3.201 \(\int \frac{x^4 (a+b \sin ^{-1}(c x))^2}{(d-c^2 d x^2)^3} \, dx\)

Optimal. Leaf size=343 \[ \frac{3 i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3}-\frac{3 i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3}-\frac{3 b^2 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac{3 b^2 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac{5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt{1-c^2 x^2}}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^5 d^3}+\frac{b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3} \]

[Out]

(b^2*x)/(12*c^4*d^3*(1 - c^2*x^2)) - (b*(a + b*ArcSin[c*x]))/(6*c^5*d^3*(1 - c^2*x^2)^(3/2)) + (5*b*(a + b*Arc
Sin[c*x]))/(4*c^5*d^3*Sqrt[1 - c^2*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/(4*c^2*d^3*(1 - c^2*x^2)^2) - (3*x*(a +
b*ArcSin[c*x])^2)/(8*c^4*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^
5*d^3) - (7*b^2*ArcTanh[c*x])/(6*c^5*d^3) + (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])
])/(c^5*d^3) - (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d^3) - (3*b^2*PolyLog[3,
(-I)*E^(I*ArcSin[c*x])])/(4*c^5*d^3) + (3*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c^5*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.536364, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {4703, 4657, 4181, 2531, 2282, 6589, 4677, 206, 266, 43, 4689, 12, 385} \[ \frac{3 i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3}-\frac{3 i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3}-\frac{3 b^2 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac{3 b^2 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac{5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt{1-c^2 x^2}}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{3 i \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^5 d^3}+\frac{b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]

[Out]

(b^2*x)/(12*c^4*d^3*(1 - c^2*x^2)) - (b*(a + b*ArcSin[c*x]))/(6*c^5*d^3*(1 - c^2*x^2)^(3/2)) + (5*b*(a + b*Arc
Sin[c*x]))/(4*c^5*d^3*Sqrt[1 - c^2*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/(4*c^2*d^3*(1 - c^2*x^2)^2) - (3*x*(a +
b*ArcSin[c*x])^2)/(8*c^4*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^
5*d^3) - (7*b^2*ArcTanh[c*x])/(6*c^5*d^3) + (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])
])/(c^5*d^3) - (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d^3) - (3*b^2*PolyLog[3,
(-I)*E^(I*ArcSin[c*x])])/(4*c^5*d^3) + (3*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c^5*d^3)

Rule 4703

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p +
1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[(b*f*n*d^IntPart[p]*(d + e*x^2
)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Gt
Q[m, 1]

Rule 4657

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

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4677

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

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 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4689

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSin[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2
, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 385

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

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}-\frac{3 \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{b \left (a+b \sin ^{-1}(c x)\right )}{2 c^5 d^3 \sqrt{1-c^2 x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac{b^2 \int \frac{-2+3 c^2 x^2}{3 c^4 \left (1-c^2 x^2\right )^2} \, dx}{2 d^3}+\frac{(3 b) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 c^3 d^3}+\frac{3 \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx}{8 c^4 d^2}\\ &=-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt{1-c^2 x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac{3 \operatorname{Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^5 d^3}+\frac{b^2 \int \frac{-2+3 c^2 x^2}{\left (1-c^2 x^2\right )^2} \, dx}{6 c^4 d^3}-\frac{\left (3 b^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{4 c^4 d^3}\\ &=\frac{b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt{1-c^2 x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{3 b^2 \tanh ^{-1}(c x)}{4 c^5 d^3}-\frac{(3 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}+\frac{(3 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}-\frac{\left (5 b^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{12 c^4 d^3}\\ &=\frac{b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt{1-c^2 x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac{3 i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{3 i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}+\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 c^5 d^3}\\ &=\frac{b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt{1-c^2 x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac{3 i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{3 i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}\\ &=\frac{b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{b \left (a+b \sin ^{-1}(c x)\right )}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{5 b \left (a+b \sin ^{-1}(c x)\right )}{4 c^5 d^3 \sqrt{1-c^2 x^2}}+\frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{3 x \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac{3 i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{3 i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac{3 b^2 \text{Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac{3 b^2 \text{Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{4 c^5 d^3}\\ \end{align*}

