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

Optimal. Leaf size=577 \[ \frac{2 i b \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{7 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{7 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2}{3 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

b^2/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) +
(a + b*ArcSin[c*x])^2/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcSin[c*x])^2/(d^2*Sqrt[d - c^2*d*x^2]) + (((14*I)
/3)*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (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]) + ((2*I)*b*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]) - (((7*I)/3)*b^2*Sqrt[1 - c
^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (((7*I)/3)*b^2*Sqrt[1 - c^2*x^2]*PolyL
og[2, I*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - ((2*I)*b*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]) - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(
d^2*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.860139, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {4705, 4713, 4709, 4183, 2531, 2282, 6589, 4657, 4181, 2279, 2391, 4655, 261} \[ \frac{2 i b \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{7 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{7 i b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (3,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2}{3 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b^2/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) +
(a + b*ArcSin[c*x])^2/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcSin[c*x])^2/(d^2*Sqrt[d - c^2*d*x^2]) + (((14*I)
/3)*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (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]) + ((2*I)*b*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]) - (((7*I)/3)*b^2*Sqrt[1 - c
^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (((7*I)/3)*b^2*Sqrt[1 - c^2*x^2]*PolyL
og[2, I*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - ((2*I)*b*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]) - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(
d^2*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

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]

Rubi steps

\begin{align*} \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 (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{d}-\frac{\left (2 b c \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{b c 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{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{d-c^2 d x^2}} \, dx}{d^2}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b c \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 (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c 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{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt{1-c^2 x^2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \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{b^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c 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{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \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{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^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 )}{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 \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^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 (2 b^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}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c 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{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \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{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{\left (2 b \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 (2 b \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 (i b^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 )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (i b^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 )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^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 (2 i b^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}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c 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{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \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{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{2 i b \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{7 i b^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{7 i b^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{2 i b \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 (2 i b^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 (2 i b^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}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c 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{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \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{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{2 i b \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{7 i b^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{7 i b^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{2 i b \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 (2 b^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 (2 b^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}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c 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{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \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{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{2 i b \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{7 i b^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{7 i b^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{2 i b \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{2 b^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{2 b^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 = 8.64443, size = 935, normalized size = 1.62 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(a^2/(3*d^3*(-1 + c^2*x^2)^2) - a^2/(d^3*(-1 + c^2*x^2))) + (a^2*Log[c*x])/d^(5/2) -
(a^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/d^(5/2) + (b^2*(1 - c^2*x^2)^(3/2)*(4 - ((-2 + ArcSin[c*x])*
ArcSin[c*x])/(-1 + c*x) + 14*ArcSin[c*x]^2 + 12*ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSi
n[c*x])]) - 28*(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])])) + (24*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] -
PolyLog[2, E^(I*ArcSin[c*x])]) + 24*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])]) + (2*Ar
cSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (2*(2 + 7*ArcSin[c*x]^2)*Sin[Arc
Sin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (2*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]
/2] + Sin[ArcSin[c*x]/2])^3 + (ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (2
*(2 + 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(12*d*(d*(1 - c^2*x^2))
^(3/2)) + (a*b*(20*ArcSin[c*x] + 12*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 18*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 -
E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] - 18*Sqrt[1 - c^2*x^2]*ArcSin
[c*x]*Log[1 + E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] + 21*Sqrt[1 - c
^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 7*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSi
n[c*x]/2]] - 21*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 7*Cos[3*ArcSin[c*x]]*Log[Cos[
ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (24*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^(I*ArcSin[c*x])] - (24*I)*(1 -
c^2*x^2)^(3/2)*PolyLog[2, E^(I*ArcSin[c*x])] - 2*Sin[2*ArcSin[c*x]]))/(12*d*(d*(1 - c^2*x^2))^(3/2))

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Maple [B]  time = 0.319, size = 1373, normalized size = 2.4 \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/(-c^2*d*x^2+d)^(5/2),x)

[Out]

1/3*a^2/d/(-c^2*d*x^2+d)^(3/2)+a^2/d^2/(-c^2*d*x^2+d)^(1/2)-a^2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2)
)/x)-7/3*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)*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))
-b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*arcsin(c*x)^2*x^2*c^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)*arcsin(c*x)^2*ln(1+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)*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-7/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2
)/d^3/(c^2*x^2-1)*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+7/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(
1/2)/d^3/(c^2*x^2-1)*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+7/3*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)*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*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)*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+2*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)*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-14/3*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)*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-2*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)*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-2*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)*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*arcsin(c*x)*x^2*c^2-
1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)*x*c+2*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-
1))^(1/2)/d^3/(c^2*x^2-1)*arcsin(c*x)*ln(1+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^2*x
^2-1)^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x*c+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*arcsin(c*x)^2-1/3*
b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*c^2*x^2+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)*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))-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)*po
lylog(3,I*c*x+(-c^2*x^2+1)^(1/2))+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2*arcsin(c*x)+1/3*b^2*(-d*(c^
2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^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/(-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^{7} - 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} - d^{3} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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