3.88 \(\int x^2 \cos ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=178 \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^3}+\frac{5 \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{144 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)} \]

[Out]

(-5*x*Sqrt[ArcCos[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcCos[a*x]])/36 - (5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(9*a
^3) - (5*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(18*a) + (x^3*ArcCos[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelC
[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^3) + (5*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(144*a^3)

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Rubi [A]  time = 0.456678, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4630, 4708, 4678, 4620, 4724, 3304, 3352, 3312} \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^3}+\frac{5 \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{144 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCos[a*x]^(5/2),x]

[Out]

(-5*x*Sqrt[ArcCos[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcCos[a*x]])/36 - (5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(9*a
^3) - (5*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(18*a) + (x^3*ArcCos[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelC
[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^3) + (5*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(144*a^3)

Rule 4630

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCos[c*x])^n)/(m
+ 1), x] + Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4708

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

Rule 4678

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

Rule 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
(x*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int x^2 \cos ^{-1}(a x)^{5/2} \, dx &=\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac{1}{6} (5 a) \int \frac{x^3 \cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac{5}{12} \int x^2 \sqrt{\cos ^{-1}(a x)} \, dx+\frac{5 \int \frac{x \cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac{5 \int \sqrt{\cos ^{-1}(a x)} \, dx}{6 a^2}-\frac{1}{72} (5 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx\\ &=-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{72 a^3}-\frac{5 \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx}{12 a}\\ &=-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac{5 \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 \sqrt{x}}+\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{72 a^3}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{12 a^3}\\ &=-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{288 a^3}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{96 a^3}+\frac{5 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{6 a^3}\\ &=-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac{5 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{6 a^3}+\frac{5 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{144 a^3}+\frac{5 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{48 a^3}\\ &=-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{16 a^3}+\frac{5 \sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{144 a^3}\\ \end{align*}

Mathematica [C]  time = 0.118874, size = 122, normalized size = 0.69 \[ -\frac{81 i \sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{7}{2},-i \cos ^{-1}(a x)\right )+81 \cos ^{-1}(a x) \text{Gamma}\left (\frac{7}{2},i \cos ^{-1}(a x)\right )+\sqrt{3} \left (i \sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{7}{2},-3 i \cos ^{-1}(a x)\right )+\cos ^{-1}(a x) \text{Gamma}\left (\frac{7}{2},3 i \cos ^{-1}(a x)\right )\right )}{648 a^3 \sqrt{i \cos ^{-1}(a x)} \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*ArcCos[a*x]^(5/2),x]

[Out]

-((81*I)*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-I)*ArcCos[a*x]] + 81*ArcCos[a*x]*Gamma[7/2, I*ArcCos[a*x]] + Sqrt[3]
*(I*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-3*I)*ArcCos[a*x]] + ArcCos[a*x]*Gamma[7/2, (3*I)*ArcCos[a*x]]))/(648*a^3*
Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]])

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Maple [A]  time = 0.09, size = 156, normalized size = 0.9 \begin{align*}{\frac{1}{864\,{a}^{3}} \left ( 216\,ax \left ( \arccos \left ( ax \right ) \right ) ^{3}+72\, \left ( \arccos \left ( ax \right ) \right ) ^{3}\cos \left ( 3\,\arccos \left ( ax \right ) \right ) +5\,\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -540\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-60\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sin \left ( 3\,\arccos \left ( ax \right ) \right ) +405\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -810\,ax\arccos \left ( ax \right ) -30\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccos(a*x)^(5/2),x)

[Out]

1/864/a^3*(216*a*x*arccos(a*x)^3+72*arccos(a*x)^3*cos(3*arccos(a*x))+5*3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1
/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))-540*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)-60*arccos(a*x)^2
*sin(3*arccos(a*x))+405*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))-810*a*
x*arccos(a*x)-30*arccos(a*x)*cos(3*arccos(a*x)))/arccos(a*x)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acos(a*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.31501, size = 491, normalized size = 2.76 \begin{align*} \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{144 \, a^{3}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{16 \, a^{3}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{8 \, a^{3}} - \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{16 \, a^{3}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{8 \, a^{3}} - \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{144 \, a^{3}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} - \frac{5 \, \sqrt{6} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{1728 \, a^{3}{\left (i - 1\right )}} - \frac{15 \, \sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{64 \, a^{3}{\left (i - 1\right )}} - \frac{5 \, \sqrt{\arccos \left (a x\right )} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{288 \, a^{3}} - \frac{15 \, \sqrt{\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{32 \, a^{3}} - \frac{15 \, \sqrt{\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{32 \, a^{3}} - \frac{5 \, \sqrt{\arccos \left (a x\right )} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{288 \, a^{3}} + \frac{5 \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{6} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{1728 \, a^{3}{\left (i - 1\right )}} + \frac{15 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{64 \, a^{3}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(5/2),x, algorithm="giac")

[Out]

5/144*i*arccos(a*x)^(3/2)*e^(3*i*arccos(a*x))/a^3 + 1/24*arccos(a*x)^(5/2)*e^(3*i*arccos(a*x))/a^3 + 5/16*i*ar
ccos(a*x)^(3/2)*e^(i*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(5/2)*e^(i*arccos(a*x))/a^3 - 5/16*i*arccos(a*x)^(3/2)
*e^(-i*arccos(a*x))/a^3 + 1/8*arccos(a*x)^(5/2)*e^(-i*arccos(a*x))/a^3 - 5/144*i*arccos(a*x)^(3/2)*e^(-3*i*arc
cos(a*x))/a^3 + 1/24*arccos(a*x)^(5/2)*e^(-3*i*arccos(a*x))/a^3 - 5/1728*sqrt(6)*sqrt(pi)*i*erf(sqrt(6)*sqrt(a
rccos(a*x))/(i - 1))/(a^3*(i - 1)) - 15/64*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*sqrt(arccos(a*x))/(i - 1))/(a^3*(i -
1)) - 5/288*sqrt(arccos(a*x))*e^(3*i*arccos(a*x))/a^3 - 15/32*sqrt(arccos(a*x))*e^(i*arccos(a*x))/a^3 - 15/32
*sqrt(arccos(a*x))*e^(-i*arccos(a*x))/a^3 - 5/288*sqrt(arccos(a*x))*e^(-3*i*arccos(a*x))/a^3 + 5/1728*sqrt(6)*
sqrt(pi)*erf(-sqrt(6)*i*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) + 15/64*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt(
arccos(a*x))/(i - 1))/(a^3*(i - 1))