Mathematica [A]  time = 6.37722, size = 667, normalized size = 1.94 \[ \frac{18 a b \left (4 i \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+i \sin ^{-1}(c x)^2+\sin ^{-1}(c x) \left (-4 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-3 i \pi \right )+2 \pi \left (-2 \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+\log \left (1+i e^{i \sin ^{-1}(c x)}\right )+2 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-\log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )\right )+18 a b \left (-4 i \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )-i \sin ^{-1}(c x)^2+\sin ^{-1}(c x) \left (4 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+i \pi \right )+2 \pi \left (2 \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+\log \left (1-i e^{i \sin ^{-1}(c x)}\right )-\log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-2 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )+8 b^2 \left (9 i \sin ^{-1}(c x) \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-9 i \sin ^{-1}(c x) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )-9 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )+9 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )-14 \tanh ^{-1}(c x)-9 i \sin ^{-1}(c x)^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )\right )+\frac{60 a^2 c x}{c^2 x^2-1}+\frac{24 a^2 c x}{\left (c^2 x^2-1\right )^2}-18 a^2 \log (1-c x)+18 a^2 \log (c x+1)-\frac{60 a b \left (\sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right )}{c x-1}+\frac{60 a b \left (\sqrt{1-c^2 x^2}+\sin ^{-1}(c x)\right )}{c x+1}+\frac{4 a b \left (\sqrt{1-c^2 x^2} (c x-2)+3 \sin ^{-1}(c x)\right )}{(c x-1)^2}-\frac{4 a b \left (\sqrt{1-c^2 x^2} (c x+2)+3 \sin ^{-1}(c x)\right )}{(c x+1)^2}+\frac{b^2 \left (\sin ^{-1}(c x) \left (74 \sqrt{1-c^2 x^2}+30 \cos \left (3 \sin ^{-1}(c x)\right )\right )+3 \left (3 c x-5 \sin \left (3 \sin ^{-1}(c x)\right )\right ) \sin ^{-1}(c x)^2+2 \left (c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )\right )}{\left (c^2 x^2-1\right )^2}}{96 c^5 d^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]

[Out]

((24*a^2*c*x)/(-1 + c^2*x^2)^2 + (60*a^2*c*x)/(-1 + c^2*x^2) - (60*a*b*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(-1
+ c*x) + (60*a*b*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(1 + c*x) + (4*a*b*((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSi
n[c*x]))/(-1 + c*x)^2 - (4*a*b*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(1 + c*x)^2 - 18*a^2*Log[1 - c*x
] + 18*a^2*Log[1 + c*x] + 18*a*b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-3*I)*Pi - 4*Log[1 + I*E^(I*ArcSin[c*x])]) +
2*Pi*(-2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])] + 2*Log[Cos[ArcSin[c*x]/2]] - Log[-Cos[
(Pi + 2*ArcSin[c*x])/4]]) + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) + 18*a*b*((-I)*ArcSin[c*x]^2 + ArcSin[c*
x]*(I*Pi + 4*Log[1 - I*E^(I*ArcSin[c*x])]) + 2*Pi*(2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 - I*E^(I*ArcSin[c*x
])] - 2*Log[Cos[ArcSin[c*x]/2]] - Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) - (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]) +
8*b^2*((-9*I)*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] - 14*ArcTanh[c*x] + (9*I)*ArcSin[c*x]*PolyLog[2, (-I)*E
^(I*ArcSin[c*x])] - (9*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] - 9*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] +
9*PolyLog[3, I*E^(I*ArcSin[c*x])]) + (b^2*(ArcSin[c*x]*(74*Sqrt[1 - c^2*x^2] + 30*Cos[3*ArcSin[c*x]]) + 3*Arc
Sin[c*x]^2*(3*c*x - 5*Sin[3*ArcSin[c*x]]) + 2*(c*x + Sin[3*ArcSin[c*x]])))/(-1 + c^2*x^2)^2)/(96*c^5*d^3)

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Maple [B]  time = 0.547, size = 903, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x)

[Out]

