3.19.37 \(\int \frac {-1+k^{3/2} x^3}{\sqrt {(1-x^2) (1-k^2 x^2)} (1+k^{3/2} x^3)} \, dx\) [1837]

Optimal. Leaf size=125 \[ -\frac {4 \text {ArcTan}\left (\frac {\sqrt {1+k+k^2} x}{1-\sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}}-\frac {2 \text {ArcTan}\left (\frac {(-1+k) x}{1+2 \sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 (-1+k)} \]

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.63, antiderivative size = 709, normalized size of antiderivative = 5.67, number of steps used = 28, number of rules used = 13, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.277, Rules used = {1976, 6857, 1117, 1738, 1224, 1712, 209, 1261, 738, 210, 1230, 1720, 212} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {(1-k) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 (1-k)}-\frac {2 \text {ArcTan}\left (\frac {\sqrt {k^2+k+1} x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {k^2+k+1}}+\frac {\text {ArcTan}\left (\frac {(1-k) \left (k x^2+1\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 (1-k)}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {k} x\right )|\frac {(k+1)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {(-1)^{2/3} \sqrt {2} \tanh ^{-1}\left (\frac {-\left (\sqrt [3]{-1} \left (k^2+1\right )+2 k\right ) k x^2+k^2+2 \sqrt [3]{-1} k+1}{\sqrt {2} \sqrt {k} \sqrt {\left (1+i \sqrt {3}\right ) \left (k^2+k+1\right )} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {\left (1+i \sqrt {3}\right ) \left (k^2+k+1\right )}}+\frac {\sqrt [3]{-1} \sqrt {2} \tanh ^{-1}\left (\frac {-\left (2 k-(-1)^{2/3} \left (k^2+1\right )\right ) k x^2+k^2-2 (-1)^{2/3} k+1}{\sqrt {2} \sqrt {k} \sqrt {\left (1-i \sqrt {3}\right ) \left (k^2+k+1\right )} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {\left (1-i \sqrt {3}\right ) \left (k^2+k+1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*ArcTan[((1 - k)*x)/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]]/(1 - k) - (2*ArcTan[(Sqrt[1 + k + k^2]*x)/Sqrt[1 -
(1 + k^2)*x^2 + k^2*x^4]])/(3*Sqrt[1 + k + k^2]) + ArcTan[((1 - k)*(1 + k*x^2))/(2*Sqrt[k]*Sqrt[1 - (1 + k^2)*
x^2 + k^2*x^4])]/(3*(1 - k)) - ((-1)^(2/3)*Sqrt[2]*ArcTanh[(1 + 2*(-1)^(1/3)*k + k^2 - k*(2*k + (-1)^(1/3)*(1
+ k^2))*x^2)/(Sqrt[2]*Sqrt[k]*Sqrt[(1 + I*Sqrt[3])*(1 + k + k^2)]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])])/(3*Sqrt
[(1 + I*Sqrt[3])*(1 + k + k^2)]) + ((-1)^(1/3)*Sqrt[2]*ArcTanh[(1 - 2*(-1)^(2/3)*k + k^2 - k*(2*k - (-1)^(2/3)
*(1 + k^2))*x^2)/(Sqrt[2]*Sqrt[k]*Sqrt[(1 - I*Sqrt[3])*(1 + k + k^2)]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])])/(3*
Sqrt[(1 - I*Sqrt[3])*(1 + k + k^2)]) + ((1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*Elliptic
F[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(3*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((1 + k*x^2)*Sqrt[(1
- (1 + k^2)*x^2 + k^2*x^4)/(1 + k*x^2)^2]*EllipticF[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(3*(1 - (-1)^(1/3))
*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]) - ((1 - (-1)^(1/3))*(1 + k*x^2)*Sqrt[(1 - (1 + k^2)*x^2 + k^2*x^4)
/(1 + k*x^2)^2]*EllipticF[2*ArcTan[Sqrt[k]*x], (1 + k)^2/(4*k)])/(3*Sqrt[k]*Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1224

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

Rule 1230

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Di
st[(c*d + a*e*q)/(c*d^2 - a*e^2), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2)
, Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a
*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1712

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B},
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 1720

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*
e*Rt[-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + b*x^2 + c*x^4)/(a*(A + B*
x^2)^2))]/(4*d*e*A*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1
/2 - b*(A/(4*a*B))], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1738

