\(\int \frac {(a+b \arcsin (c x))^2}{x^3 (d-c^2 d x^2)^{3/2}} \, dx\) [252]

Optimal result

Integrand size = 29, antiderivative size = 634 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {4789, 4793, 4803, 4268, 2611, 2320, 6724, 4749, 4266, 2317, 2438, 272, 65, 214} \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 i b c^2 \sqrt {1-c^2 x^2} \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

-((b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(d*x*Sqrt[d - c^2*d*x^2])) + (3*c^2*(a + b*ArcSin[c*x])^2)/(2*d*
Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(2*d*x^2*Sqrt[d - c^2*d*x^2]) + ((4*I)*b*c^2*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (3*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*
x^2]])/(d*Sqrt[d - c^2*d*x^2]) + ((3*I)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*
x])])/(d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d*Sqrt[d
- c^2*d*x^2]) + ((2*I)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (
(3*I)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (3*
b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) + (3*b^2*c^2*Sqrt[1 - c^2*x^
2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2])

Rule 65

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 214

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

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 4266

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

Rule 4268

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 4749

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

Rule 4789

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 + 1)*((a + b*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(m
+ 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4793

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 + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(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]) /; Fre
eQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rule 4803

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

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*} \text {integral}& = -\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {1}{2} \left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx}{2 d}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\arcsin (c x)\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 7.96 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a^2}{2 d^2 x^2}-\frac {a^2 c^2}{d^2 \left (-1+c^2 x^2\right )}\right )+\frac {3 a^2 c^2 \log (x)}{2 d^{3/2}}-\frac {3 a^2 c^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{2 d^{3/2}}+\frac {a b c \left (6 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-6 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-\frac {-2 \arcsin (c x)+6 \arcsin (c x) \cos (2 \arcsin (c x))+3 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )-3 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )+2 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-2 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+\sqrt {1-c^2 x^2} \left (-3 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+3 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+2 \sin (2 \arcsin (c x))}{c x}\right )}{4 d x \sqrt {d \left (1-c^2 x^2\right )}}+\frac {b^2 c^2 \sqrt {1-c^2 x^2} \left (8 \arcsin (c x)^2-4 \arcsin (c x) \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x)^2 \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )+8 \log \left (\tan \left (\frac {1}{2} \arcsin (c x)\right )\right )-16 \left (\arcsin (c x) \left (\log \left (1-i e^{i \arcsin (c x)}\right )-\log \left (1+i e^{i \arcsin (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )\right )+12 \left (\arcsin (c x)^2 \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )+2 i \arcsin (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )+2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )+\arcsin (c x)^2 \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )+\frac {8 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )}-\frac {8 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}-4 \arcsin (c x) \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{8 d \sqrt {d \left (1-c^2 x^2\right )}} \]

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a^2/(d^2*x^2) - (a^2*c^2)/(d^2*(-1 + c^2*x^2))) + (3*a^2*c^2*Log[x])/(2*d^(3/2
)) - (3*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(3/2)) + (a*b*c*((6*I)*PolyLog[2, -E^(I*ArcSi
n[c*x])]*Sin[2*ArcSin[c*x]] - (6*I)*PolyLog[2, E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - (-2*ArcSin[c*x] + 6*Arc
Sin[c*x]*Cos[2*ArcSin[c*x]] + 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] - 3*ArcSin[c*x]*Cos[
3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] + 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]
- 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + Sqrt[1 - c^2*x^2]*(-3*ArcSin[c*x]*Log[1
- E^(I*ArcSin[c*x])] + 3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2
]] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*Sin[2*ArcSin[c*x]])/(c*x)))/(4*d*x*Sqrt[d*(1 - c^2*x^
2)]) + (b^2*c^2*Sqrt[1 - c^2*x^2]*(8*ArcSin[c*x]^2 - 4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcS
in[c*x]/2]^2 + 8*Log[Tan[ArcSin[c*x]/2]] - 16*(ArcSin[c*x]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcS
in[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])])) + 12*(ArcSin[c*x]^2*(Lo
g[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + (2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - P
olyLog[2, E^(I*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])])) + ArcSin[
c*x]^2*Sec[ArcSin[c*x]/2]^2 + (8*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) -
(8*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 4*ArcSin[c*x]*Tan[ArcSin[c*x
]/2]))/(8*d*Sqrt[d*(1 - c^2*x^2)])

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.37

method	result	size
default	\(a^{2} \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )-\frac {i a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\)	\(869\)
parts	\(a^{2} \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )-\frac {i a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\)	\(869\)

[In]

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

[Out]

a^2*(-1/2/d/x^2/(-c^2*d*x^2+d)^(1/2)+3/2*c^2*(1/d/(-c^2*d*x^2+d)^(1/2)-1/d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2
+d)^(1/2))/x)))+b^2*(-1/2*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)/x^2*arcsin(c*x)*(3*c^2*x^2*arcsin(c*x)-2*c*x*
(-c^2*x^2+1)^(1/2)-arcsin(c*x))+1/2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/d^2*(3*arcsin(c*x)^2
*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-3*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-6*I*arcsin(c*x)*polylog(2,-I*c*
x-(-c^2*x^2+1)^(1/2))+6*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-4*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^
2+1)^(1/2)))+4*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+4*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-4*I*di
log(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-6*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))+6*p
olylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2)))*c^2)-I*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*
x^2+1)^(1/2)*(3*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^4*c^4+3*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4
+4*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+3*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+3*I*(-c^2*x^2+1)^(1/2)
*arcsin(c*x)*c^2*x^2-3*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^2*c^2+I*x^3*c^3-3*dilog(I*c*x+(-c^2*x^2+
1)^(1/2))*c^2*x^2-4*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-3*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-I*(-c
^2*x^2+1)^(1/2)*arcsin(c*x)-I*c*x)/d^2/(c^4*x^4-2*c^2*x^2+1)/x^2

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2
*x^3), x)

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

-1/2*(3*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(3/2) - 3*c^2/(sqrt(-c^2*d*x^2 + d)*d) +
1/(sqrt(-c^2*d*x^2 + d)*d*x^2))*a^2 + sqrt(d)*integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2
*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d
^2*x^3), x)

Giac [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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