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

Optimal. Leaf size=1092 \[ \text{result too large to display} \]

[Out]

(b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^
(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)^
2) - (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt
[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*ArcTa
nh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2)) + (3*b*c*
ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) + (b
*c^3*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2))
+ (3*b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d
+ e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)
^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])]
)/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^
2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d]
+ Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sq
rt[-d] - Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*
Sqrt[-d] - Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I
*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/
(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 1.24787, antiderivative size = 1092, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4667, 4743, 731, 725, 206, 4741, 4521, 2190, 2279, 2391} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c^3}{16 d \sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c^3}{16 d \sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c}{16 d^2 \sqrt{e} \sqrt{d c^2+e}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right ) c}{16 d^2 \sqrt{e} \sqrt{d c^2+e}}+\frac{b \sqrt{1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b \sqrt{1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{e} x+\sqrt{-d}\right )^2}+\frac{3 \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{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^
(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)^
2) - (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt
[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*ArcTa
nh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2)) + (3*b*c*
ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) + (b
*c^3*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2))
+ (3*b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d
+ e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)
^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])]
)/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^
2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d]
+ Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sq
rt[-d] - Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*
Sqrt[-d] - Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I
*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/
(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/((-d)^(5/2)*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 731

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

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 )^3} \, dx &=\int \left (-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-e x\right )^3}-\frac{3 e \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+e x\right )^3}-\frac{3 e \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{3 e \left (a+b \sin ^{-1}(c x)\right )}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{(3 e) \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{16 d^2}-\frac{(3 e) \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{16 d^2}-\frac{(3 e) \int \frac{a+b \sin ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{8 d^2}-\frac{e^{3/2} \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^3} \, dx}{8 (-d)^{3/2}}-\frac{e^{3/2} \int \frac{a+b \sin ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^3} \, dx}{8 (-d)^{3/2}}\\ &=-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{(3 b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 d^2}-\frac{(3 b c) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 d^2}+\frac{\left (b c \sqrt{e}\right ) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2 \sqrt{1-c^2 x^2}} \, dx}{16 (-d)^{3/2}}-\frac{\left (b c \sqrt{e}\right ) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2 \sqrt{1-c^2 x^2}} \, dx}{16 (-d)^{3/2}}-\frac{(3 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}{8 d^2}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{16 (-d)^{5/2}}-\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{16 (-d)^{5/2}}-\frac{(3 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 )}{16 d^2}+\frac{(3 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 )}{16 d^2}+\frac{\left (b c^3\right ) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}-e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 d \left (c^2 d+e\right )}-\frac{\left (b c^3\right ) \int \frac{1}{\left (\sqrt{-d} \sqrt{e}+e x\right ) \sqrt{1-c^2 x^2}} \, dx}{16 d \left (c^2 d+e\right )}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac{\left (b c^3\right ) \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 )}{16 d \left (c^2 d+e\right )}+\frac{\left (b c^3\right ) \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 )}{16 d \left (c^2 d+e\right )}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}-\frac{(3 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 )}{16 (-d)^{5/2}}-\frac{(3 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 )}{16 (-d)^{5/2}}-\frac{(3 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 )}{16 (-d)^{5/2}}-\frac{(3 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 )}{16 (-d)^{5/2}}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}\\ &=\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{b c \sqrt{1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{e}+c^2 \sqrt{-d} x}{\sqrt{c^2 d+e} \sqrt{1-c^2 x^2}}\right )}{16 d \sqrt{e} \left (c^2 d+e\right )^{3/2}}+\frac{3 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 )}{16 d^2 \sqrt{e} \sqrt{c^2 d+e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 6.06254, size = 1055, normalized size = 0.97 \[ \frac{3 a x}{8 d^2 \left (e x^2+d\right )}+\frac{a x}{4 d \left (e x^2+d\right )^2}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}+b \left (\frac{3 i \left (\frac{\sin ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-\frac{c \tan ^{-1}\left (\frac{\sqrt{d} x c^2+i \sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{\sqrt{d c^2+e}}\right )}{16 d^2 \sqrt{e}}-\frac{3 \left (-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \tanh ^{-1}\left (\frac{i \sqrt{d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right )}{\sqrt{d c^2+e}}\right )}{16 d^2 \sqrt{e}}+\frac{i \left (-\frac{i \sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (-i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt{e} x \sqrt{d}\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x-i \sqrt{d}\right )^2}\right )}{16 d^{3/2}}-\frac{i \left (\frac{i \sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt{d} \sqrt{e} x\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left (d c^2+e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x+i \sqrt{d}\right )^2}\right )}{16 d^{3/2}}-\frac{3 \left (\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{c \sqrt{d}-\sqrt{d c^2+e}}+1\right )+\log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d} c+\sqrt{d c^2+e}}+1\right )\right )\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )\right )}{32 d^{5/2} \sqrt{e}}+\frac{3 \left (\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-c \sqrt{d}}+1\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )\right )\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right )\right )}{32 d^{5/2} \sqrt{e}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]
) + b*((((3*I)/16)*(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]))/(d^2*Sqrt[e]) - (3*(-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcT
anh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(16*d^2*Sqrt[e]) + ((I
/16)*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*((-I)*Sqrt[d] +
Sqrt[e]*x)^2) - (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*
Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*(-((c*S
qrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + (
I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])
)/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) - (3*(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]))]))/(32*d^(5/2)*Sqrt[e]) + (3*(ArcS
in[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*Sqr
t[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(32*d^(5/
2)*Sqrt[e]))