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x
^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; Free
Q[{a, b, c, d, e}, x]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, 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^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {-1+k^{3/2} x^3}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (1+k^{3/2} x^3\right )} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (\frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {2}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (1+k^{3/2} x^3\right )}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (1+k^{3/2} x^3\right )} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (-\frac {1}{3 \left (-1-\sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {1}{3 \left (-1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}-\frac {1}{3 \left (-1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (-1-\sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (-1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\left (-1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-(-1)^{2/3} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1-k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1-(-1)^{2/3} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 (-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x}{\sqrt {1-x^2} \left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1-k^2 x^2}} \, dx}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} (1-k x) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (1-(-1)^{2/3} k x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left ((-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \left (1+\sqrt [3]{-1} k x\right ) \sqrt {1-k^2 x}} \, dx,x,x^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+k-\left (k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 \sqrt [3]{-1} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+(-1)^{2/3} k-\left ((-1)^{2/3} k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 (-1)^{2/3} \sqrt {k} \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-1-\sqrt [3]{-1} k-\left (-\sqrt [3]{-1} k-k^2\right ) x^2} \, dx,x,\frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tan ^{-1}\left (\frac {\sqrt {k} \sqrt {1-x^2}}{\sqrt {1-k^2 x^2}}\right )}{3 (1-k) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 (-1)^{2/3} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {\sqrt [3]{-1}+k} \sqrt {1-x^2}}{\sqrt {1+\sqrt [3]{-1} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {\sqrt [3]{-1}+k} \sqrt {1+\sqrt [3]{-1} k} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 \sqrt [3]{-1} \sqrt {1-x^2} \sqrt {1-k^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {k} \sqrt {-(-1)^{2/3}+k} \sqrt {1-x^2}}{\sqrt {1-(-1)^{2/3} k} \sqrt {1-k^2 x^2}}\right )}{3 \sqrt {-(-1)^{2/3}+k} \sqrt {1-(-1)^{2/3} k} \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left ((-1)^{2/3} k;\sin ^{-1}(x)|k^2\right )}{3 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 10.26, size = 558, normalized size = 4.46 \begin {gather*} \frac {(-1-i) \sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \left (-\frac {(1-i) \text {ArcTan}\left (\frac {\sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-1+x^2}}\right )}{-1+k}+\frac {\left (-1-i \sqrt {3}+k-i \sqrt {3} k\right ) \text {ArcTan}\left (\frac {(1+i) \sqrt {1+k+k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-\sqrt {2+2 i \sqrt {3}}-4 i k+\left (-i+\sqrt {3}\right ) k^2} \sqrt {-1+x^2}}\right )}{\sqrt {1+k+k^2} \sqrt {-\sqrt {2+2 i \sqrt {3}}-4 i k+\left (-i+\sqrt {3}\right ) k^2}}+\frac {\left (1-i \sqrt {3}+\left (-1-i \sqrt {3}\right ) k\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {1+k+k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-\sqrt {2-2 i \sqrt {3}}+4 i k+\left (i+\sqrt {3}\right ) k^2} \sqrt {-1+x^2}}\right )}{\sqrt {1+k+k^2} \sqrt {-\sqrt {2-2 i \sqrt {3}}+4 i k+\left (i+\sqrt {3}\right ) k^2}}\right )+3 \sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\text {ArcSin}(x)\left |k^2\right .\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (k;\text {ArcSin}(x)\left |k^2\right .\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (-\sqrt [3]{-1} k;\text {ArcSin}(x)\left |k^2\right .\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \Pi \left (\frac {1}{2} i \left (i+\sqrt {3}\right ) k;\text {ArcSin}(x)\left |k^2\right .\right )}{3 \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1 - I)*Sqrt[-1 + x^2]*Sqrt[-1 + k^2*x^2]*(((-1 + I)*ArcTan[Sqrt[-1 + k^2*x^2]/(Sqrt[k]*Sqrt[-1 + x^2])])/(-
1 + k) + ((-1 - I*Sqrt[3] + k - I*Sqrt[3]*k)*ArcTan[((1 + I)*Sqrt[1 + k + k^2]*Sqrt[-1 + k^2*x^2])/(Sqrt[k]*Sq
rt[-Sqrt[2 + (2*I)*Sqrt[3]] - (4*I)*k + (-I + Sqrt[3])*k^2]*Sqrt[-1 + x^2])])/(Sqrt[1 + k + k^2]*Sqrt[-Sqrt[2
+ (2*I)*Sqrt[3]] - (4*I)*k + (-I + Sqrt[3])*k^2]) + ((1 - I*Sqrt[3] + (-1 - I*Sqrt[3])*k)*ArcTanh[((1 + I)*Sqr
t[1 + k + k^2]*Sqrt[-1 + k^2*x^2])/(Sqrt[k]*Sqrt[-Sqrt[2 - (2*I)*Sqrt[3]] + (4*I)*k + (I + Sqrt[3])*k^2]*Sqrt[
-1 + x^2])])/(Sqrt[1 + k + k^2]*Sqrt[-Sqrt[2 - (2*I)*Sqrt[3]] + (4*I)*k + (I + Sqrt[3])*k^2])) + 3*Sqrt[1 - x^
2]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[x], k^2] - 2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[k, ArcSin[x], k^
2] - 2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticPi[-((-1)^(1/3)*k), ArcSin[x], k^2] - 2*Sqrt[1 - x^2]*Sqrt[1 -
k^2*x^2]*EllipticPi[(I/2)*(I + Sqrt[3])*k, ArcSin[x], k^2])/(3*Sqrt[(-1 + x^2)*(-1 + k^2*x^2)])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.26, size = 1988, normalized size = 15.90

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1659\)
default	\(\text {Expression too large to display}\)	\(1988\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 0.78, size = 212, normalized size = 1.70 \begin {gather*} -\frac {2 \, \sqrt {k^{2} + k + 1} {\left (k - 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {k^{2} + k + 1} {\left ({\left (k^{2} + 2 \, k + 1\right )} x + {\left (k x^{2} + 1\right )} \sqrt {k}\right )}}{k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} + 4 \, k^{2} + 4 \, k + 1\right )} x^{2} + k}\right ) - {\left (k^{2} + k + 1\right )} \arctan \left (-\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left ({\left (k^{3} + k^{2} - k - 1\right )} x - 2 \, {\left ({\left (k^{2} - k\right )} x^{2} + k - 1\right )} \sqrt {k}\right )}}{4 \, k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} - 2 \, k^{2} + 4 \, k + 1\right )} x^{2} + 4 \, k}\right )}{3 \, {\left (k^{3} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\sqrt {k} x - 1\right ) \left (\sqrt {k} x + k x^{2} + 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (\sqrt {k} x + 1\right ) \left (- \sqrt {k} x + k x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {k^{3/2}\,x^3-1}{\left (k^{3/2}\,x^3+1\right )\,\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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