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

Optimal. Leaf size=757 \[ -\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}} \]

[Out]

-(a + b*ArcSin[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[
e]*x)) + (b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d*Sqrt[e]*Sqrt[c^2*d
+ e]) + (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d*Sqrt[e]*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])])/(4*(-d)^(
3/2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4
*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e
])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^
2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^
2*d + e]))])/((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d
+ e])])/((-d)^(3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d +
e]))])/((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]
)])/((-d)^(3/2)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 0.99458, antiderivative size = 757, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4667, 4743, 725, 206, 4741, 4521, 2190, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*ArcSin[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[
e]*x)) + (b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d*Sqrt[e]*Sqrt[c^2*d
+ e]) + (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d*Sqrt[e]*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])])/(4*(-d)^(
3/2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4
*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e
])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^
2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^
2*d + e]))])/((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d
+ e])])/((-d)^(3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d +
e]))])/((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]
)])/((-d)^(3/2)*Sqrt[e])

Rule 4667

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

Rule 4743

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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{a+b \sin ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 d}-\frac{(b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{4 d}-\frac{e \int \left (-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{3/2}}+\frac{\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{3/2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{-e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 d}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{c^2 d e+e^2-x^2} \, dx,x,\frac{e+c^2 \sqrt{-d} \sqrt{e} x}{\sqrt{1-c^2 x^2}}\right )}{4 d}\\ &=-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \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 )}{4 (-d)^{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 )}{4 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \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 )}{4 (-d)^{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 )}{4 (-d)^{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 )}{4 (-d)^{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 )}{4 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{e}}+\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 )}{4 (-d)^{3/2} \sqrt{e}}-\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 )}{4 (-d)^{3/2} \sqrt{e}}+\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 )}{4 (-d)^{3/2} \sqrt{e}}-\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 )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{e}}-\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 )}{4 (-d)^{3/2} \sqrt{e}}+\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 )}{4 (-d)^{3/2} \sqrt{e}}-\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 )}{4 (-d)^{3/2} \sqrt{e}}+\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 )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{4 d \sqrt{e} \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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{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 )}{4 (-d)^{3/2} \sqrt{e}}-\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 )}{4 (-d)^{3/2} \sqrt{e}}+\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 )}{4 (-d)^{3/2} \sqrt{e}}-\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 )}{4 (-d)^{3/2} \sqrt{e}}+\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 )}{4 (-d)^{3/2} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 1.69092, size = 591, normalized size = 0.78 \[ \frac{1}{2} \left (\frac{b \left (\text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )-\text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+i \sqrt{d} \left (\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left (\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}\right )+\sqrt{d} \left (\frac{c \tanh ^{-1}\left (\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right )}{\sqrt{c^2 d+e}}+\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right )+i \sin ^{-1}(c x) \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )-i \sin ^{-1}(c x) \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )}{2 d^{3/2} \sqrt{e}}+\frac{a x}{d^2+d e x^2}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2} \sqrt{e}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.547, size = 1687, normalized size = 2.2 \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))/(e*x^2+d)^2,x)

[Out]

1/2*c^2*a*x/d/(c^2*e*x^2+c^2*d)+1/2*a/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/2*c^2*b*arcsin(c*x)*x/d/(c^2*e*x
^2+c^2*d)-c^5*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*
d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)*d-c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2
)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)*(c^2*d*(
c^2*d+e))^(1/2)-c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-
2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)/e^2-1/2*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e
)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d/(c^2*d+e)/e^2*
(c^2*d*(c^2*d+e))^(1/2)+c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1
/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3+c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2
)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d/e^3*(c^2*d*(c^2*d+e)
)^(1/2)+1/2*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*
d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d/e^2-c^5*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e
*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)*d+c^3*b*((2*c^2*d+2
*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+
e)*e)^(1/2))/e^3/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arcta
nh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)/e^2+1/2*c*b*((2*c^2
*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1
/2)+e)*e)^(1/2))/d/(c^2*d+e)/e^2*(c^2*d*(c^2*d+e))^(1/2)+c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)
*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3-c*b*((2*c^2*d+2*(c^
2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e
)^(1/2))/d/e^3*(c^2*d*(c^2*d+e))^(1/2)+1/2*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh(e*(I*c*
x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d/e^2+1/4*c*b/d*sum(1/_R1/(_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=Ro
otOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))+1/4*c*b/d*sum(_R1/(_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(-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))/(e*x^2+d)^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{b \arcsin \left (c x\right ) + a}{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((a+b*arcsin(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asin(c*x))/(d + e*x**2)**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 )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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