\(\int \frac {-1+k^3 x^3}{\sqrt {(1-x) x (1-k^2 x)} (1+k^3 x^3)} \, dx\) [1572]

Optimal result

Integrand size = 40, antiderivative size = 108 \[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {(1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{3 (1+k)}-\frac {4 \arctan \left (\frac {\sqrt {1-k+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{3 \sqrt {1-k+k^2}} \]

[Out]

-2*arctan((1+k)*x/(x+(-k^2-1)*x^2+k^2*x^3)^(1/2))/(3+3*k)-4/3*arctan((k^2-k+1)^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^
(1/2)/(-1+x)/(k^2*x-1))/(k^2-k+1)^(1/2)

Rubi [C] (verified)

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

Time = 1.50 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.37, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {6850, 6820, 6857, 728, 116, 948, 12, 174, 551} \[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (-\frac {1}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1}}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (-\frac {(-1)^{2/3}}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \]

[In]

Int[(-1 + k^3*x^3)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 + k^3*x^3)),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)] + (4*(1 -
x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[-k^(-1), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(3*Sqrt[-k^2]*Sq
rt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + (4*(1 - x)*Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[(-1)^(1/3)/k
, ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(3*Sqrt[-k^2]*Sqrt[(1 - x)*x*(1 - k^2*x)]*Sqrt[x - x^2]) + (4*(1 - x)*
Sqrt[-x]*Sqrt[x]*Sqrt[1 - k^2*x]*EllipticPi[-((-1)^(2/3)/k), ArcSin[Sqrt[-k^2]*Sqrt[-x]], k^(-2)])/(3*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*} \text {integral}& = \frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^3 x^3}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (1+k^3 x^3\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^3 x^3}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1+k^3 x^3\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^3 x^3\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^3 x^3\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}{3 (-1-k x) \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {1}{3 \left (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {1}{3 \left (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x} \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} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\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}{(-1-k x) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{3 \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}{\left (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{3 \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}{\left (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )}} \\ & = \frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \sqrt {2} \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} (-1-k x) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \sqrt {2} \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 (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \sqrt {2} \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 (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \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} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \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} (-1-k x) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \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 (-1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \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 (-1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x}} \, dx}{3 \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} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \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 (-1+k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (4 \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 (-1-\sqrt [3]{-1} k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{3 \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (4 \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 (-1+(-1)^{2/3} k x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{3 \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} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (-\frac {1}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1}}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {4 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \operatorname {EllipticPi}\left (-\frac {(-1)^{2/3}}{k},\arcsin \left (\sqrt {-k^2} \sqrt {-x}\right ),\frac {1}{k^2}\right )}{3 \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.73 \[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {(1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{3 (1+k)}-\frac {4 \arctan \left (\frac {\sqrt {1-k+k^2} x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{3 \sqrt {1-k+k^2}} \]

[In]

Integrate[(-1 + k^3*x^3)/(Sqrt[(1 - x)*x*(1 - k^2*x)]*(1 + k^3*x^3)),x]

[Out]

(-2*ArcTan[((1 + k)*x)/Sqrt[(-1 + x)*x*(-1 + k^2*x)]])/(3*(1 + k)) - (4*ArcTan[(Sqrt[1 - k + k^2]*x)/Sqrt[(-1
+ x)*x*(-1 + k^2*x)]])/(3*Sqrt[1 - k + k^2])

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {2 \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right )}{3+3 k}+\frac {4 \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {k^{2}-k +1}}\right )}{3 \sqrt {k^{2}-k +1}}\)	\(76\)
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right )}{3+3 k}+\frac {4 \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {k^{2}-k +1}}\right )}{3 \sqrt {k^{2}-k +1}}\)	\(76\)
elliptic	\(\text {Expression too large to display}\)	\(907\)

[In]

int((k^3*x^3-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^3*x^3+1),x,method=_RETURNVERBOSE)

[Out]

2*arctan(((-1+x)*x*(k^2*x-1))^(1/2)/(1+k)/x)/(3+3*k)+4/3/(k^2-k+1)^(1/2)*arctan(((-1+x)*x*(k^2*x-1))^(1/2)/x/(
k^2-k+1)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (94) = 188\).

Time = 0.33 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.01 \[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=\frac {2 \, \sqrt {k^{2} - k + 1} {\left (k + 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - {\left (2 \, k^{2} - k + 2\right )} x + 1\right )} \sqrt {k^{2} - k + 1}}{2 \, {\left ({\left (k^{4} - k^{3} + k^{2}\right )} x^{3} - {\left (k^{4} - k^{3} + 2 \, k^{2} - k + 1\right )} x^{2} + {\left (k^{2} - k + 1\right )} x\right )}}\right ) + {\left (k^{2} - k + 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} + k^{2}\right )} x^{3} - {\left (k^{3} + k^{2} + k + 1\right )} x^{2} + {\left (k + 1\right )} x\right )}}\right )}{3 \, {\left (k^{3} + 1\right )}} \]

[In]

integrate((k^3*x^3-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^3*x^3+1),x, algorithm="fricas")

[Out]

1/3*(2*sqrt(k^2 - k + 1)*(k + 1)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - (2*k^2 - k + 2)*x + 1
)*sqrt(k^2 - k + 1)/((k^4 - k^3 + k^2)*x^3 - (k^4 - k^3 + 2*k^2 - k + 1)*x^2 + (k^2 - k + 1)*x)) + (k^2 - k +
1)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2*(k^2 + k + 1)*x + 1)/((k^3 + k^2)*x^3 - (k^3 + k^
2 + k + 1)*x^2 + (k + 1)*x)))/(k^3 + 1)

Sympy [F(-1)]

Timed out. \[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((k**3*x**3-1)/((1-x)*x*(-k**2*x+1))**(1/2)/(k**3*x**3+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=\int { \frac {k^{3} x^{3} - 1}{{\left (k^{3} x^{3} + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]

[In]

integrate((k^3*x^3-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^3*x^3+1),x, algorithm="maxima")

[Out]

integrate((k^3*x^3 - 1)/((k^3*x^3 + 1)*sqrt((k^2*x - 1)*(x - 1)*x)), x)

Giac [F]

\[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=\int { \frac {k^{3} x^{3} - 1}{{\left (k^{3} x^{3} + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]

[In]

integrate((k^3*x^3-1)/((1-x)*x*(-k^2*x+1))^(1/2)/(k^3*x^3+1),x, algorithm="giac")

[Out]

integrate((k^3*x^3 - 1)/((k^3*x^3 + 1)*sqrt((k^2*x - 1)*(x - 1)*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+k^3 x^3}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1+k^3 x^3\right )} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}