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

Optimal. Leaf size=573 \[ -\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{e \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}-\frac{b c \sqrt{1-c^2 x^2}}{2 d x} \]

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(2*d*x) - (a + b*ArcSin[c*x])/(2*d*x^2) + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(
I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcS
in[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x
]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I
*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) - (e*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/d^2 - ((I/2)
*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/d^2 - ((I/2)*b*e*PolyLog[2,
(Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^2 - ((I/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*Arc
Sin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^2 - ((I/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sq
rt[-d] + Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^2

________________________________________________________________________________________

Rubi [A]  time = 0.988345, antiderivative size = 573, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {4733, 4627, 264, 4625, 3717, 2190, 2279, 2391, 4741, 4521} \[ -\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{i b e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 d^2}-\frac{e \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}-\frac{b c \sqrt{1-c^2 x^2}}{2 d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(2*d*x) - (a + b*ArcSin[c*x])/(2*d*x^2) + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(
I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcS
in[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x
]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I
*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) - (e*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/d^2 - ((I/2)
*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/d^2 - ((I/2)*b*e*PolyLog[2,
(Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^2 - ((I/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*Arc
Sin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^2 - ((I/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sq
rt[-d] + Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^2

Rule 4733

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[
ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^
2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 4627

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

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 4625

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

Rule 3717

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

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 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 4741

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

Rule 4521

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^
2, 2] + b*E^(I*(c + d*x))), x], x] + Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + b*E^
(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (d+e x^2\right )} \, dx &=\int \left (\frac{a+b \sin ^{-1}(c x)}{d x^3}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}+\frac{e^2 x \left (a+b \sin ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \sin ^{-1}(c x)}{x^3} \, dx}{d}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx}{d^2}+\frac{e^2 \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^2}\\ &=-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}+\frac{(b c) \int \frac{1}{x^2 \sqrt{1-c^2 x^2}} \, dx}{2 d}-\frac{e \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}+\frac{e^2 \int \left (-\frac{a+b \sin ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{d^2}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}+\frac{i e \left (a+b \sin ^{-1}(c x)\right )^2}{2 b d^2}+\frac{(2 i e) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac{e^{3/2} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 d^2}+\frac{e^{3/2} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 d^2}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}+\frac{i e \left (a+b \sin ^{-1}(c x)\right )^2}{2 b d^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac{(b e) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac{(i b e) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{\left (i e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}-\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}-\frac{\left (i e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}+\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{\left (i e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}-\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{\left (i e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}+\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac{i b e \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}+\frac{i b e \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{(i b e) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{(i b e) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{(i b e) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{(i b e) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{d^2}-\frac{i b e \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{i b e \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{i b e \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}-\frac{i b e \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 d^2}+\frac{i b e \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^2}\\ \end{align*}

