3.24.67 \(\int \frac {-x+x^2}{\sqrt {(1-x) x (1-k^2 x)} (1-2 x+k^2 x^2)} \, dx\) [2367]

Optimal. Leaf size=189 \[ -\frac {i \text {ArcTan}\left (\frac {\sqrt {-2+k^2-2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{-1+k^2 x}\right )}{2 \sqrt {-1+k^2} \sqrt {-2+k^2-2 i \sqrt {-1+k^2}}}+\frac {i \text {ArcTan}\left (\frac {\sqrt {-2+k^2+2 i \sqrt {-1+k^2}} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{-1+k^2 x}\right )}{2 \sqrt {-1+k^2} \sqrt {-2+k^2+2 i \sqrt {-1+k^2}}} \]

[Out]

-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)

________________________________________________________________________________________

Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.60, antiderivative size = 455, normalized size of antiderivative = 2.41, number of steps used = 26, number of rules used = 14, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.342, Rules used = {1607, 6850, 21, 6820, 6860, 936, 948, 12, 174, 551, 857, 728, 111, 116} \begin {gather*} \frac {\left (-k^2-\sqrt {1-k^2}+1\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{1-\sqrt {1-k^2}};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (-k^2+\sqrt {1-k^2}+1\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{\sqrt {1-k^2}+1};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (-k^2-\sqrt {1-k^2}+2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\text {ArcSin}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (-k^2+\sqrt {1-k^2}+2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\text {ArcSin}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x + x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 - 2*x + k^2*x^2)),x]

[Out]

-(((2 - k^2 - Sqrt[1 - k^2])*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticF[ArcSin[Sqrt[x]], k^2])/(k^2*Sqrt[1
- k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)])) + ((2 - k^2 + Sqrt[1 - k^2])*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 - k^2*x]*Elliptic
F[ArcSin[Sqrt[x]], k^2])/(k^2*Sqrt[1 - k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]) + ((1 - k^2 - Sqrt[1 - k^2])*(1 - x)*
Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[(1 - Sqrt[1 - k^2])^(-1), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/((
-k^2)^(3/2)*Sqrt[1 - k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) - ((1 - k^2 + Sqrt[1 - k^2])*(1 - x)*Sqrt
[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[(1 + Sqrt[1 - k^2])^(-1), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/((-k^2
)^(3/2)*Sqrt[1 - k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (GtQ[-b/d, 0] || LtQ[-b/f, 0])

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 728

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c/b)*x]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[
c, 0] && RationalQ[b]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 936

Int[Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]), x_Symbol] :> Dist
[(c*d^2 - b*d*e + a*e^2)/e^2, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] - Dist[1/e^2, Int[
(c*d - b*e - c*e*x)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*
f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 948

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]), Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6850

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n*z^q)^FracP
art[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p]))), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !FreeQ[z, x]

Rule 6860

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] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \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 {(-1+x) x}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {(-1+x) \sqrt {x}}{\sqrt {1-x} \sqrt {1-k^2 x} \left (1-2 x+k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {1-x} \sqrt {x}}{\sqrt {1-k^2 x} \left (1-2 x+k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {x-x^2}}{\sqrt {1-k^2 x} \left (1-2 x+k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (-\frac {k^2 \sqrt {x-x^2}}{\sqrt {1-k^2} \left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}}-\frac {k^2 \sqrt {x-x^2}}{\sqrt {1-k^2} \sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {x-x^2}}{\left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}} \, dx}{\sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {x-x^2}}{\sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{\sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-2+2 k^2-2 \sqrt {1-k^2}-2 k^2 x}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {2-2 k^2-2 \sqrt {1-k^2}+2 k^2 x}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {2} \left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\sqrt {2} \left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=-\frac {\left (\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=-\frac {\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \left (-2 \left (1-\sqrt {1-k^2}\right )-2 k^2 x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (2 \left (1+\sqrt {1-k^2}\right )+2 k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=-\frac {\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (1-k^2-\sqrt {1-k^2}\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{1-\sqrt {1-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (1-k^2+\sqrt {1-k^2}\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{1+\sqrt {1-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 7.26, size = 227, normalized size = 1.20 \begin {gather*} -\frac {i \sqrt {-1+x} \sqrt {x} \sqrt {\frac {-1+k^2 x}{-1+k^2}} \left (2 \sqrt {1-k^2} F\left (i \sinh ^{-1}\left (\sqrt {-1+x}\right )|\frac {k^2}{-1+k^2}\right )-\left (1+\sqrt {1-k^2}\right ) \Pi \left (\frac {k^2}{-1+k^2-\sqrt {1-k^2}};i \sinh ^{-1}\left (\sqrt {-1+x}\right )|\frac {k^2}{-1+k^2}\right )-\left (-1+\sqrt {1-k^2}\right ) \Pi \left (\frac {k^2}{-1+k^2+\sqrt {1-k^2}};i \sinh ^{-1}\left (\sqrt {-1+x}\right )|\frac {k^2}{-1+k^2}\right )\right )}{k^2 \sqrt {1-k^2} \sqrt {(-1+x) x \left (-1+k^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + x^2)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 - 2*x + k^2*x^2)),x]

[Out]

((-I)*Sqrt[-1 + x]*Sqrt[x]*Sqrt[(-1 + k^2*x)/(-1 + k^2)]*(2*Sqrt[1 - k^2]*EllipticF[I*ArcSinh[Sqrt[-1 + x]], k
^2/(-1 + k^2)] - (1 + Sqrt[1 - k^2])*EllipticPi[k^2/(-1 + k^2 - Sqrt[1 - k^2]), I*ArcSinh[Sqrt[-1 + x]], k^2/(
-1 + k^2)] - (-1 + Sqrt[1 - k^2])*EllipticPi[k^2/(-1 + k^2 + Sqrt[1 - k^2]), I*ArcSinh[Sqrt[-1 + x]], k^2/(-1
+ k^2)]))/(k^2*Sqrt[1 - k^2]*Sqrt[(-1 + x)*x*(-1 + k^2*x)])

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.23, size = 1121, normalized size = 5.93

method	result	size
default	\(\text {Expression too large to display}\)	\(1121\)
elliptic	\(\text {Expression too large to display}\)	\(1132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-x)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

-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)*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)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-x)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(k-1>0)', see `assume?` for mor
e details)Is

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 467, normalized size = 2.47 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-x)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1),x, algorithm="fricas")

[Out]

[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)]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-x)/((1-x)*x*(-k**2*x+1))**(1/2)/(k**2*x**2-2*x+1),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-x)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^2*x^2-2*x+1),x, algorithm="giac")

[Out]

integrate((x^2 - x)/((k^2*x^2 - 2*x + 1)*sqrt((k^2*x - 1)*(x - 1)*x)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - x^2)/((k^2*x^2 - 2*x + 1)*(x*(k^2*x - 1)*(x - 1))^(1/2)),x)

[Out]

int(-(x - x^2)/((k^2*x^2 - 2*x + 1)*(x*(k^2*x - 1)*(x - 1))^(1/2)), x)

________________________________________________________________________________________