3.6.22 \(\int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {(1-x^2) (1-k^2 x^2)} (-1+k^2 x^2)} \, dx\) [522]

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

[Out]

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

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Rubi [A]
time = 0.33, antiderivative size = 63, normalized size of antiderivative = 1.58, number of steps used = 4, number of rules used = 4, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1976, 1697, 1704, 1604} \begin {gather*} \frac {k^2 x \left (1-x^2\right )}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}{x} \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*x*(1 - x^2))/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4] + Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]/x

Rule 1604

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x,
r])), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rule 1697

Int[((Px_)*(x_)^(m_))/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A =
Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, Simp[A*x^(m + 1)*(Sqrt[a + b*x^2 + c*x^4]/(a*d*(m
+ 1))), x] + Dist[1/(a*d*(m + 1)), Int[(x^(m + 2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]))*Simp[a*B*d*(m + 1) -
A*(a*e*(m + 1) + b*d*(m + 2)) + (a*C*d*(m + 1) - A*(b*e*(m + 2) + c*d*(m + 3)))*x^2 - A*c*e*(m + 3)*x^4, x],
x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && ILtQ[m/2, 0]

Rule 1704

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2
+ c*x^4)^FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + c*(x^2/e))^FracPart[p]), Int[Px*(d + e*x^2)^(p + q)*(a/d
+ (c/e)*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] &&  !IntegerQ[p] && (PolyQ[Px, x^2] || MatchQ[Px, ((f_) + (g_.)*x^2)^(r_.) /; FreeQ[{f, g, r}, 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]

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 = 7.41, size = 30, normalized size = 0.75 \begin {gather*} \frac {1-x^2}{x \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\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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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 2.
time = 0.39, size = 551, normalized size = 13.78

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	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\left (-\frac {k^{2}}{2}+\frac {1}{2}\right ) \left (\frac {k^{2} x^{3}-k^{2} x +k \,x^{2}-k}{\left (k^{2}-1\right ) \sqrt {\left (x -\frac {1}{k}\right ) \left (k^{2} x^{3}-k^{2} x +k \,x^{2}-k \right )}}+\frac {\left (-\frac {k^{2}-2}{2 \left (k^{2}-1\right )}+\frac {k^{2}}{2 k^{2}-2}\right ) \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\EllipticF \left (x , k\right )-\EllipticE \left (x , k\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )+\left (\frac {k^{2}}{2}-\frac {1}{2}\right ) \left (-\frac {k^{2} x^{3}-k^{2} x -k \,x^{2}+k}{\left (k^{2}-1\right ) \sqrt {\left (x +\frac {1}{k}\right ) \left (k^{2} x^{3}-k^{2} x -k \,x^{2}+k \right )}}+\frac {\left (\frac {k^{2}-2}{2 k^{2}-2}-\frac {k^{2}}{2 \left (k^{2}-1\right )}\right ) \sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \EllipticF \left (x , k\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\EllipticF \left (x , k\right )-\EllipticE \left (x , k\right )\right )}{\left (k^{2}-1\right ) \sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\right )+\frac {\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}{x}-\frac {\sqrt {-x^{2}+1}\, \sqrt {-k^{2} x^{2}+1}\, \left (\EllipticF \left (x , k\right )-\EllipticE \left (x , k\right )\right )}{\sqrt {k^{2} x^{4}-k^{2} x^{2}-x^{2}+1}}\)	\(551\)

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]

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

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Maxima [A]
time = 0.34, 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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Fricas [A]
time = 0.37, 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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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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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^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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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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