3.224 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^2}{x^3} \, dx\)

Optimal. Leaf size=590 \[ -\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}} \]

[Out]

2*b^2*c^2*d*Sqrt[d - c^2*d*x^2] + (3*a*b*c^3*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (3*b^2*c^3*d*x*Sqrt[
d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(x*Sqrt[1 - c^
2*x^2]) - (b*c^3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] - (3*c^2*d*Sqrt[d - c^2*d*x^2]
*(a + b*ArcSin[c*x])^2)/2 - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^
2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (b^2*c^2*d*Sqrt[d - c^2*d*x^2]*ArcTan
h[Sqrt[1 - c^2*x^2]])/Sqrt[1 - c^2*x^2] - ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E
^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*
ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (3*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c
^2*x^2] - (3*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

________________________________________________________________________________________

Rubi [A]  time = 0.612378, antiderivative size = 590, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 15, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.517, Rules used = {4695, 4697, 4709, 4183, 2531, 2282, 6589, 4619, 261, 14, 4687, 446, 80, 63, 208} \[ -\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^3,x]

[Out]

2*b^2*c^2*d*Sqrt[d - c^2*d*x^2] + (3*a*b*c^3*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (3*b^2*c^3*d*x*Sqrt[
d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(x*Sqrt[1 - c^
2*x^2]) - (b*c^3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] - (3*c^2*d*Sqrt[d - c^2*d*x^2]
*(a + b*ArcSin[c*x])^2)/2 - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^
2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (b^2*c^2*d*Sqrt[d - c^2*d*x^2]*ArcTan
h[Sqrt[1 - c^2*x^2]])/Sqrt[1 - c^2*x^2] - ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E
^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*
ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (3*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c
^2*x^2] - (3*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

Rule 4695

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

Rule 4697

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

Rule 4709

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

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rule 4619

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 446

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

Rule 80

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] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac{1}{2} \left (3 c^2 d\right ) \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac{\left (3 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{-1-c^2 x^2}{x \sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 b c^3 d \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac{\left (3 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{-1-c^2 x}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^3 d \sqrt{d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt{1-c^2 x^2}}\\ &=-b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (3 b c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (3 b c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 \sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 c^4 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (3 i b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (3 i b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}}-\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{\left (3 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}}-\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}-\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 7.05624, size = 854, normalized size = 1.45 \[ -\frac{3}{2} a^2 d^{3/2} \log (x) c^2+\frac{3}{2} a^2 d^{3/2} \log \left (d+\sqrt{-d \left (c^2 x^2-1\right )} \sqrt{d}\right ) c^2-2 a b d \sqrt{d \left (1-c^2 x^2\right )} \left (-\frac{c x}{\sqrt{1-c^2 x^2}}+\sin ^{-1}(c x)+\frac{\sin ^{-1}(c x) \left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt{1-c^2 x^2}}+\frac{i \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt{1-c^2 x^2}}\right ) c^2-b^2 d \sqrt{d \left (1-c^2 x^2\right )} \left (\frac{\left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right ) \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\sin ^{-1}(c x)^2+\frac{2 i \left (\text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )\right ) \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{2 \left (\text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )-\text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )\right )}{\sqrt{1-c^2 x^2}}-2\right ) c^2+\frac{a b d^2 \sqrt{1-c^2 x^2} \left (-\sin ^{-1}(c x) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+\sin ^{-1}(c x) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )-2 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )-4 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+4 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-4 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+4 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) c^2}{4 \sqrt{d \left (1-c^2 x^2\right )}}+\frac{b^2 d^2 \sqrt{1-c^2 x^2} \left (-\csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2+\sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)^2-4 \log \left (1-e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)^2+4 \log \left (1+e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)^2-4 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)-8 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)+8 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)-4 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sin ^{-1}(c x)+8 \log \left (\tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+8 \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )-8 \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )\right ) c^2}{8 \sqrt{d \left (1-c^2 x^2\right )}}+\left (-c^2 d a^2-\frac{d a^2}{2 x^2}\right ) \sqrt{-d \left (c^2 x^2-1\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^3,x]

[Out]

