3.226 \(\int \frac{x^3 \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=177 \[ -\frac{2 x^2 \sqrt{1-a x} \sqrt{a x+1}}{27 a^2}-\frac{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^4}-\frac{40 \sqrt{1-a x} \sqrt{a x+1}}{27 a^4}-\frac{4 x \sqrt{a x-1} \cosh ^{-1}(a x)}{3 a^3 \sqrt{1-a x}}-\frac{2 x^3 \sqrt{a x-1} \cosh ^{-1}(a x)}{9 a \sqrt{1-a x}} \]

[Out]

(-40*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(27*a^4) - (2*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(27*a^2) - (4*x*Sqrt[-1 + a*x
]*ArcCosh[a*x])/(3*a^3*Sqrt[1 - a*x]) - (2*x^3*Sqrt[-1 + a*x]*ArcCosh[a*x])/(9*a*Sqrt[1 - a*x]) - (2*Sqrt[1 -
a^2*x^2]*ArcCosh[a*x]^2)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(3*a^2)

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Rubi [A]  time = 0.591863, antiderivative size = 237, normalized size of antiderivative = 1.34, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 5759, 5718, 5654, 74, 5662, 100, 12} \[ -\frac{2 x^2 (1-a x) (a x+1)}{27 a^2 \sqrt{1-a^2 x^2}}-\frac{40 (1-a x) (a x+1)}{27 a^4 \sqrt{1-a^2 x^2}}-\frac{2 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 a^3 \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

(-40*(1 - a*x)*(1 + a*x))/(27*a^4*Sqrt[1 - a^2*x^2]) - (2*x^2*(1 - a*x)*(1 + a*x))/(27*a^2*Sqrt[1 - a^2*x^2])
- (4*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*a^3*Sqrt[1 - a^2*x^2]) - (2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a
*x]*ArcCosh[a*x])/(9*a*Sqrt[1 - a^2*x^2]) - (2*(1 - a*x)*(1 + a*x)*ArcCosh[a*x]^2)/(3*a^4*Sqrt[1 - a^2*x^2]) -
(x^2*(1 - a*x)*(1 + a*x)*ArcCosh[a*x]^2)/(3*a^2*Sqrt[1 - a^2*x^2])

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 100

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int \frac{x^3 \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^3 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int x^2 \cosh ^{-1}(a x) \, dx}{3 a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 \sqrt{1-a^2 x^2}}-\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \cosh ^{-1}(a x) \, dx}{3 a^3 \sqrt{1-a^2 x^2}}\\ &=-\frac{2 x^2 (1-a x) (1+a x)}{27 a^2 \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 a^3 \sqrt{1-a^2 x^2}}-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{2 x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{3 a^2 \sqrt{1-a^2 x^2}}\\ &=-\frac{4 (1-a x) (1+a x)}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (1+a x)}{27 a^2 \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 a^3 \sqrt{1-a^2 x^2}}-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a^2 \sqrt{1-a^2 x^2}}\\ &=-\frac{40 (1-a x) (1+a x)}{27 a^4 \sqrt{1-a^2 x^2}}-\frac{2 x^2 (1-a x) (1+a x)}{27 a^2 \sqrt{1-a^2 x^2}}-\frac{4 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 a^3 \sqrt{1-a^2 x^2}}-\frac{2 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a \sqrt{1-a^2 x^2}}-\frac{2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^4 \sqrt{1-a^2 x^2}}-\frac{x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{3 a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.150591, size = 123, normalized size = 0.69 \[ \left (-\frac{2 x^2}{27 a^2}-\frac{40}{27 a^4}\right ) \sqrt{1-a^2 x^2}-\frac{\sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^2}{3 a^4}+\frac{2 x \sqrt{1-a^2 x^2} \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)}{9 a^3 \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

