\(\int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x (1-k^2 x)} (-1+2 x+(-2+k^2) x^2)} \, dx\) [699]

Optimal result

Integrand size = 48, antiderivative size = 55 \[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-2+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{-1+k^2 x}\right )}{\sqrt {-2+k^2}} \]

[Out]

2*arctan((k^2-2)^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/(k^2*x-1))/(k^2-2)^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 3.63 (sec) , antiderivative size = 396, normalized size of antiderivative = 7.20, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6850, 6860, 116, 174, 552, 551} \[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=-\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {k^2-1} \left (k^2+2 \sqrt {k^2-1}\right ) \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \operatorname {EllipticPi}\left (\frac {2-k^2}{-k^2-\sqrt {k^2-1}+1},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (-k^2-\sqrt {k^2-1}+1\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {2 \sqrt {k^2-1} \left (k^2-2 \sqrt {k^2-1}\right ) \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \operatorname {EllipticPi}\left (\frac {2-k^2}{-k^2+\sqrt {k^2-1}+1},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (-k^2+\sqrt {k^2-1}+1\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \]

[In]

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

[Out]

(-2*k^2*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticF[ArcSin[Sqrt[x]], k^2])/((2 - k^2)*Sqrt[(1 - x)*x*(1 - k^
2*x)]) + (2*Sqrt[-1 + k^2]*(k^2 + 2*Sqrt[-1 + k^2])*Sqrt[1 + (k^2*(1 - x))/(1 - k^2)]*Sqrt[1 - x]*Sqrt[x]*Elli
pticPi[(2 - k^2)/(1 - k^2 - Sqrt[-1 + k^2]), ArcSin[Sqrt[1 - x]], -(k^2/(1 - k^2))])/((2 - k^2)*(1 - k^2 - Sqr
t[-1 + k^2])*Sqrt[(1 - x)*x*(1 - k^2*x)]) - (2*Sqrt[-1 + k^2]*(k^2 - 2*Sqrt[-1 + k^2])*Sqrt[1 + (k^2*(1 - x))/
(1 - k^2)]*Sqrt[1 - x]*Sqrt[x]*EllipticPi[(2 - k^2)/(1 - k^2 + Sqrt[-1 + k^2]), ArcSin[Sqrt[1 - x]], -(k^2/(1
- k^2))])/((2 - k^2)*(1 - k^2 + Sqrt[-1 + k^2])*Sqrt[(1 - x)*x*(1 - k^2*x)])

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 552

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

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*} \text {integral}& = \frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1-2 x+k^2 x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+2 x+\left (-2+k^2\right ) 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}{\left (2-k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}}+\frac {2 \left (-1+k^2\right ) (1-2 x)}{\left (-2+k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+2 x+\left (-2+k^2\right ) x^2\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 {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1-2 x}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = -\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {-2+\frac {k^2}{\sqrt {-1+k^2}}}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (2-2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )}+\frac {-2-\frac {k^2}{\sqrt {-1+k^2}}}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (2+2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )}\right ) \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = -\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \left (-2-\frac {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} \left (2+2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \left (-2+\frac {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} \left (2-2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = -\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2+k^2 x^2} \left (-2 \left (1-k^2-\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2+k^2 x^2} \left (-2 \left (1-k^2+\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = -\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {k^2 x^2}{1-k^2}} \left (-2 \left (1-k^2-\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {k^2 x^2}{1-k^2}} \left (-2 \left (1-k^2+\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = -\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {-1+k^2} \left (k^2+2 \sqrt {-1+k^2}\right ) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \operatorname {EllipticPi}\left (\frac {2-k^2}{1-k^2-\sqrt {-1+k^2}},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (1-k^2-\sqrt {-1+k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {2 \left (2-k^2 \left (2-\sqrt {-1+k^2}\right )\right ) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \operatorname {EllipticPi}\left (\frac {2-k^2}{1-k^2+\sqrt {-1+k^2}},\arcsin \left (\sqrt {1-x}\right ),-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (1-k^2+\sqrt {-1+k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-2+k^2} (-1+x) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{\sqrt {-2+k^2}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 1.43 (sec) , antiderivative size = 2704, normalized size of antiderivative = 49.16

method	result	size
default	\(\text {Expression too large to display}\)	\(2704\)
elliptic	\(\text {Expression too large to display}\)	\(2727\)

