3.21.88 \(\int \frac {-1+k^4 x^4}{\sqrt {(1-x) x (1-k^2 x)} (1+k^4 x^4)} \, dx\) [2088]

Optimal. Leaf size=151 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {1-\sqrt {2} k+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{\sqrt {1-\sqrt {2} k+k^2}}-\frac {\text {ArcTan}\left (\frac {\sqrt {1+\sqrt {2} k+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{\sqrt {1+\sqrt {2} k+k^2}} \]

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.72, antiderivative size = 487, normalized size of antiderivative = 3.23, number of steps used = 28, number of rules used = 9, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {6850, 6820, 6857, 728, 116, 948, 12, 174, 551} \begin {gather*} \frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\text {ArcSin}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {k^2}{\left (-k^4\right )^{3/4}};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt [4]{-k^4}}{k^2};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {\sqrt {-\sqrt {-k^4}}}{k^2};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {\sqrt {-\sqrt {-k^4}}}{k^2};\text {ArcSin}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 + k^4*x^4)),x]

[Out]

(2*Sqrt[1 - x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticF[ArcSin[Sqrt[x]], k^2])/Sqrt[(1 - x)*x*(1 - k^2*x)] + ((1 - x)
*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[k^2/(-k^4)^(3/4), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(Sqrt[-k^
2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + ((1 - x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[(-k^4)^(1
/4)/k^2, ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(Sqrt[-k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + ((1 -
x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[-(Sqrt[-Sqrt[-k^4]]/k^2), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])
/(Sqrt[-k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + ((1 - x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi
[Sqrt[-Sqrt[-k^4]]/k^2, ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(Sqrt[-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 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 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 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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 33.59, size = 10871, normalized size = 71.99 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-1 + k^4*x^4)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 + k^4*x^4)),x]

[Out]

Result too large to show

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.11, size = 255, normalized size = 1.69

method	result	size
default	\(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (k^{4} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{4}+1}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (k^{4}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{k^{4}}\)	\(255\)
elliptic	\(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (k^{4} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1\right ) \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {k^{6} \underline {\hspace {1.25 ex}}\alpha ^{3}+k^{4} \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{4}+1}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (k^{4}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}}{k^{4}}\)	\(255\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((k^4*x^4-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4+1),x,method=_RETURNVERBOSE)

[Out]

-2/k^2*(-(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^4*sum(1/_alpha^3*(_alpha^3*k^6+_alpha^2*k^4+_alpha*k^2+1)/(
k^4+1)*(-(x-1/k^2)*k^2)^(1/2)*((-1+x)/(1/k^2-1))^(1/2)*(k^2*x)^(1/2)/(x*(k^2*x^2-k^2*x-x+1))^(1/2)*EllipticPi(
(-(x-1/k^2)*k^2)^(1/2),(_alpha^3*k^6+_alpha^2*k^4+_alpha*k^2+1)/(k^4+1),(1/k^2/(1/k^2-1))^(1/2)),_alpha=RootOf
(_Z^4*k^4+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((k^4*x^4-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4+1),x, algorithm="maxima")

[Out]

integrate((k^4*x^4 - 1)/((k^4*x^4 + 1)*sqrt((k^2*x - 1)*(x - 1)*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1197 vs. \(2 (129) = 258\).
time = 0.44, size = 1197, normalized size = 7.93 \begin {gather*} -\frac {1}{4} \, \sqrt {-\frac {k^{2} + 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{2} - 2 \, {\left (k^{4} + k^{2}\right )} x^{3} + 4 \, \sqrt {\frac {1}{2}} {\left ({\left (k^{6} + k^{2}\right )} x^{3} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x^{2} + {\left (k^{4} + 1\right )} x\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 2 \, {\left (k^{2} + 1\right )} x + 2 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (2 \, k^{2} x - {\left (k^{4} + k^{2}\right )} x^{2} - k^{2} + 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + {\left (k^{6} + k^{2}\right )} x^{2} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 1\right )} \sqrt {-\frac {k^{2} + 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} + 1}{k^{4} x^{4} + 1}\right ) + \frac {1}{4} \, \sqrt {-\frac {k^{2} + 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{2} - 2 \, {\left (k^{4} + k^{2}\right )} x^{3} + 4 \, \sqrt {\frac {1}{2}} {\left ({\left (k^{6} + k^{2}\right )} x^{3} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x^{2} + {\left (k^{4} + 1\right )} x\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 2 \, {\left (k^{2} + 1\right )} x - 2 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (2 \, k^{2} x - {\left (k^{4} + k^{2}\right )} x^{2} - k^{2} + 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + {\left (k^{6} + k^{2}\right )} x^{2} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 1\right )} \sqrt {-\frac {k^{2} + 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} + 1}{k^{4} x^{4} + 1}\right ) - \frac {1}{4} \, \sqrt {-\frac {k^{2} - 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{2} - 2 \, {\left (k^{4} + k^{2}\right )} x^{3} - 4 \, \sqrt {\frac {1}{2}} {\left ({\left (k^{6} + k^{2}\right )} x^{3} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x^{2} + {\left (k^{4} + 1\right )} x\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 2 \, {\left (k^{2} + 1\right )} x + 2 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (2 \, k^{2} x - {\left (k^{4} + k^{2}\right )} x^{2} - k^{2} - 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + {\left (k^{6} + k^{2}\right )} x^{2} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 1\right )} \sqrt {-\frac {k^{2} - 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} + 1}{k^{4} x^{4} + 1}\right ) + \frac {1}{4} \, \sqrt {-\frac {k^{2} - 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{2} - 2 \, {\left (k^{4} + k^{2}\right )} x^{3} - 4 \, \sqrt {\frac {1}{2}} {\left ({\left (k^{6} + k^{2}\right )} x^{3} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x^{2} + {\left (k^{4} + 1\right )} x\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 2 \, {\left (k^{2} + 1\right )} x - 2 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (2 \, k^{2} x - {\left (k^{4} + k^{2}\right )} x^{2} - k^{2} - 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + {\left (k^{6} + k^{2}\right )} x^{2} - {\left (k^{6} + k^{4} + k^{2} + 1\right )} x + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} - 1\right )} \sqrt {-\frac {k^{2} - 2 \, \sqrt {\frac {1}{2}} {\left (k^{4} + 1\right )} \sqrt {\frac {k^{2}}{k^{8} + 2 \, k^{4} + 1}} + 1}{k^{4} + 1}} + 1}{k^{4} x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((k^4*x^4-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4+1),x, algorithm="fricas")

