({\left (k^{4} - k^{2}\right )} x^{3} - {\left (k^{4} - 1\right )} x^{2} + {\left (k^{2} - 1\right )} x\right )}}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}\right ] \]
[1/4*((k^2 - 1)*log((k^4*x^4 + 4*k^2*x^3 - 2*(3*k^2 + 2)*x^2 - 4*sqrt(k^2* x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 1) + 4*x + 1)/(k^4*x^4 - 4*k^2*x^3 + 2 *(k^2 + 2)*x^2 - 4*x + 1)) - sqrt(-k^2 + 1)*log((k^4*x^4 - 4*(2*k^4 - k^2) *x^3 + 2*(4*k^4 + k^2 - 2)*x^2 - 4*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2* x^2 - 2*k^2*x + 1)*sqrt(-k^2 + 1) - 4*(2*k^2 - 1)*x + 1)/(k^4*x^4 - 4*k^2* x^3 + 2*(k^2 + 2)*x^2 - 4*x + 1)))/(k^4 - k^2), 1/4*((k^2 - 1)*log((k^4*x^ 4 + 4*k^2*x^3 - 2*(3*k^2 + 2)*x^2 - 4*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k ^2*x^2 - 1) + 4*x + 1)/(k^4*x^4 - 4*k^2*x^3 + 2*(k^2 + 2)*x^2 - 4*x + 1)) + 2*sqrt(k^2 - 1)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2*k^2*x + 1)*sqrt(k^2 - 1)/((k^4 - k^2)*x^3 - (k^4 - 1)*x^2 + (k^2 - 1)*x) ))/(k^4 - k^2)]
Timed out. \[ \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(k-1>0)', see `assume?` for more details)Is
\[ \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\int { \frac {x^{2} - x}{{\left (k^{2} x^{2} - 2 \, x + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
Timed out. \[ \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx=\int -\frac {x-x^2}{\left (k^2\,x^2-2\,x+1\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}} \,d x \]