[In]

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

[Out]

-2/(k^2-2)*(-(x-1/k^2)*k^2)^(1/2)*((x-1)/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)*Elliptic
F((-(x-1/k^2)*k^2)^(1/2),(1/k^2/(1/k^2-1))^(1/2))+1/(k^2-2)*(-8/(-2*k^2/(k^2-2)+2/(k^2-2)*k^2*(k^2-1)^(1/2)+4/
(k^2-2)-4/(k^2-2)*(k^2-1)^(1/2)+2)*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k
^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)-1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(
-1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)+8/(-2*k^2/(k^2-2)+2/(k^2-2)*k^2*(k^2-1)^(1/2)+4/(k
^2-2)-4/(k^2-2)*(k^2-1)^(1/2)+2)/k^2*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(k^2*x^3
-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)-1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2
-(-1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)+8/(-2*k^2/(k^2-2)+2/(k^2-2)*k^2*(k^2-1)^(1/2)+4/
(k^2-2)-4/(k^2-2)*(k^2-1)^(1/2)+2)*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k
^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)-1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2-(
-1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)*(k^2-1)^(1/2)-8/(-2*k^2/(k^2-2)+2/(k^2-2)*k^2*(k^2
-1)^(1/2)+4/(k^2-2)-4/(k^2-2)*(k^2-1)^(1/2)+2)/k^2*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k^2*x)^
(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)-1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2
),1/k^2/(1/k^2-(-1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)*(k^2-1)^(1/2)-4/(-2*k^2/(k^2-2)+2/
(k^2-2)*k^2*(k^2-1)^(1/2)+4/(k^2-2)-4/(k^2-2)*(k^2-1)^(1/2)+2)*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1
/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)-1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^
2)*k^2)^(1/2),1/k^2/(1/k^2-(-1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))+4/(-2*k^2/(k^2-2)+2/(k^2-2)*k^
2*(k^2-1)^(1/2)+4/(k^2-2)-4/(k^2-2)*(k^2-1)^(1/2)+2)/k^2*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k
^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)-1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2
)^(1/2),1/k^2/(1/k^2-(-1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))-8/(-2*k^2/(k^2-2)-2/(k^2-2)*k^2*(k^2
-1)^(1/2)+4/(k^2-2)+4/(k^2-2)*(k^2-1)^(1/2)+2)*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k^2*x)^(1/2
)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)+1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/
k^2/(1/k^2+(1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)+8/(-2*k^2/(k^2-2)-2/(k^2-2)*k^2*(k^2-1)
^(1/2)+4/(k^2-2)+4/(k^2-2)*(k^2-1)^(1/2)+2)/k^2*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k^2*x)^(1/
2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)+1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1
/k^2/(1/k^2+(1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)-8/(-2*k^2/(k^2-2)-2/(k^2-2)*k^2*(k^2-1
)^(1/2)+4/(k^2-2)+4/(k^2-2)*(k^2-1)^(1/2)+2)*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/
(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)+1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*k^2)^(1/2),1/k^
2/(1/k^2+(1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)*(k^2-1)^(1/2)+8/(-2*k^2/(k^2-2)-2/(k^2-2)
*k^2*(k^2-1)^(1/2)+4/(k^2-2)+4/(k^2-2)*(k^2-1)^(1/2)+2)/k^2*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(1/2)
*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)+1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k^2)*
k^2)^(1/2),1/k^2/(1/k^2+(1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))/(k^2-2)*(k^2-1)^(1/2)-4/(-2*k^2/(k
^2-2)-2/(k^2-2)*k^2*(k^2-1)^(1/2)+4/(k^2-2)+4/(k^2-2)*(k^2-1)^(1/2)+2)*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^
2-1))^(1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)+1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((
-(x-1/k^2)*k^2)^(1/2),1/k^2/(1/k^2+(1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2))+4/(-2*k^2/(k^2-2)-2/(k^
2-2)*k^2*(k^2-1)^(1/2)+4/(k^2-2)+4/(k^2-2)*(k^2-1)^(1/2)+2)/k^2*(-k^2*x+1)^(1/2)*(1/(1/k^2-1)*x-1/(1/k^2-1))^(
1/2)*(k^2*x)^(1/2)/(k^2*x^3-k^2*x^2-x^2+x)^(1/2)/(1/k^2+1/(k^2-2)+1/(k^2-2)*(k^2-1)^(1/2))*EllipticPi((-(x-1/k
^2)*k^2)^(1/2),1/k^2/(1/k^2+(1+(k^2-1)^(1/2))/(k^2-2)),(1/k^2/(1/k^2-1))^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.89 \[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=\left [-\frac {\sqrt {-k^{2} + 2} \log \left (\frac {{\left (k^{4} - 4 \, k^{2} + 4\right )} x^{4} - 4 \, {\left (2 \, k^{4} - 5 \, k^{2} + 2\right )} x^{3} + 2 \, {\left (4 \, k^{4} - 5 \, k^{2} - 4\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left ({\left (k^{2} - 2\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + 1\right )} \sqrt {-k^{2} + 2} - 4 \, {\left (2 \, k^{2} - 3\right )} x + 1}{{\left (k^{4} - 4 \, k^{2} + 4\right )} x^{4} + 4 \, {\left (k^{2} - 2\right )} x^{3} - 2 \, {\left (k^{2} - 4\right )} x^{2} - 4 \, x + 1}\right )}{2 \, {\left (k^{2} - 2\right )}}, \frac {\arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left ({\left (k^{2} - 2\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + 1\right )} \sqrt {k^{2} - 2}}{2 \, {\left ({\left (k^{4} - 2 \, k^{2}\right )} x^{3} - {\left (k^{4} - k^{2} - 2\right )} x^{2} + {\left (k^{2} - 2\right )} x\right )}}\right )}{\sqrt {k^{2} - 2}}\right ] \]

