3.2.81 \(\int \frac {(d-c^2 d x^2)^3 (a+b \text {ArcSin}(c x))^2}{x^3} \, dx\) [181]

Optimal. Leaf size=371 \[ -\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{x}+\frac {3}{32} c^2 d^3 (a+b \text {ArcSin}(c x))^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{2 x^2}+\frac {i c^2 d^3 (a+b \text {ArcSin}(c x))^3}{b}-3 c^2 d^3 (a+b \text {ArcSin}(c x))^2 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )-\frac {3}{2} b^2 c^2 d^3 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right ) \]

[Out]

-21/32*b^2*c^4*d^3*x^2+1/32*b^2*c^6*d^3*x^4-7/8*b*c^3*d^3*x*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))-b*c*d^3*(-c^2
*x^2+1)^(5/2)*(a+b*arcsin(c*x))/x+3/32*c^2*d^3*(a+b*arcsin(c*x))^2-3/2*c^2*d^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))^
2-3/4*c^2*d^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2-1/2*d^3*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2/x^2+I*c^2*d^3*(a+b
*arcsin(c*x))^3/b-3*c^2*d^3*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+b^2*c^2*d^3*ln(x)+3*I*b*c^2
*d^3*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-3/2*b^2*c^2*d^3*polylog(3,(I*c*x+(-c^2*x^2+1)^(
1/2))^2)+3/16*b*c^3*d^3*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.49, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 15, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4785, 4787, 4721, 3798, 2221, 2611, 2320, 6724, 4741, 4737, 30, 4743, 14, 272, 45} \begin {gather*} 3 i b c^2 d^3 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{2 x^2}+\frac {i c^2 d^3 (a+b \text {ArcSin}(c x))^3}{b}+\frac {3}{32} c^2 d^3 (a+b \text {ArcSin}(c x))^2-3 c^2 d^3 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {21}{32} b^2 c^4 d^3 x^2+b^2 c^2 d^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*b^2*c^4*d^3*x^2)/32 + (b^2*c^6*d^3*x^4)/32 + (3*b*c^3*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/16 - (
7*b*c^3*d^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/8 - (b*c*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x
+ (3*c^2*d^3*(a + b*ArcSin[c*x])^2)/32 - (3*c^2*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/2 - (3*c^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)/(2*x^2) + (I*c^2*d^3*(a + b*
ArcSin[c*x])^3)/b - 3*c^2*d^3*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] + b^2*c^2*d^3*Log[x] + (3*I
)*b*c^2*d^3*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (3*b^2*c^2*d^3*PolyLog[3, E^((2*I)*ArcSin[
c*x])])/2

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 30

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 3798

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 4721

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

Rule 4737

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

Rule 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 4743

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcS
in[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]

Rule 4785

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/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c
^2*x^2)^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 4787

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/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(
1 - c^2*x^2)^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]

Rule 6724

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]