Mathematica [A]  time = 2.27454, size = 483, normalized size = 0.84 \[ \frac{2 b \left (-i e \text{PolyLog}\left (2,\frac{\left (-2 \sqrt{c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{2 i \sin ^{-1}(c x)}}{e}\right )-i e \text{PolyLog}\left (2,\frac{\left (2 \sqrt{c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{2 i \sin ^{-1}(c x)}}{e}\right )+2 i e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+4 i e \sin ^{-1}\left (\sqrt{-\frac{c^2 d}{e}}\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d \left (c^2 d+e\right )}}{c d \sqrt{1-c^2 x^2}}\right )+2 e \sin ^{-1}\left (\sqrt{-\frac{c^2 d}{e}}\right ) \log \left (1-\frac{\left (-2 \sqrt{c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{2 i \sin ^{-1}(c x)}}{e}\right )+2 e \sin ^{-1}(c x) \log \left (1-\frac{\left (-2 \sqrt{c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{2 i \sin ^{-1}(c x)}}{e}\right )-2 e \sin ^{-1}\left (\sqrt{-\frac{c^2 d}{e}}\right ) \log \left (1-\frac{\left (2 \sqrt{c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{2 i \sin ^{-1}(c x)}}{e}\right )+2 e \sin ^{-1}(c x) \log \left (1-\frac{\left (2 \sqrt{c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{2 i \sin ^{-1}(c x)}}{e}\right )-\frac{2 c d \sqrt{1-c^2 x^2}}{x}-\frac{2 d \sin ^{-1}(c x)}{x^2}-4 e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )\right )+4 a e \log \left (d+e x^2\right )-\frac{4 a d}{x^2}-8 a e \log (x)}{8 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*a*d)/x^2 - 8*a*e*Log[x] + 4*a*e*Log[d + e*x^2] + 2*b*((-2*c*d*Sqrt[1 - c^2*x^2])/x - (2*d*ArcSin[c*x])/x^
2 + (4*I)*e*ArcSin[Sqrt[-((c^2*d)/e)]]*ArcTan[(Sqrt[c^2*d*(c^2*d + e)]*x)/(c*d*Sqrt[1 - c^2*x^2])] - 4*e*ArcSi
n[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*e*ArcSin[Sqrt[-((c^2*d)/e)]]*Log[1 - ((2*c^2*d + e - 2*Sqrt[c^2*d*(c
^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] + 2*e*ArcSin[c*x]*Log[1 - ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])*E^(
(2*I)*ArcSin[c*x]))/e] - 2*e*ArcSin[Sqrt[-((c^2*d)/e)]]*Log[1 - ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^(
(2*I)*ArcSin[c*x]))/e] + 2*e*ArcSin[c*x]*Log[1 - ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*
x]))/e] + (2*I)*e*PolyLog[2, E^((2*I)*ArcSin[c*x])] - I*e*PolyLog[2, ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)]
)*E^((2*I)*ArcSin[c*x]))/e] - I*e*PolyLog[2, ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))
/e]))/(8*d^2)

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Maple [C]  time = 0.237, size = 419, normalized size = 0.7 \begin{align*}{\frac{ae\ln \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }{2\,{d}^{2}}}-{\frac{a}{2\,d{x}^{2}}}-{\frac{ae\ln \left ( cx \right ) }{{d}^{2}}}+{\frac{{\frac{i}{2}}{c}^{2}b}{d}}-{\frac{bc}{2\,dx}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) }{2\,d{x}^{2}}}-{\frac{ibe}{{d}^{2}}{\it dilog} \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{be\arcsin \left ( cx \right ) }{{d}^{2}}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{ibe}{{d}^{2}}{\it dilog} \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{i}{4}}be}{{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}e-4\,{c}^{2}d-e}{{{\it \_R1}}^{2}e-2\,{c}^{2}d-e} \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }}-{\frac{{\frac{i}{4}}b{e}^{2}}{{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}-1}{{{\it \_R1}}^{2}e-2\,{c}^{2}d-e} \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*e/d^2*ln(c^2*e*x^2+c^2*d)-1/2*a/d/x^2-a/d^2*e*ln(c*x)+1/2*I*c^2*b/d-1/2*b*c*(-c^2*x^2+1)^(1/2)/d/x-1/2*b
/d*arcsin(c*x)/x^2-I*b/d^2*e*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-b/d^2*e*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2)
)+I*b/d^2*e*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-1/4*I*b/d^2*e*sum((_R1^2*e-4*c^2*d-e)/(_R1^2*e-2*c^2*d-e)*(I*arc
sin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(
-4*c^2*d-2*e)*_Z^2+e))-1/4*I*b/d^2*e^2*sum((_R1^2-1)/(_R1^2*e-2*c^2*d-e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^
2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{e \log \left (e x^{2} + d\right )}{d^{2}} - \frac{2 \, e \log \left (x\right )}{d^{2}} - \frac{1}{d x^{2}}\right )} + b \int \frac{\arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e x^{5} + d x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arcsin(c*x) + a)/(e*x^5 + d*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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