[Out]

-1/4*sqrt(-(k^2 + 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1))*log((k^4*x^4 + 4*k^2*x^2 -
2*(k^4 + k^2)*x^3 + 4*sqrt(1/2)*((k^6 + k^2)*x^3 - (k^6 + k^4 + k^2 + 1)*x^2 + (k^4 + 1)*x)*sqrt(k^2/(k^8 + 2
*k^4 + 1)) - 2*(k^2 + 1)*x + 2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(2*k^2*x - (k^4 + k^2)*x^2 - k^2 + 2*sqrt(1/2
)*(k^4 + (k^6 + k^2)*x^2 - (k^6 + k^4 + k^2 + 1)*x + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 1)*sqrt(-(k^2 + 2*sqrt(1
/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1)) + 1)/(k^4*x^4 + 1)) + 1/4*sqrt(-(k^2 + 2*sqrt(1/2)*(
k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1))*log((k^4*x^4 + 4*k^2*x^2 - 2*(k^4 + k^2)*x^3 + 4*sqrt(1/2
)*((k^6 + k^2)*x^3 - (k^6 + k^4 + k^2 + 1)*x^2 + (k^4 + 1)*x)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 2*(k^2 + 1)*x - 2*
sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(2*k^2*x - (k^4 + k^2)*x^2 - k^2 + 2*sqrt(1/2)*(k^4 + (k^6 + k^2)*x^2 - (k^6
+ k^4 + k^2 + 1)*x + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 1)*sqrt(-(k^2 + 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2
*k^4 + 1)) + 1)/(k^4 + 1)) + 1)/(k^4*x^4 + 1)) - 1/4*sqrt(-(k^2 - 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4
+ 1)) + 1)/(k^4 + 1))*log((k^4*x^4 + 4*k^2*x^2 - 2*(k^4 + k^2)*x^3 - 4*sqrt(1/2)*((k^6 + k^2)*x^3 - (k^6 + k^4
+ k^2 + 1)*x^2 + (k^4 + 1)*x)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 2*(k^2 + 1)*x + 2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 +
x)*(2*k^2*x - (k^4 + k^2)*x^2 - k^2 - 2*sqrt(1/2)*(k^4 + (k^6 + k^2)*x^2 - (k^6 + k^4 + k^2 + 1)*x + 1)*sqrt(k
^2/(k^8 + 2*k^4 + 1)) - 1)*sqrt(-(k^2 - 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1)) + 1)
/(k^4*x^4 + 1)) + 1/4*sqrt(-(k^2 - 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1))*log((k^4*
x^4 + 4*k^2*x^2 - 2*(k^4 + k^2)*x^3 - 4*sqrt(1/2)*((k^6 + k^2)*x^3 - (k^6 + k^4 + k^2 + 1)*x^2 + (k^4 + 1)*x)*
sqrt(k^2/(k^8 + 2*k^4 + 1)) - 2*(k^2 + 1)*x - 2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(2*k^2*x - (k^4 + k^2)*x^2 -
k^2 - 2*sqrt(1/2)*(k^4 + (k^6 + k^2)*x^2 - (k^6 + k^4 + k^2 + 1)*x + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) - 1)*sqrt
(-(k^2 - 2*sqrt(1/2)*(k^4 + 1)*sqrt(k^2/(k^8 + 2*k^4 + 1)) + 1)/(k^4 + 1)) + 1)/(k^4*x^4 + 1))

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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((k**4*x**4-1)/((1-x)*x*(-k**2*x+1))**(1/2)/(k**4*x**4+1),x)

[Out]

Timed out

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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((k^4*x^4-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^4*x^4+1),x, algorithm="giac")

[Out]

integrate((k^4*x^4 - 1)/((k^4*x^4 + 1)*sqrt((k^2*x - 1)*(x - 1)*x)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {k^4\,x^4-1}{\left (k^4\,x^4+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((k^4*x^4 - 1)/((k^4*x^4 + 1)*(x*(k^2*x - 1)*(x - 1))^(1/2)),x)

[Out]

int((k^4*x^4 - 1)/((k^4*x^4 + 1)*(x*(k^2*x - 1)*(x - 1))^(1/2)), x)

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