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Maple [C]  time = 0.734, size = 3110, normalized size = 2.9 \begin{align*} \text{output 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)^3,x)

[Out]

3/8*a/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-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)^2*(c^2*d*(c^2*d+e))
^(1/2)+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)^2*(c^2*d*(c^2*d+e))^(1/2)-c^7*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)^2*d-c^7*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)^2*d+5/4*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^2/d/(c^2*
d+e)+5/4*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^2/d/(c^2*d+e)-3/4*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))/d/(c^2*d+e)^2/e+3/8*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))/e/d^2/(c^2*d+e)-3/4*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))/d/(c^2*d+e)^2/e+3/8*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))/e/d^2/(c^2*d+e)+3/8*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^2/(c^2*d+e)^2/e*(c^2*d*(c^2*d+e))^(1/2)-3/4
*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))/e^2/d^2/(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))/e^3/d/(c
^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+5/4*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))/(c^2*d+e)^2/d/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/d/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-5/4*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)^2/d/e^2*(c^2*d*(c^2*d+e))^(1/2)+3/8*c^6*b*e/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x^3+1/8*c^5*b/d/(c^
2*d+e)/(c^2*e*x^2+c^2*d)^2*(-c^2*x^2+1)^(1/2)*x^2*e+3/8*c^4*b/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x^
3*e^2+5/8*c^4*b/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x*e-3/8*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^2/(c^2*d+e)
^2/e*(c^2*d*(c^2*d+e))^(1/2)+3/4*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))/e^2/d^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+3/16*c
^3*b/d/(c^2*d+e)*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))+3/8*c^2*a/d^2*x/(c^2*e*x^2+c^2*d)+1/
4*c^4*a*x/d/(c^2*e*x^2+c^2*d)^2+1/8*c^5*b/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(-c^2*x^2+1)^(1/2)+3/16*c^3*b/d/(c^2*d
+e)*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=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))-7/4*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))/(c^2*d+e)^2/
e^2-7/4*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))/(c^2*d+e)^2/e^2+c^5*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*a
rctan(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^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)+5/8*c^6*b/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)*x+3/16*c*b/d^2/(c^2*d+e)*
e*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))+3/16*c*b/d^2/(c^2*d+e)*e*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=RootO
f(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)^3,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^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral((b*arcsin(c*x) + a)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^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))/(e*x**2+d)**3,x)

[Out]

Timed out

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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