(-40/(27*a^4) - (2*x^2)/(27*a^2))*Sqrt[1 - a^2*x^2] + (2*x*Sqrt[1 - a^2*x^2]*(6 + a^2*x^2)*ArcCosh[a*x])/(9*a^
3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCosh[a*x]^2)/(3*a^4)

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Maple [B]  time = 0.192, size = 343, normalized size = 1.9 \begin{align*} -{\frac{9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}-6\,{\rm arccosh} \left (ax\right )+2}{216\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}+4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}-3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}-6\,{\rm arccosh} \left (ax\right )+6}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( \sqrt{ax+1}\sqrt{ax-1}ax+{a}^{2}{x}^{2}-1 \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )+6}{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }-{\frac{9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )+2}{216\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,{x}^{4}{a}^{4}-5\,{a}^{2}{x}^{2}-4\,{a}^{3}{x}^{3}\sqrt{ax-1}\sqrt{ax+1}+3\,\sqrt{ax+1}\sqrt{ax-1}ax+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216*(-a^2*x^2+1)^(1/2)*(4*x^4*a^4-5*a^2*x^2+4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-3*(a*x+1)^(1/2)*(a*x-1)^(
1/2)*a*x+1)*(9*arccosh(a*x)^2-6*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*((a*x+1)^(1/2)*(a*x-1)^
(1/2)*a*x+a^2*x^2-1)*(arccosh(a*x)^2-2*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-3/8*(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x+1)
^(1/2)*(a*x-1)^(1/2)*a*x-1)*(arccosh(a*x)^2+2*arccosh(a*x)+2)/a^4/(a^2*x^2-1)-1/216*(-a^2*x^2+1)^(1/2)*(4*x^4*
a^4-5*a^2*x^2-4*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)+3*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x+1)*(9*arccosh(a*x)^2+6*a
rccosh(a*x)+2)/a^4/(a^2*x^2-1)

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Maxima [C]  time = 1.74646, size = 142, normalized size = 0.8 \begin{align*} -\frac{1}{3} \,{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname{arcosh}\left (a x\right )^{2} + \frac{2 \,{\left (-i \, \sqrt{a^{2} x^{2} - 1} x^{2} - \frac{20 i \, \sqrt{a^{2} x^{2} - 1}}{a^{2}}\right )}}{27 \, a^{2}} + \frac{2 \,{\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname{arcosh}\left (a x\right )}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccosh(a*x)^2 + 2/27*(-I*sqrt(a^2*x^2 - 1)*x^2 -
20*I*sqrt(a^2*x^2 - 1)/a^2)/a^2 + 2/9*(I*a^2*x^3 + 6*I*x)*arccosh(a*x)/a^3

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Fricas [A]  time = 2.20258, size = 324, normalized size = 1.83 \begin{align*} -\frac{9 \,{\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + 2 \,{\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt{-a^{2} x^{2} + 1}}{27 \,{\left (a^{6} x^{2} - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(9*(a^4*x^4 + a^2*x^2 - 2)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 6*(a^3*x^3 + 6*a*x)*sqrt(
a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1)) + 2*(a^4*x^4 + 19*a^2*x^2 - 20)*sqrt(-a^2*x^2 + 1
))/(a^6*x^2 - a^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{acosh}^{2}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*acosh(a*x)**2/sqrt(-(a*x - 1)*(a*x + 1)), x)

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Giac [C]  time = 1.19231, size = 161, normalized size = 0.91 \begin{align*} \frac{{\left ({\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{-a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{3 \, a^{4}} + \frac{3 \,{\left (-2 i \, a^{2} x^{3} - 12 i \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{-2 i \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 42 i \, \sqrt{a^{2} x^{2} - 1}}{a}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 - 1))^2/a^4 + 1/27*(3*(-2*I*a^2*x^3 -
12*I*x)*log(a*x + sqrt(a^2*x^2 - 1)) - (-2*I*(a^2*x^2 - 1)^(3/2) - 42*I*sqrt(a^2*x^2 - 1))/a)/a^3