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

Optimal. Leaf size=574 \[ -\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 e^2 \sqrt{c^2 d+e}} \]

[Out]

(d*(a + b*ArcSin[c*x]))/(2*e^2*(d + e*x^2)) - ((I/2)*(a + b*ArcSin[c*x])^2)/(b*e^2) - (b*c*Sqrt[d]*ArcTan[(Sqr
t[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*e^2*Sqrt[c^2*d + 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*e^2) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*Arc
Sin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]
))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*
Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) - ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt
[c^2*d + e]))])/e^2 - ((I/2)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/e^2 -
((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/e^2 - ((I/2)*b*PolyLog[
2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/e^2

________________________________________________________________________________________

Rubi [A]  time = 0.956311, antiderivative size = 574, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4733, 4729, 377, 205, 4741, 4521, 2190, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 e^2}+\frac{d \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 \left (d+e x^2\right )}-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b e^2}-\frac{b c \sqrt{d} \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 e^2 \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(a + b*ArcSin[c*x]))/(2*e^2*(d + e*x^2)) - ((I/2)*(a + b*ArcSin[c*x])^2)/(b*e^2) - (b*c*Sqrt[d]*ArcTan[(Sqr
t[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*e^2*Sqrt[c^2*d + 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*e^2) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*Arc
Sin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]
))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*
Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) - ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt
[c^2*d + e]))])/e^2 - ((I/2)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/e^2 -
((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/e^2 - ((I/2)*b*PolyLog[
2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/e^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 4729

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

Rule 377

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

Rule 205

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

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]

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]

Rubi steps

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

Mathematica [A]  time = 1.03385, size = 593, normalized size = 1.03 \[ \frac{b \left (-i \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )-i \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )+\sqrt{d} \left (\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left (\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}\right )-i \sqrt{d} \left (-\frac{c \tanh ^{-1}\left (\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right )\right )+\frac{2 a d}{d+e x^2}+2 a \log \left (d+e x^2\right )}{4 e^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*a*d)/(d + e*x^2) + 2*a*Log[d + e*x^2] + b*(Sqrt[d]*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqr
t[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - I*Sqrt[d]*(-(ArcSin[c*x]/(I*Sqr
t[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d +
e]) - I*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])]
+ Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*
x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d +
e]))]) - I*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d +
e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcS
in[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d +
e])])))/(4*e^2)

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Maple [C]  time = 0.523, size = 2907, normalized size = 5.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*c^2*b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*d/e^2/(c^2*d
+e)-2*I*c^2*b*arcsin(c*x)^2*d/e^4*(c^2*d*(c^2*d+e))^(1/2)+5/4*I*c^2*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2
/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^2/(c^2*d+e)*d+5/2*I*c^2*b*arcsin(c*x)^2*d/e^2/(c^2*d+e)+4*I*c^4*b*ar
csin(c*x)^2/e^3/(c^2*d+e)*d^2+2*I*c^6*b*d^3*arcsin(c*x)^2/e^4/(c^2*d+e)+2*I*c^4*b*d^2*polylog(2,e*(I*c*x+(-c^2
*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^3/(c^2*d+e)-I*c^2*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^
(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*d/e^4*(c^2*d*(c^2*d+e))^(1/2)+1/2*b*ln(1-e*(I*c*x+(-c^2*x^2+1)
^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^2-1/2*I*b/e^2*sum((_R1^2*e-4*c^2*d-2*e)/(_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))-I*b*arcsin(c*x)^2/e^2-1/4*I*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))
^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^2+1/2*a/e^2*ln(c^2*e*x^2+c^2*d)+1/2*c^2*b*arcsin(c*x)/e^2*d/(c^2*e
*x^2+c^2*d)+2*c^2*b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^3
*d+2*c^4*b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*d^2/e^4+I*b*
(c^2*d*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arcsin(c*x)^2-3/2*b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*
d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+1/2*b*(c^2*d*(c^2*d+e))^(1/2)/e^2/(c^
2*d+e)*arcsin(c*x)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))-2*I*c^2*b*arcsin
(c*x)^2/e^3*d-2*I*c^4*b*arcsin(c*x)^2*d^2/e^4-I*c^4*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2
*d*(c^2*d+e))^(1/2)+e))*d^2/e^4-I*c^2*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^
(1/2)+e))*d/e^3+1/2*I*b*(c^2*d*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arctanh(1/4*(2*(I*c*x+(-c^2*x^2+1)^(1/2))^2*e-4*
c^2*d-2*e)/(c^4*d^2+c^2*d*e)^(1/2))+3/4*I*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+
e))^(1/2)+e))/e^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-1/4*I*b*(c^2*d*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*polylog(2,e*
(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))+I*c^6*b*d^3*polylog(2,e*(I*c*x+(-c^2*x^2+1
)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^4/(c^2*d+e)+2*c^2*b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2
*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)*d/e^4*(c^2*d*(c^2*d+e))^(1/2)-2*c^6*b*d^3*ln(1-e*(I*c*x+(-c^2
*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^4/(c^2*d+e)-4*c^4*b*d^2*ln(1-e*(I*c*x+(-
c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^3/(c^2*d+e)+1/8*I/c^2*b*polylog(2,e*(
I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/d/e/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+3/2*I
*c^2*b*d*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^3/(c^2*d+e)*(c^2*d*
(c^2*d+e))^(1/2)-1/8*I/c^2*b*(c^2*d*(c^2*d+e))^(1/2)/e/d/(c^2*d+e)*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2
*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e))-3*c^2*b*d*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e)
)^(1/2)+e))*arcsin(c*x)/e^3/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-2*c^4*b*d^2*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/
(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^4/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+1/4/c^2*b*(c^2*d*(c^2
*d+e))^(1/2)/e/d/(c^2*d+e)*arcsin(c*x)*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+
e))-1/4/c^2*b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/d/e/(c^2*
d+e)*(c^2*d*(c^2*d+e))^(1/2)+2*I*c^4*b*d^2*arcsin(c*x)^2/e^4/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+3*I*c^2*b*arcsi
n(c*x)^2/e^3/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)*d+I*c^4*b*d^2*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d
-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^4/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*
c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e^3*(c^2*d*(c^2*d+e))^(1/2)+1/2*c^2*a/e^2*d/(c^2*e*x^2+c^2*d)-
1/2*I*b*polylog(2,e*(I*c*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e^3*(c^2*d*(c^2*d+e))^
(1/2)+1/2*I*b*arcsin(c*x)^2/e/(c^2*d+e)-I*b*arcsin(c*x)^2/e^3*(c^2*d*(c^2*d+e))^(1/2)+1/4*I*b*polylog(2,e*(I*c
*x+(-c^2*x^2+1)^(1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))/e/(c^2*d+e)-1/2*b*ln(1-e*(I*c*x+(-c^2*x^2+1)^(
1/2))^2/(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e))*arcsin(c*x)/e/(c^2*d+e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{d}{e^{3} x^{2} + d e^{2}} + \frac{\log \left (e x^{2} + d\right )}{e^{2}}\right )} + b \int \frac{x^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \arcsin \left (c x\right ) + a x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3*arcsin(c*x) + a*x^3)/(e^2*x^4 + 2*d*e*x^2 + d^2), 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(x**3*(a+b*asin(c*x))/(e*x**2+d)**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 )} x^{3}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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