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

Optimal. Leaf size=354 \[ -i b d^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac{19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{7}{144} b^2 c^4 d^3 x^4+\frac{71}{144} b^2 c^2 d^3 x^2-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3 \]

[Out]

(71*b^2*c^2*d^3*x^2)/144 - (7*b^2*c^4*d^3*x^4)/144 - (b^2*d^3*(1 - c^2*x^2)^3)/108 - (19*b*c*d^3*x*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/24 - (7*b*c*d^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/36 - (b*c*d^3*x*(1 - c^
2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/18 - (19*d^3*(a + b*ArcSin[c*x])^2)/48 + (d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*
x])^2)/2 + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/4 + (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/6 - ((I
/3)*d^3*(a + b*ArcSin[c*x])^3)/b + d^3*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*d^3*(a + b*A
rcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] + (b^2*d^3*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2

________________________________________________________________________________________

Rubi [A]  time = 0.658373, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {4699, 4625, 3717, 2190, 2531, 2282, 6589, 4647, 4641, 30, 4649, 14, 261} \[ -i b d^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac{19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{7}{144} b^2 c^4 d^3 x^4+\frac{71}{144} b^2 c^2 d^3 x^2-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(71*b^2*c^2*d^3*x^2)/144 - (7*b^2*c^4*d^3*x^4)/144 - (b^2*d^3*(1 - c^2*x^2)^3)/108 - (19*b*c*d^3*x*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x]))/24 - (7*b*c*d^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/36 - (b*c*d^3*x*(1 - c^
2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/18 - (19*d^3*(a + b*ArcSin[c*x])^2)/48 + (d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*
x])^2)/2 + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/4 + (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/6 - ((I
/3)*d^3*(a + b*ArcSin[c*x])^3)/b + d^3*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*d^3*(a + b*A
rcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] + (b^2*d^3*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2

Rule 4699

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 + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(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 + 2*p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, 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 4647

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

Rule 4641

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4649

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

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 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]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac{1}{3} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac{1}{18} \left (5 b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac{1}{2} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{18} \left (b^2 c^2 d^3\right ) \int x \left (1-c^2 x^2\right )^2 \, dx\\ &=-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac{1}{24} \left (5 b c d^3\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac{1}{8} \left (3 b c d^3\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (b c d^3\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{72} \left (5 b^2 c^2 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac{1}{8} \left (b^2 c^2 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx\\ &=-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \operatorname{Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{48} \left (5 b c d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{16} \left (3 b c d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{2} \left (b c d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{72} \left (5 b^2 c^2 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx+\frac{1}{48} \left (5 b^2 c^2 d^3\right ) \int x \, dx+\frac{1}{8} \left (b^2 c^2 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx+\frac{1}{16} \left (3 b^2 c^2 d^3\right ) \int x \, dx+\frac{1}{2} \left (b^2 c^2 d^3\right ) \int x \, dx\\ &=\frac{71}{144} b^2 c^2 d^3 x^2-\frac{7}{144} b^2 c^4 d^3 x^4-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\left (2 i d^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2-\frac{7}{144} b^2 c^4 d^3 x^4-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (2 b d^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2-\frac{7}{144} b^2 c^4 d^3 x^4-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (i b^2 d^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2-\frac{7}{144} b^2 c^4 d^3 x^4-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2-\frac{7}{144} b^2 c^4 d^3 x^4-\frac{1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} b^2 d^3 \text{Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.82619, size = 448, normalized size = 1.27 \[ \frac{d^3 \left (-3456 i a b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+3456 i b^2 \sin ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(c x)}\right )+1728 b^2 \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(c x)}\right )-576 a^2 c^6 x^6+2592 a^2 c^4 x^4-5184 a^2 c^2 x^2+3456 a^2 \log (c x)-192 a b c^5 x^5 \sqrt{1-c^2 x^2}+1056 a b c^3 x^3 \sqrt{1-c^2 x^2}-3600 a b c x \sqrt{1-c^2 x^2}-1152 a b c^6 x^6 \sin ^{-1}(c x)+5184 a b c^4 x^4 \sin ^{-1}(c x)-10368 a b c^2 x^2 \sin ^{-1}(c x)-3456 i a b \sin ^{-1}(c x)^2+3600 a b \sin ^{-1}(c x)+6912 a b \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+1152 i b^2 \sin ^{-1}(c x)^3-1566 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )-108 b^2 \sin ^{-1}(c x) \sin \left (4 \sin ^{-1}(c x)\right )-6 b^2 \sin ^{-1}(c x) \sin \left (6 \sin ^{-1}(c x)\right )+3456 b^2 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )-783 b^2 \cos \left (2 \sin ^{-1}(c x)\right )+1566 b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )-27 b^2 \cos \left (4 \sin ^{-1}(c x)\right )+216 b^2 \sin ^{-1}(c x)^2 \cos \left (4 \sin ^{-1}(c x)\right )-b^2 \cos \left (6 \sin ^{-1}(c x)\right )+18 b^2 \sin ^{-1}(c x)^2 \cos \left (6 \sin ^{-1}(c x)\right )-144 i \pi ^3 b^2\right )}{3456} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^3*((-144*I)*b^2*Pi^3 - 5184*a^2*c^2*x^2 + 2592*a^2*c^4*x^4 - 576*a^2*c^6*x^6 - 3600*a*b*c*x*Sqrt[1 - c^2*x^
2] + 1056*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 192*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 3600*a*b*ArcSin[c*x] - 10368*a*b
*c^2*x^2*ArcSin[c*x] + 5184*a*b*c^4*x^4*ArcSin[c*x] - 1152*a*b*c^6*x^6*ArcSin[c*x] - (3456*I)*a*b*ArcSin[c*x]^
2 + (1152*I)*b^2*ArcSin[c*x]^3 - 783*b^2*Cos[2*ArcSin[c*x]] + 1566*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 27*b
^2*Cos[4*ArcSin[c*x]] + 216*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] - b^2*Cos[6*ArcSin[c*x]] + 18*b^2*ArcSin[c*x]
^2*Cos[6*ArcSin[c*x]] + 3456*b^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 6912*a*b*ArcSin[c*x]*Log[1 -
E^((2*I)*ArcSin[c*x])] + 3456*a^2*Log[c*x] + (3456*I)*b^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (34
56*I)*a*b*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 1728*b^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] - 1566*b^2*ArcSin[c*
x]*Sin[2*ArcSin[c*x]] - 108*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]] - 6*b^2*ArcSin[c*x]*Sin[6*ArcSin[c*x]]))/3456

