3.6.22 \(\int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {(1-x^2) (1-k^2 x^2)} (-1+k^2 x^2)} \, dx\)

Optimal. Leaf size=40 \[ -\frac {\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}{x \left (k^2 x^2-1\right )} \]

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Rubi [C]  time = 1.71, antiderivative size = 451, normalized size of antiderivative = 11.28, number of steps used = 14, number of rules used = 10, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6719, 21, 6742, 414, 424, 472, 583, 524, 419, 471} \begin {gather*} \frac {k^2 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {k^2 \left (1-x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \left (1-x^2\right ) \left (1-k^2 x^2\right )}{\left (1-k^2\right ) x \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 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(x)|k^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {k \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(k x)|\frac {1}{k^2}\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(x)|k^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 k^4 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(k^2*(1 - x^2))/((1 - k^2)*x*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) + (k^2*x*(1 - x^2))/((1 - k^2)*Sqrt[(1 - x^2)*(1 -
k^2*x^2)]) - (2*k^4*x*(1 - x^2))/((1 - k^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) + ((1 - 2*k^2)*(1 - x^2)*(1 - k^2*
x^2))/((1 - k^2)*x*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) + (2*k^2*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticE[ArcSin[x]
, k^2])/((1 - k^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) + ((1 - 2*k^2)*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticE[Arc
Sin[x], k^2])/((1 - k^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) - (k*Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticE[ArcSin[
k*x], k^(-2)])/((1 - k^2)*Sqrt[(1 - x^2)*(1 - k^2*x^2)]) - (Sqrt[1 - x^2]*Sqrt[1 - k^2*x^2]*EllipticF[ArcSin[x
], k^2])/Sqrt[(1 - x^2)*(1 - k^2*x^2)]

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 419

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 524

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

Rule 583

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 30, normalized size = 0.75 \begin {gather*} \frac {1-x^2}{x \sqrt {\left (x^2-1\right ) \left (k^2 x^2-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 5.63, size = 40, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}{x \left (-1+k^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 36, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1}}{k^{2} x^{3} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k^{2} x^{4} - 2 \, k^{2} x^{2} + 1}{{\left (k^{2} x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.16, size = 29, normalized size = 0.72

method	result	size
gosper	\(-\frac {\left (-1+x \right ) \left (1+x \right )}{\sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}\, x}\)	\(29\)
elliptic	\(\frac {-x^{2}+1}{x \sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}\, k^{2}}\)	\(35\)
trager	\(-\frac {\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}{x \left (k^{2} x^{2}-1\right )}\)	\(41\)
risch	\(\frac {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}{x \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}}-\frac {\left (1+x \right ) \left (-1+x \right ) k^{2} x}{\sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}}\)	\(66\)
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\left (\frac {k^{2}}{2}-\frac {1}{2}\right ) \left (-\frac {k^{2} x^{3}-k^{2} x -k \,x^{2}+k}{\left (k^{2}-1\right ) \sqrt {\left (x +\frac {1}{k}\right ) \left (k^{2} x^{3}-k^{2} x -k \,x^{2}+k \right )}}+\frac {\left (\frac {k^{2}-2}{2 k^{2}-2}-\frac {k^{2}}{2 \left (k^{2}-1\right )}\right ) \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\EllipticF \left (x , k\right )-\EllipticE \left (x , k\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )+\frac {\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}{x}-\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\EllipticF \left (x , k\right )-\EllipticE \left (x , k\right )\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\left (-\frac {k^{2}}{2}+\frac {1}{2}\right ) \left (\frac {k^{2} x^{3}-k^{2} x +k \,x^{2}-k}{\left (k^{2}-1\right ) \sqrt {\left (x -\frac {1}{k}\right ) \left (k^{2} x^{3}-k^{2} x +k \,x^{2}-k \right )}}+\frac {\left (-\frac {k^{2}-2}{2 \left (k^{2}-1\right )}+\frac {k^{2}}{2 k^{2}-2}\right ) \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\EllipticF \left (x , k\right )-\EllipticE \left (x , k\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )\)	\(551\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.70, size = 34, normalized size = 0.85 \begin {gather*} -\frac {x^{2} - 1}{\sqrt {k x + 1} \sqrt {k x - 1} \sqrt {x + 1} \sqrt {x - 1} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.33, size = 33, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}}{x\,\left (k^2\,x^2-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x^2 - 1)*(k^2*x^2 - 1))^(1/2)/(x*(k^2*x^2 - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k^{2} x^{4} - 2 k^{2} x^{2} + 1}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x - 1\right ) \left (k x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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