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^3} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d^2\right ) \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b^2 c^2 d^3\right ) \int \frac {\left (1-c^2 x^2\right )^2}{x} \, dx+\frac {1}{2} \left (3 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (5 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\frac {1}{2} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {\left (1-c^2 x\right )^2}{x} \, dx,x,x^2\right )+\frac {1}{8} \left (9 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\left (3 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {1}{4} \left (15 b c^3 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^2 c^4 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac {1}{4} \left (5 b^2 c^4 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx\\ &=\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d^3\right ) \text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{2} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (-2 c^2+\frac {1}{x}+c^4 x\right ) \, dx,x,x^2\right )+\frac {1}{16} \left (9 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{2} \left (3 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{8} \left (15 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{8} \left (3 b^2 c^4 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac {1}{16} \left (9 b^2 c^4 d^3\right ) \int x \, dx+\frac {1}{4} \left (5 b^2 c^4 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac {1}{2} \left (3 b^2 c^4 d^3\right ) \int x \, dx+\frac {1}{8} \left (15 b^2 c^4 d^3\right ) \int x \, dx\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}+b^2 c^2 d^3 \log (x)+\left (6 i c^2 d^3\right ) \text {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 {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+\left (6 b c^2 d^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\left (3 i b^2 c^2 d^3\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 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 (3 b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 556, normalized size = 1.50 \begin {gather*} -\frac {d^3 \left (128 a^2-32 i b^2 c^2 \pi ^3 x^2-384 a^2 c^4 x^4+64 a^2 c^6 x^6+256 a b c x \sqrt {1-c^2 x^2}-336 a b c^3 x^3 \sqrt {1-c^2 x^2}+32 a b c^5 x^5 \sqrt {1-c^2 x^2}+256 a b \text {ArcSin}(c x)-768 a b c^4 x^4 \text {ArcSin}(c x)+128 a b c^6 x^6 \text {ArcSin}(c x)+256 b^2 c x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+128 b^2 \text {ArcSin}(c x)^2-768 i a b c^2 x^2 \text {ArcSin}(c x)^2+256 i b^2 c^2 x^2 \text {ArcSin}(c x)^3+672 a b c^2 x^2 \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )-80 b^2 c^2 x^2 \cos (2 \text {ArcSin}(c x))+160 b^2 c^2 x^2 \text {ArcSin}(c x)^2 \cos (2 \text {ArcSin}(c x))-b^2 c^2 x^2 \cos (4 \text {ArcSin}(c x))+8 b^2 c^2 x^2 \text {ArcSin}(c x)^2 \cos (4 \text {ArcSin}(c x))+768 b^2 c^2 x^2 \text {ArcSin}(c x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c x)}\right )+1536 a b c^2 x^2 \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+768 a^2 c^2 x^2 \log (x)-256 b^2 c^2 x^2 \log (c x)+768 i b^2 c^2 x^2 \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c x)}\right )-768 i a b c^2 x^2 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )+384 b^2 c^2 x^2 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c x)}\right )-160 b^2 c^2 x^2 \text {ArcSin}(c x) \sin (2 \text {ArcSin}(c x))-4 b^2 c^2 x^2 \text {ArcSin}(c x) \sin (4 \text {ArcSin}(c x))\right )}{256 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/256*(d^3*(128*a^2 - (32*I)*b^2*c^2*Pi^3*x^2 - 384*a^2*c^4*x^4 + 64*a^2*c^6*x^6 + 256*a*b*c*x*Sqrt[1 - c^2*x
^2] - 336*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 32*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 256*a*b*ArcSin[c*x] - 768*a*b*c^4
*x^4*ArcSin[c*x] + 128*a*b*c^6*x^6*ArcSin[c*x] + 256*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 128*b^2*ArcSin[c*
x]^2 - (768*I)*a*b*c^2*x^2*ArcSin[c*x]^2 + (256*I)*b^2*c^2*x^2*ArcSin[c*x]^3 + 672*a*b*c^2*x^2*ArcTan[(c*x)/(-
1 + Sqrt[1 - c^2*x^2])] - 80*b^2*c^2*x^2*Cos[2*ArcSin[c*x]] + 160*b^2*c^2*x^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]]
- b^2*c^2*x^2*Cos[4*ArcSin[c*x]] + 8*b^2*c^2*x^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 768*b^2*c^2*x^2*ArcSin[c*
x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 1536*a*b*c^2*x^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 768*a^2*c
^2*x^2*Log[x] - 256*b^2*c^2*x^2*Log[c*x] + (768*I)*b^2*c^2*x^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])]
- (768*I)*a*b*c^2*x^2*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 384*b^2*c^2*x^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] -
160*b^2*c^2*x^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - 4*b^2*c^2*x^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]]))/x^2

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (375 ) = 750\).
time = 0.73, size = 819, normalized size = 2.21

method	result	size
derivativedivides	\(c^{2} \left (-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+d^{3} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-2 d^{3} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+d^{3} b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{256}-3 d^{3} a^{2} \ln \left (c x \right )-3 d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i d^{3} b^{2} \arcsin \left (c x \right )^{3}-\frac {d^{3} a b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {5 d^{3} b^{2}}{16}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {5 d^{3} b^{2} \arcsin \left (c x \right )^{2} c^{2} x^{2}}{4}-\frac {5 b^{2} c^{2} d^{3} x^{2}}{8}-\frac {d^{3} a b \arcsin \left (c x \right )}{c^{2} x^{2}}+\frac {5 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{4}+\frac {5 d^{3} a b \sqrt {-c^{2} x^{2}+1}\, c x}{4}+\frac {5 d^{3} a b \arcsin \left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} a b \sqrt {-c^{2} x^{2}+1}}{c x}-6 d^{3} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 d^{3} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{32}+\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 d^{3} a^{2} c^{2} x^{2}}{2}-\frac {d^{3} a^{2} c^{4} x^{4}}{4}+\frac {d^{3} a b \sin \left (4 \arcsin \left (c x \right )\right )}{64}-6 d^{3} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 i d^{3} a b \arcsin \left (c x \right )^{2}+6 i d^{3} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i d^{3} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 d^{3} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i d^{3} b^{2} \arcsin \left (c x \right )+i d^{3} a b -\frac {5 d^{3} b^{2} \arcsin \left (c x \right )^{2}}{8}-\frac {d^{3} a^{2}}{2 c^{2} x^{2}}-\frac {5 d^{3} a b \arcsin \left (c x \right )}{4}\right )\)	\(819\)
default	\(c^{2} \left (-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+d^{3} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-2 d^{3} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+d^{3} b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{256}-3 d^{3} a^{2} \ln \left (c x \right )-3 d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i d^{3} b^{2} \arcsin \left (c x \right )^{3}-\frac {d^{3} a b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {5 d^{3} b^{2}}{16}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {5 d^{3} b^{2} \arcsin \left (c x \right )^{2} c^{2} x^{2}}{4}-\frac {5 b^{2} c^{2} d^{3} x^{2}}{8}-\frac {d^{3} a b \arcsin \left (c x \right )}{c^{2} x^{2}}+\frac {5 d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{4}+\frac {5 d^{3} a b \sqrt {-c^{2} x^{2}+1}\, c x}{4}+\frac {5 d^{3} a b \arcsin \left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} a b \sqrt {-c^{2} x^{2}+1}}{c x}-6 d^{3} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 d^{3} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{32}+\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 d^{3} a^{2} c^{2} x^{2}}{2}-\frac {d^{3} a^{2} c^{4} x^{4}}{4}+\frac {d^{3} a b \sin \left (4 \arcsin \left (c x \right )\right )}{64}-6 d^{3} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 i d^{3} a b \arcsin \left (c x \right )^{2}+6 i d^{3} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i d^{3} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 d^{3} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i d^{3} b^{2} \arcsin \left (c x \right )+i d^{3} a b -\frac {5 d^{3} b^{2} \arcsin \left (c x \right )^{2}}{8}-\frac {d^{3} a^{2}}{2 c^{2} x^{2}}-\frac {5 d^{3} a b \arcsin \left (c x \right )}{4}\right )\)	\(819\)

