3.262 \(\int \frac{(a+b \sin ^{-1}(c x))^2}{x^3 (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=752 \[ \frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

(b^2*c^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(d^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]
) + (2*b*c^3*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcSin[c*x])
^2)/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcSin[c*x])^2/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcSin
[c*x])^2)/(2*d^2*Sqrt[d - c^2*d*x^2]) + (((26*I)/3)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*Ar
cSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*
x])])/(d^2*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^2*Sqrt[d - c^2*d*x
^2]) + ((5*I)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*
x^2]) - (((13*I)/3)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) +
(((13*I)/3)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - ((5*I)*b*c^
2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*b^2*c^2*
Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (5*b^2*c^2*Sqrt[1 - c^2*x^2]*Pol
yLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.25643, antiderivative size = 752, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 18, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.621, Rules used = {4701, 4705, 4713, 4709, 4183, 2531, 2282, 6589, 4657, 4181, 2279, 2391, 4655, 261, 266, 51, 63, 208} \[ \frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(b^2*c^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(d^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]
) + (2*b*c^3*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcSin[c*x])
^2)/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcSin[c*x])^2/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcSin
[c*x])^2)/(2*d^2*Sqrt[d - c^2*d*x^2]) + (((26*I)/3)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*Ar
cSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*
x])])/(d^2*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^2*Sqrt[d - c^2*d*x
^2]) + ((5*I)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*
x^2]) - (((13*I)/3)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) +
(((13*I)/3)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - ((5*I)*b*c^
2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*b^2*c^2*
Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (5*b^2*c^2*Sqrt[1 - c^2*x^2]*Pol
yLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

Rule 4701

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*d^IntPart[p]*(d + e*x^2)^F
racPart[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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && Inte
gerQ[m]

Rule 4705

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*d^IntPart[p]*(d + e*x^2)^Frac
Part[p])/(2*f*(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] && LtQ[p, -1] &&  !GtQ
[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rule 4713

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[
Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2], Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &&  !GtQ[d, 0] && (IntegerQ[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 4657

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 4181

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 2279

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 2391

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 4655

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcSin[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(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] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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 266

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{1}{2} \left (5 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (5 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{2 d}+\frac{\left (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{d-c^2 d x^2}} \, dx}{2 d^2}+\frac{\left (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^4 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 c^4 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{1-c^2 x^2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 \sqrt{1-c^2 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 )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 10.64, size = 1090, normalized size = 1.45 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(-a^2/(2*d^3*x^2) + (a^2*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a^2*c^2)/(d^3*(-1 + c^2*
x^2))) + (5*a^2*c^2*Log[x])/(2*d^(5/2)) - (5*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) +
(a*b*c^2*Sqrt[1 - c^2*x^2]*((-2*(-1 + ArcSin[c*x]))/(-1 + c*x) + 52*ArcSin[c*x] - 6*Cot[ArcSin[c*x]/2] - 3*Ar
cSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 60*ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + 52
*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (60*I)*(Poly
Log[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c
*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos
[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*
x]/2])^3 + (2*(1 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (52*ArcSin[c*x]*Sin[ArcSin[c*x]
/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 6*Tan[ArcSin[c*x]/2]))/(12*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b^2*
c^2*Sqrt[1 - c^2*x^2]*(8 - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) + 52*ArcSin[c*x]^2 - 12*ArcSin[c*x]*C
ot[ArcSin[c*x]/2] - 3*ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 + 24*Log[Tan[ArcSin[c*x]/2]] - 104*(ArcSin[c*x]*(Log[
1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - PolyLog[2,
I*E^(I*ArcSin[c*x])])) + 60*(ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + (2*I)*A
rcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcSin[c*x]
)] + PolyLog[3, E^(I*ArcSin[c*x])])) + 3*ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]
/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (4*(2 + 13*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c
*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^
3 + (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (4*(2 + 13*ArcSin[c*x]^2)*
Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 12*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(24*d^2*Sq
rt[d*(1 - c^2*x^2)])

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Maple [B]  time = 0.468, size = 1876, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-5/2*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^2*arcsin(c*x)^2*c^4+b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^
2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+5*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(
1/2)/d^3/(c^2*x^2-1)*c^2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))-5*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/
d^3/(c^2*x^2-1)*c^2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))-b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2
*x^2-1)*c^2*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+20/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*
x)*c^2-a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*arcsin(c*x)+5*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*
x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-26/3*I*a*b*(-c^2*x^2+1)^(1/2)*(-d
*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-5*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x
^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-5*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(
1/2)/d^3/(c^2*x^2-1)*c^2*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-5*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d
^3/(c^2*x^2-1)*c^2*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+5*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1)
)^(1/2)/d^3/(c^2*x^2-1)*c^2*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3
/(c^4*x^4-2*c^2*x^2+1)*x^2*c^4+10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2*c^2-1/2
*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*arcsin(c*x)^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^
4*x^4-2*c^2*x^2+1)*c^2+5/2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*arcsin(c*x)^2*ln(
1+I*c*x+(-c^2*x^2+1)^(1/2))-5/2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*arcsin(c*x)^
2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-13/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^2*arcsin
(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+13/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c^
2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+13/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2
*x^2-1)*c^2*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-13/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^
2*x^2-1)*c^2*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-5*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^2*
arcsin(c*x)*c^4+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x*(-c^2*x^2+1)^(1/2)*c^3-a*b*(-d*(c^2
*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x*(-c^2*x^2+1)^(1/2)*c+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*
c^2*x^2+1)*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3-b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x*arcsin(
c*x)*(-c^2*x^2+1)^(1/2)*c+5/6*a^2*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/2*a^2*c^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a^2*c^2/
d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-1/2*a^2/d/x^2/(-c^2*d*x^2+d)^(3/2)

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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((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),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{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),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^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*
c^2*d^3*x^5 - d^3*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**2/x**3/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2/((-c^2*d*x^2 + d)^(5/2)*x^3), x)