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Maple [B]  time = 0.382, size = 743, normalized size = 2.1 \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*(a+b*arcsin(c*x))^2/x,x)

[Out]

-811/3456*d^3*b^2+25/48*b^2*c^2*d^3*x^2-11/144*b^2*c^4*d^3*x^4-I*d^3*a*b*arcsin(c*x)^2-1/6*d^3*b^2*arcsin(c*x)
^2*c^6*x^6+3/4*d^3*b^2*arcsin(c*x)^2*c^4*x^4-3/2*d^3*b^2*arcsin(c*x)^2*c^2*x^2-2*I*d^3*b^2*arcsin(c*x)*polylog
(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*d^3*b^2*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+2*d^3*a*b*arcsin(c*x
)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*d^3*a*b*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*d^3*a*b*polylog(2,I*
c*x+(-c^2*x^2+1)^(1/2))-2*I*d^3*a*b*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-25/24*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1
)^(1/2)*c*x-1/18*d^3*a*b*(-c^2*x^2+1)^(1/2)*c^5*x^5+11/36*d^3*a*b*(-c^2*x^2+1)^(1/2)*c^3*x^3-25/24*d^3*a*b*(-c
^2*x^2+1)^(1/2)*c*x-1/3*d^3*a*b*arcsin(c*x)*c^6*x^6+3/2*d^3*a*b*arcsin(c*x)*c^4*x^4-3*d^3*a*b*arcsin(c*x)*c^2*
x^2-1/18*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5*x^5+11/36*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^3+d
^3*a^2*ln(c*x)+25/48*d^3*b^2*arcsin(c*x)^2+2*d^3*b^2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+2*d^3*b^2*polylog(3,
I*c*x+(-c^2*x^2+1)^(1/2))+25/24*d^3*a*b*arcsin(c*x)+d^3*b^2*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+d^3*b
^2*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-1/3*I*d^3*b^2*arcsin(c*x)^3-1/6*d^3*a^2*c^6*x^6+3/4*d^3*a^2*c^
4*x^4-3/2*d^3*a^2*c^2*x^2+1/108*d^3*b^2*c^6*x^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{6} \, a^{2} c^{6} d^{3} x^{6} + \frac{3}{4} \, a^{2} c^{4} d^{3} x^{4} - \frac{3}{2} \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} \log \left (x\right ) - \int \frac{{\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a^2*c^6*d^3*x^6 + 3/4*a^2*c^4*d^3*x^4 - 3/2*a^2*c^2*d^3*x^2 + a^2*d^3*log(x) - integrate(((b^2*c^6*d^3*x^
6 - 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 - b^2*d^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^6
*d^3*x^6 - 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^2 - a*b*d^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a^{2} c^{6} d^{3} x^{6} - 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} - a^{2} d^{3} +{\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arcsin \left (c x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c^6*d^3*x^6 - 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 - a^2*d^3 + (b^2*c^6*d^3*x^6 - 3*b^2*c^4*d^
3*x^4 + 3*b^2*c^2*d^3*x^2 - b^2*d^3)*arcsin(c*x)^2 + 2*(a*b*c^6*d^3*x^6 - 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^
2 - a*b*d^3)*arcsin(c*x))/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - d^{3} \left (\int - \frac{a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int - 3 a^{2} c^{4} x^{3}\, dx + \int a^{2} c^{6} x^{5}\, dx + \int - \frac{b^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x}\, dx + \int - \frac{2 a b \operatorname{asin}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int - 3 b^{2} c^{4} x^{3} \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{5} \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname{asin}{\left (c x \right )}\, dx + \int - 6 a b c^{4} x^{3} \operatorname{asin}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{5} \operatorname{asin}{\left (c x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d**3*(Integral(-a**2/x, x) + Integral(3*a**2*c**2*x, x) + Integral(-3*a**2*c**4*x**3, x) + Integral(a**2*c**6
*x**5, x) + Integral(-b**2*asin(c*x)**2/x, x) + Integral(-2*a*b*asin(c*x)/x, x) + Integral(3*b**2*c**2*x*asin(
c*x)**2, x) + Integral(-3*b**2*c**4*x**3*asin(c*x)**2, x) + Integral(b**2*c**6*x**5*asin(c*x)**2, x) + Integra
l(6*a*b*c**2*x*asin(c*x), x) + Integral(-6*a*b*c**4*x**3*asin(c*x), x) + Integral(2*a*b*c**6*x**5*asin(c*x), 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 )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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