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^3,x,method=_RETURNVERBOSE)

[Out]

c^2*(-d^3*b^2*arcsin(c*x)/c/x*(-c^2*x^2+1)^(1/2)-6*d^3*b^2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))-6*d^3*b^2*polyl
og(3,-I*c*x-(-c^2*x^2+1)^(1/2))+1/256*d^3*b^2*cos(4*arcsin(c*x))-3*d^3*a^2*ln(c*x)-6*d^3*a*b*arcsin(c*x)*ln(1-
I*c*x-(-c^2*x^2+1)^(1/2))-6*d^3*a*b*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-1/16*d^3*a*b*arcsin(c*x)*cos(4*
arcsin(c*x))+5/16*d^3*b^2+6*I*d^3*b^2*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+6*I*d^3*b^2*arcsin(c*x)*
polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+3*I*d^3*a*b*arcsin(c*x)^2+6*I*d^3*a*b*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))
+6*I*d^3*a*b*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-1/2*d^3*b^2*arcsin(c*x)^2/c^2/x^2+5/4*d^3*b^2*arcsin(c*x)^2*
c^2*x^2-5/8*b^2*c^2*d^3*x^2-d^3*a*b*arcsin(c*x)/c^2/x^2+5/4*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+5/4*d^3
*a*b*(-c^2*x^2+1)^(1/2)*c*x+5/2*d^3*a*b*arcsin(c*x)*c^2*x^2-d^3*a*b/c/x*(-c^2*x^2+1)^(1/2)-3*d^3*b^2*arcsin(c*
x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-3*d^3*b^2*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-1/32*d^3*b^2*arcsin
(c*x)^2*cos(4*arcsin(c*x))+1/64*d^3*b^2*arcsin(c*x)*sin(4*arcsin(c*x))+3/2*d^3*a^2*c^2*x^2-1/4*d^3*a^2*c^4*x^4
+1/64*d^3*a*b*sin(4*arcsin(c*x))+d^3*b^2*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-2*d^3*b^2*ln(I*c*x+(-c^2*x^2+1)^(1/2))
-5/8*d^3*b^2*arcsin(c*x)^2+d^3*b^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-1/2*d^3*a^2/c^2/x^2+I*d^3*b^2*arcsin(c*x)^3+
I*d^3*b^2*arcsin(c*x)-5/4*d^3*a*b*arcsin(c*x)+I*d^3*a*b)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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^3,x, algorithm="maxima")

[Out]

-1/4*a^2*c^6*d^3*x^4 + 3/2*a^2*c^4*d^3*x^2 - 3*a^2*c^2*d^3*log(x) - a*b*d^3*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c
*x)/x^2) - 1/2*a^2*d^3/x^2 - 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)*ar
ctan2(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)*arcta
n2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/x^3, x)

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d^{3} \left (\int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int \left (- 3 a^{2} c^{4} x\right )\, dx + \int a^{2} c^{6} x^{3}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b^{2} c^{4} x \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{3} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx + \int \left (- 6 a b c^{4} x \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{3} \operatorname {asin}{\left (c x \right )}\, dx\right ) \end {gather*}

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**3,x)

[Out]

-d**3*(Integral(-a**2/x**3, x) + Integral(3*a**2*c**2/x, x) + Integral(-3*a**2*c**4*x, x) + Integral(a**2*c**6
*x**3, x) + Integral(-b**2*asin(c*x)**2/x**3, x) + Integral(-2*a*b*asin(c*x)/x**3, x) + Integral(3*b**2*c**2*a
sin(c*x)**2/x, x) + Integral(-3*b**2*c**4*x*asin(c*x)**2, x) + Integral(b**2*c**6*x**3*asin(c*x)**2, x) + Inte
gral(6*a*b*c**2*asin(c*x)/x, x) + Integral(-6*a*b*c**4*x*asin(c*x), x) + Integral(2*a*b*c**6*x**3*asin(c*x), 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((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2/x^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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