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

Optimal. Leaf size=632 \[ -\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{i b e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{2 e \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{a+b \sin ^{-1}(c x)}{2 d^2 x^2}+\frac{b c e \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt{c^2 d+e}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x} \]

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(2*d^2*x) - (a + b*ArcSin[c*x])/(2*d^2*x^2) - (e*(a + b*ArcSin[c*x]))/(2*d^2*(d + e*x
^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]) + (e*(a + b
*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*
x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1
- (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]
*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 - (2*e*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[
c*x])])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/d^3 - (I*b*e
*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*
E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^3 - (I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*
c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (I*b*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^3

________________________________________________________________________________________

Rubi [A]  time = 1.0447, antiderivative size = 632, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {4733, 4627, 264, 4625, 3717, 2190, 2279, 2391, 4729, 377, 205, 4741, 4521} \[ -\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{i b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{i b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{i b e \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{d^3}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{2 e \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{a+b \sin ^{-1}(c x)}{2 d^2 x^2}+\frac{b c e \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt{c^2 d+e}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(2*d^2*x) - (a + b*ArcSin[c*x])/(2*d^2*x^2) - (e*(a + b*ArcSin[c*x]))/(2*d^2*(d + e*x
^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]) + (e*(a + b
*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*
x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1
- (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]
*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 - (2*e*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[
c*x])])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/d^3 - (I*b*e
*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*
E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^3 - (I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*
c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (I*b*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^3

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 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]

Rubi steps

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

Mathematica [F]  time = 5.8612, size = 0, normalized size = 0. \[ \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [C]  time = 0.346, size = 679, normalized size = 1.1 \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))/x^3/(e*x^2+d)^2,x)

[Out]

-1/2*c^2*a*e/d^2/(c^2*e*x^2+c^2*d)+a*e/d^3*ln(c^2*e*x^2+c^2*d)-1/2*a/d^2/x^2-2*a/d^3*e*ln(c*x)-1/2*I*b*(c^2*d*
(c^2*d+e))^(1/2)/(c^2*d+e)/d^3*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))*e-2*I*b/d^3*e*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-1/2*c^3*b*x/(c^2*e*x^2+c^2*d)/d^2*(-c^2*x^2+1)^(1/2)*e-1/2*
c^3*b/x/(c^2*e*x^2+c^2*d)/d*(-c^2*x^2+1)^(1/2)-c^2*b*arcsin(c*x)*e/d^2/(c^2*e*x^2+c^2*d)-1/2*c^2*b/x^2/(c^2*e*
x^2+c^2*d)/d*arcsin(c*x)-1/2*I*b/d^3*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))+1/2*I*
c^4*b/(c^2*e*x^2+c^2*d)/d-2*b/d^3*e*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-1/2*I*b/d^3*e*sum((_R1^2*e-4*c^
2*d-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))+1/2*I*c^4*b*x^2/(c^2*e*x^2+c^2*d)/d^2*e+2*I*b/d^3*e*di
log(1+I*c*x+(-c^2*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a{\left (\frac{2 \, e x^{2} + d}{d^{2} e x^{4} + d^{3} x^{2}} - \frac{2 \, e \log \left (e x^{2} + d\right )}{d^{3}} + \frac{4 \, e \log \left (x\right )}{d^{3}}\right )} + b \int \frac{\arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e^{2} x^{7} + 2 \, d e x^{5} + d^{2} 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)^2,x, algorithm="maxima")

[Out]

-1/2*a*((2*e*x^2 + d)/(d^2*e*x^4 + d^3*x^2) - 2*e*log(e*x^2 + d)/d^3 + 4*e*log(x)/d^3) + b*integrate(arctan2(c
*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^2*x^7 + 2*d*e*x^5 + d^2*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^{2} x^{7} + 2 \, d e x^{5} + d^{2} 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)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [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*arcsin(c*x))/x^3/(e*x^2+d)^2,x, algorithm="giac")

[Out]

Timed out