3.632 \(\int \frac{a+b \sin ^{-1}(c x)}{x^4 (d+e x^2)} \, dx\)

Optimal. Leaf size=649 \[ \frac{i b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{i b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{b c e \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d} \]

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(6*d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) + (e*(a + b*ArcSin[c*x]))/(d^2*x) - (b*c^3*
ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (b*c*e*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 + (e^(3/2)*(a + b*ArcSin[c*x])*Log[
1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x
])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*Arc
Sin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a
+ b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + ((I/2
)*b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(-d)^(5/2) - ((I/2)*b
*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(-d)^(5/2) + ((I/2)*b*e^(3/
2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(-d)^(5/2) - ((I/2)*b*e^(3/2)*
PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(-d)^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 0.962196, antiderivative size = 649, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4733, 4627, 266, 51, 63, 208, 4667, 4741, 4521, 2190, 2279, 2391} \[ \frac{i b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{i b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{b c e \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])/(x^4*(d + e*x^2)),x]

[Out]

-(b*c*Sqrt[1 - c^2*x^2])/(6*d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) + (e*(a + b*ArcSin[c*x]))/(d^2*x) - (b*c^3*
ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (b*c*e*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 + (e^(3/2)*(a + b*ArcSin[c*x])*Log[
1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x
])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*Arc
Sin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a
+ b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + ((I/2
)*b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(-d)^(5/2) - ((I/2)*b
*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(-d)^(5/2) + ((I/2)*b*e^(3/
2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(-d)^(5/2) - ((I/2)*b*e^(3/2)*
PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(-d)^(5/2)

Rule 4733

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

Rule 4627

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

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 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)}{x^4 \left (d+e x^2\right )} \, dx &=\int \left (\frac{a+b \sin ^{-1}(c x)}{d x^4}-\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x^2}+\frac{e^2 \left (a+b \sin ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \sin ^{-1}(c x)}{x^4} \, dx}{d}-\frac{e \int \frac{a+b \sin ^{-1}(c x)}{x^2} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \sin ^{-1}(c x)}{d+e x^2} \, dx}{d^2}\\ &=-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}+\frac{(b c) \int \frac{1}{x^3 \sqrt{1-c^2 x^2}} \, dx}{3 d}-\frac{(b c e) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{d^2}+\frac{e^2 \int \left (\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \sin ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{d^2}\\ &=-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{6 d}-\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d^2}-\frac{e^2 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 (-d)^{5/2}}-\frac{e^2 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 (-d)^{5/2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 d}+\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c d^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}-\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac{e^2 \operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{c \sqrt{-d}+\sqrt{e} \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}+\frac{b c e \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{6 d}-\frac{\left (i e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}-\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac{\left (i e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}+\sqrt{c^2 d+e}-\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac{\left (i e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}-\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac{\left (i e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{i c \sqrt{-d}+\sqrt{c^2 d+e}+\sqrt{e} e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}-\frac{b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d}+\frac{b c e \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{\left (b e^{3/2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}+\frac{\left (b e^{3/2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}-\frac{\left (b e^{3/2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}+\frac{\left (b e^{3/2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^{i x}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 (-d)^{5/2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}-\frac{b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d}+\frac{b c e \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac{\left (i b e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 (-d)^{5/2}}-\frac{\left (i b e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 (-d)^{5/2}}+\frac{\left (i b e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 (-d)^{5/2}}-\frac{\left (i b e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 (-d)^{5/2}}\\ &=-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \sin ^{-1}(c x)\right )}{d^2 x}-\frac{b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d}+\frac{b c e \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^2}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac{i b e^{3/2} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac{i b e^{3/2} \text{Li}_2\left (-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{Li}_2\left (\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.4353, size = 531, normalized size = 0.82 \[ b \left (-\frac{e^{3/2} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )}{4 d^{5/2}}+\frac{e^{3/2} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )}{4 d^{5/2}}-\frac{e \left (-c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-\frac{\sin ^{-1}(c x)}{x}\right )}{d^2}-\frac{c x \sqrt{1-c^2 x^2}+c^3 x^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+2 \sin ^{-1}(c x)}{6 d x^3}\right )+\frac{a e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2}}+\frac{a e}{d^2 x}-\frac{a}{3 d x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSin[c*x])/(x^4*(d + e*x^2)),x]

[Out]

-a/(3*d*x^3) + (a*e)/(d^2*x) + (a*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(5/2) + b*(-((e*(-(ArcSin[c*x]/x) - c
*ArcTanh[Sqrt[1 - c^2*x^2]]))/d^2) - (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2
]])/(6*d*x^3) - (e^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - S
qrt[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]))]))/(4*d^(5/2)) + (e^(3/2)*(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*ArcS
in[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(4*d^(5/2)))

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Maple [C]  time = 0.544, size = 472, normalized size = 0.7 \begin{align*}{\frac{a{e}^{2}}{{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{a}{3\,d{x}^{3}}}+{\frac{ae}{{d}^{2}x}}-{\frac{bc}{6\,d{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arcsin \left ( cx \right ) e}{{d}^{2}x}}-{\frac{b\arcsin \left ( cx \right ) }{3\,d{x}^{3}}}-{\frac{b{e}^{2}}{8\,c{d}^{3}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}e-4\,{c}^{2}d-e}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e-2\,{c}^{2}d-e \right ) } \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }}+{\frac{{c}^{3}b}{6\,d}\ln \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1}-1 \right ) }-{\frac{{c}^{3}b}{6\,d}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{b{e}^{2}}{8\,c{d}^{3}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( -4\,{c}^{2}d-2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{4\,{{\it \_R1}}^{2}{c}^{2}d+{{\it \_R1}}^{2}e-e}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e-2\,{c}^{2}d-e \right ) } \left ( i\arcsin \left ( cx \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) \right ) }}-{\frac{bce}{{d}^{2}}\ln \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1}-1 \right ) }+{\frac{bce}{{d}^{2}}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e^2/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/3*a/d/x^3+a/d^2*e/x-1/6*b*c*(-c^2*x^2+1)^(1/2)/d/x^2+b*arcsin(
c*x)/d^2*e/x-1/3*b*arcsin(c*x)/d/x^3-1/8/c*b/d^3*e^2*sum((_R1^2*e-4*c^2*d-e)/_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))+1/6*c^3*b/d*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-1/6*c^3*b/d*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+1/8/
c*b/d^3*e^2*sum((4*_R1^2*c^2*d+_R1^2*e-e)/_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))-c*b/d^2*e*ln(I*
c*x+(-c^2*x^2+1)^(1/2)-1)+c*b/d^2*e*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))

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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))/x^4/(e*x^2+d),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 x^{6} + d x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(e*x^2+d),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asin}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))/x**4/(e*x**2+d),x)

[Out]

Integral((a + b*asin(c*x))/(x**4*(d + e*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))/x^4/(e*x^2+d),x, algorithm="giac")

[Out]

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