13/12/c^5*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(-c^2*x^2+1)^(1/2)-3/4/c^5*a*b/d^3*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2
+1)^(1/2)))+3/4/c^5*a*b/d^3*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+13/12/c^5*b^2/d^3/(c^4*x^4-2*c^2*x^
2+1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+3/4*I/c^5*a*b/d^3*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/4*I/c^5*a*b/d^3*
dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/4*I/c^5*b^2/d^3*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3
/4*I/c^5*b^2/d^3*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5/8/c^2*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arc
sin(c*x)^2*x^3-3/8/c^4*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2*x-3/4*b^2*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^
(1/2)))/c^5/d^3+3/4*b^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^3-5/4/c^3*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*
x^2*(-c^2*x^2+1)^(1/2)-3/4/c^4*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*x-5/4/c^3*b^2/d^3/(c^4*x^4-2*c^2*x^2+
1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2+5/4/c^2*a*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*x^3+3/8/c^5*b^2/d^3*ar
csin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+7/3*I/c^5*b^2/d^3*arctan(I*c*x+(-c^2*x^2+1)^(1/2))+1/12/c^4*b^2
/d^3/(c^4*x^4-2*c^2*x^2+1)*x-1/12/c^2*b^2/d^3/(c^4*x^4-2*c^2*x^2+1)*x^3-3/8/c^5*b^2/d^3*arcsin(c*x)^2*ln(1+I*(
I*c*x+(-c^2*x^2+1)^(1/2)))+1/16/c^5*a^2/d^3/(c*x-1)^2+5/16/c^5*a^2/d^3/(c*x-1)-1/16/c^5*a^2/d^3/(c*x+1)^2+5/16
/c^5*a^2/d^3/(c*x+1)-3/16/c^5*a^2/d^3*ln(c*x-1)+3/16/c^5*a^2/d^3*ln(c*x+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{16} \, a^{2}{\left (\frac{2 \,{\left (5 \, c^{2} x^{3} - 3 \, x\right )}}{c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}} + \frac{3 \, \log \left (c x + 1\right )}{c^{5} d^{3}} - \frac{3 \, \log \left (c x - 1\right )}{c^{5} d^{3}}\right )} + \frac{3 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (c x + 1\right ) - 3 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (-c x + 1\right ) + 2 \,{\left (5 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} - 2 \,{\left (c^{9} d^{3} x^{4} - 2 \, c^{7} d^{3} x^{2} + c^{5} d^{3}\right )} \int \frac{16 \, a b c^{4} x^{4} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (3 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 3 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) + 2 \,{\left (5 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{10} d^{3} x^{6} - 3 \, c^{8} d^{3} x^{4} + 3 \, c^{6} d^{3} x^{2} - c^{4} d^{3}}\,{d x}}{16 \,{\left (c^{9} d^{3} x^{4} - 2 \, c^{7} d^{3} x^{2} + c^{5} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

1/16*a^2*(2*(5*c^2*x^3 - 3*x)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) + 3*log(c*x + 1)/(c^5*d^3) - 3*log(c*x -
1)/(c^5*d^3)) + 1/16*(3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(
c*x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) +
2*(5*b^2*c^3*x^3 - 3*b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 16*(c^9*d^3*x^4 - 2*c^7*d^3*x^2 +
c^5*d^3)*integrate(-1/8*(16*a*b*c^4*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (3*(b^2*c^4*x^4 - 2*b^2*
c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)
*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) + 2*(5*b^2*c^3*x^3 - 3*b^2*c*x)*arctan2(c*x, sqrt(c*
x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^10*d^3*x^6 - 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 - c^4*d^3)
, x))/(c^9*d^3*x^4 - 2*c^7*d^3*x^2 + c^5*d^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{2} x^{4} \arcsin \left (c x\right )^{2} + 2 \, a b x^{4} \arcsin \left (c x\right ) + a^{2} x^{4}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a^{2} x^{4}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac{b^{2} x^{4} \operatorname{asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac{2 a b x^{4} \operatorname{asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**3,x)

[Out]

-(Integral(a**2*x**4/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(b**2*x**4*asin(c*x)**2/(c**6*x
**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(2*a*b*x**4*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**
2 - 1), x))/d**3

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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(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

Timed out