(-(a^2*c^2*d) - (a^2*d)/(2*x^2))*Sqrt[-(d*(-1 + c^2*x^2))] - (3*a^2*c^2*d^(3/2)*Log[x])/2 + (3*a^2*c^2*d^(3/2)
*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 2*a*b*c^2*d*Sqrt[d*(1 - c^2*x^2)]*(-((c*x)/Sqrt[1 - c^2*x^2])
+ ArcSin[c*x] + (ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (
I*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]) - b^2*c^2*d*Sqrt[d*(1 -
c^2*x^2)]*(-2 - (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + ArcSin[c*x]^2 + (ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c
*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + ((2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] -
PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcS
in[c*x])]))/Sqrt[1 - c^2*x^2]) + (a*b*c^2*d^2*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSi
n[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*Poly
Log[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[Ar
cSin[c*x]/2]))/(4*Sqrt[d*(1 - c^2*x^2)]) + (b^2*c^2*d^2*Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] -
ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]^2*Log[1 + E^(
I*ArcSin[c*x])] + 8*Log[Tan[ArcSin[c*x]/2]] - (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (8*I)*ArcSin[
c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 8*PolyLog[3, -E^(I*ArcSin[c*x])] - 8*PolyLog[3, E^(I*ArcSin[c*x])] + ArcS
in[c*x]^2*Sec[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(8*Sqrt[d*(1 - c^2*x^2)])

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Maple [B]  time = 0.398, size = 1372, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3,x)

[Out]

-2*a*b*(-d*(c^2*x^2-1))^(1/2)*c^4*d/(c^2*x^2-1)*arcsin(c*x)*x^2+a*b*d*(-d*(c^2*x^2-1))^(1/2)/x/(c^2*x^2-1)*(-c
^2*x^2+1)^(1/2)*c-2*a*b*(-d*(c^2*x^2-1))^(1/2)*c^3*d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-2*b^2*(-d*(c^2*x^2-1))^(
1/2)*c^3*d/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x+b^2*d*arcsin(c*x)*(-d*(c^2*x^2-1))^(1/2)/x/(c^2*x^2-1)
*(-c^2*x^2+1)^(1/2)*c-3/2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(c^2*x^2-1)*arcsin(c*x)^2*ln(1+I
*c*x+(-c^2*x^2+1)^(1/2))+3/2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(c^2*x^2-1)*arcsin(c*x)^2*ln(
1-I*c*x-(-c^2*x^2+1)^(1/2))-1/2*a^2/d/x^2*(-c^2*d*x^2+d)^(5/2)+3/2*a^2*c^2*d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x
^2+d)^(1/2))/x)-3/2*a^2*c^2*(-c^2*d*x^2+d)^(1/2)*d-2*b^2*(-d*(c^2*x^2-1))^(1/2)*c^2*d/(c^2*x^2-1)+2*b^2*(-d*(c
^2*x^2-1))^(1/2)*c^4*d/(c^2*x^2-1)*x^2+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*c^2*d/(c^2*x^2-1)*arcsin(c*x)^2+1/2*b^2*
d*arcsin(c*x)^2*(-d*(c^2*x^2-1))^(1/2)/x^2/(c^2*x^2-1)-6*I*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d
/(2*c^2*x^2-2)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(c^
2*x^2-1)*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^
2*d/(c^2*x^2-1)*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2
)*c^2*d/(2*c^2*x^2-2)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+6*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/
2)*c^2*d/(2*c^2*x^2-2)*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+6*I*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^
(1/2)*c^2*d/(2*c^2*x^2-2)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-1/2*a^2*c^2*(-c^2*d*x^2+d)^(3/2)+2*b^2*(-d*(c^2
*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(c^2*x^2-1)*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))-3*b^2*(-d*(c^2*x^2-1))^(
1/2)*(-c^2*x^2+1)^(1/2)*c^2*d/(c^2*x^2-1)*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-
c^2*x^2+1)^(1/2)*c^2*d/(c^2*x^2-1)*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))-b^2*(-d*(c^2*x^2-1))^(1/2)*c^4*d/(c^2*x
^2-1)*arcsin(c*x)^2*x^2+a*b*(-d*(c^2*x^2-1))^(1/2)*c^2*d/(c^2*x^2-1)*arcsin(c*x)+a*b*d*arcsin(c*x)*(-d*(c^2*x^
2-1))^(1/2)/x^2/(c^2*x^2-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="fricas")

[Out]

integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*
x))*sqrt(-c^2*d*x^2 + d)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2/x**3,x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^2/x^3, x)