[In]

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

[Out]

[-1/2*sqrt(-k^2 + 2)*log(((k^4 - 4*k^2 + 4)*x^4 - 4*(2*k^4 - 5*k^2 + 2)*x^3 + 2*(4*k^4 - 5*k^2 - 4)*x^2 - 4*sq
rt(k^2*x^3 - (k^2 + 1)*x^2 + x)*((k^2 - 2)*x^2 - 2*(k^2 - 1)*x + 1)*sqrt(-k^2 + 2) - 4*(2*k^2 - 3)*x + 1)/((k^
4 - 4*k^2 + 4)*x^4 + 4*(k^2 - 2)*x^3 - 2*(k^2 - 4)*x^2 - 4*x + 1))/(k^2 - 2), arctan(1/2*sqrt(k^2*x^3 - (k^2 +
1)*x^2 + x)*((k^2 - 2)*x^2 - 2*(k^2 - 1)*x + 1)*sqrt(k^2 - 2)/((k^4 - 2*k^2)*x^3 - (k^4 - k^2 - 2)*x^2 + (k^2
- 2)*x))/sqrt(k^2 - 2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [F]

\[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=\int { \frac {k^{2} x^{2} - 2 \, x + 1}{\sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} {\left ({\left (k^{2} - 2\right )} x^{2} + 2 \, x - 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.53 \[ \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx=\frac {\ln \left (\frac {x\,2{}\mathrm {i}+k^2\,x^2\,1{}\mathrm {i}-k^2\,x\,2{}\mathrm {i}-x^2\,2{}\mathrm {i}-2\,\sqrt {k^2-2}\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}+1{}\mathrm {i}}{2\,k^2\,x^2-4\,x^2+4\,x-2}\right )\,1{}\mathrm {i}}{\sqrt {k^2-2}} \]

[In]

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

[Out]

(log((x*2i + k^2*x^2*1i - k^2*x*2i - x^2*2i - 2*(k^2 - 2)^(1/2)*(x*(k^2*x - 1)*(x - 1))^(1/2) + 1i)/(4*x + 2*k
^2*x^2 - 4*x^2 - 2))*1i)/(k^2 - 2)^(1/2)