\(\int \frac {(a+b \arcsin (c x))^2}{x^4 (d-c^2 d x^2)^3} \, dx\) [209]

Optimal result

Integrand size = 27, antiderivative size = 572 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3} \]

[Out]

-1/2*b^2*c^2/d^3/x+1/6*b^2*c^2/d^3/x/(-c^2*x^2+1)-1/12*b^2*c^4*x/d^3/(-c^2*x^2+1)+1/6*b*c^3*(a+b*arcsin(c*x))/
d^3/(-c^2*x^2+1)^(3/2)-1/3*b*c*(a+b*arcsin(c*x))/d^3/x^2/(-c^2*x^2+1)^(3/2)-1/3*(a+b*arcsin(c*x))^2/d^3/x^3/(-
c^2*x^2+1)^2-7/3*c^2*(a+b*arcsin(c*x))^2/d^3/x/(-c^2*x^2+1)^2+35/12*c^4*x*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)
^2+35/8*c^4*x*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)+35/4*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^
2+1)^(1/2)))/d^3-38/3*b*c^3*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/d^3+17/6*b^2*c^3*arctanh(c*x)/
d^3-19/3*I*b^2*c^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))/d^3-35/4*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,I*(I*c*x+(
-c^2*x^2+1)^(1/2)))/d^3+19/3*I*b^2*c^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d^3-35/4*I*c^3*(a+b*arcsin(c*x))^2
*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d^3-35/4*b^2*c^3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3+35/4*b^2*c^3*p
olylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3-29/12*b*c^3*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {4789, 4747, 4749, 4266, 2611, 2320, 6724, 4767, 212, 205, 4793, 4803, 4268, 2317, 2438, 296, 331} \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {35 i c^3 \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{4 d^3}-\frac {38 b c^3 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 d^3}+\frac {35 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{4 d^3}-\frac {35 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{4 d^3}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^2}{2 d^3 x}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )} \]

[In]

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

[Out]

-1/2*(b^2*c^2)/(d^3*x) + (b^2*c^2)/(6*d^3*x*(1 - c^2*x^2)) - (b^2*c^4*x)/(12*d^3*(1 - c^2*x^2)) + (b*c^3*(a +
b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2)) - (b*c*(a + b*ArcSin[c*x]))/(3*d^3*x^2*(1 - c^2*x^2)^(3/2)) - (29*
b*c^3*(a + b*ArcSin[c*x]))/(12*d^3*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7
*c^2*(a + b*ArcSin[c*x])^2)/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x])^2)/(12*d^3*(1 - c^2*x^2)
^2) + (35*c^4*x*(a + b*ArcSin[c*x])^2)/(8*d^3*(1 - c^2*x^2)) - (((35*I)/4)*c^3*(a + b*ArcSin[c*x])^2*ArcTan[E^
(I*ArcSin[c*x])])/d^3 - (38*b*c^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(3*d^3) + (17*b^2*c^3*ArcTan
h[c*x])/(6*d^3) + (((19*I)/3)*b^2*c^3*PolyLog[2, -E^(I*ArcSin[c*x])])/d^3 + (((35*I)/4)*b*c^3*(a + b*ArcSin[c*
x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^3 - (((35*I)/4)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c
*x])])/d^3 - (((19*I)/3)*b^2*c^3*PolyLog[2, E^(I*ArcSin[c*x])])/d^3 - (35*b^2*c^3*PolyLog[3, (-I)*E^(I*ArcSin[
c*x])])/(4*d^3) + (35*b^2*c^3*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*d^3)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

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

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2317

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 2320

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

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 4266

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 4268

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

Rule 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(p
+ 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a +
b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(
p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rule 4749

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 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4789

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(m
+ 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d + e*x
^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4793

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(p + 1)))*Simp[(d + e*
x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; Fre
eQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rule 4803

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

Rule 6724

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (7 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x^3 \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3} \\ & = -\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (35 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )^2} \, dx}{3 d^3}+\frac {\left (5 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3} \\ & = \frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}+\frac {19 b c^3 (a+b \arcsin (c x))}{9 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{2 d^3}+\frac {\left (5 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{9 d^3}-\frac {\left (14 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{9 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{6 d^3}+\frac {\left (35 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {19 b^2 c^4 x}{18 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {19 b c^3 (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {\left (5 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{18 d^3}+\frac {\left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{2 d^3}-\frac {\left (7 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{9 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{18 d^3}-\frac {\left (14 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 d^3}+\frac {\left (35 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{8 d^2} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {62 b^2 c^3 \text {arctanh}(c x)}{9 d^3}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arcsin (c x)\right )}{8 d^3}+\frac {\left (5 b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{3 d^3}+\frac {\left (14 b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{3 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{36 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{4 d^3} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}-\frac {\left (35 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3}+\frac {\left (35 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3}-\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3}+\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3}-\frac {\left (14 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3}+\frac {\left (14 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {\left (5 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {\left (5 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {\left (14 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {\left (14 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {\left (35 i b^2 c^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3}+\frac {\left (35 i b^2 c^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {\left (35 b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {\left (35 b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{4 d^3} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1657\) vs. \(2(572)=1144\).

Time = 12.71 (sec) , antiderivative size = 1657, normalized size of antiderivative = 2.90 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {a^2}{3 d^3 x^3}-\frac {3 a^2 c^2}{d^3 x}+\frac {a^2 c^4 x}{4 d^3 \left (-1+c^2 x^2\right )^2}-\frac {11 a^2 c^4 x}{8 d^3 \left (-1+c^2 x^2\right )}-\frac {35 a^2 c^3 \log (1-c x)}{16 d^3}+\frac {35 a^2 c^3 \log (1+c x)}{16 d^3}-\frac {2 a b \left (\frac {c^3 \left ((2-c x) \sqrt {1-c^2 x^2}-3 \arcsin (c x)\right )}{48 (-1+c x)^2}-\frac {11 c^3 \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{16 (-1+c x)}+\frac {11 c^4 \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{16 \left (c+c^2 x\right )}+\frac {c^3 \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{48 (1+c x)^2}-3 c^2 \left (-\frac {\arcsin (c x)}{x}-c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )+\frac {c x \sqrt {1-c^2 x^2}+2 \arcsin (c x)+c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 x^3}+\frac {35}{16} c^4 \left (\frac {3 i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}-\frac {\pi \log \left (1+i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}+\frac {\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c}\right )-\frac {35}{16} c^4 \left (\frac {i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}+\frac {\pi \log \left (1-i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}-\frac {\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c}\right )\right )}{d^3}-\frac {b^2 c^3 \left (-\frac {19}{3} i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+\frac {19}{3} i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+\frac {1}{24} \left (68 \arcsin (c x)+35 \arcsin (c x)^3-105 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )-105 \pi \arcsin (c x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \arcsin (c x)} \left (1-i e^{i \arcsin (c x)}\right )\right )+105 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )+105 \arcsin (c x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \arcsin (c x)} \left (-i+e^{i \arcsin (c x)}\right )\right )-105 \pi \arcsin (c x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \arcsin (c x)} \left (-i+e^{i \arcsin (c x)}\right )\right )-105 \arcsin (c x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \arcsin (c x)} \left ((1+i)+(1-i) e^{i \arcsin (c x)}\right )\right )+105 \pi \arcsin (c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+68 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-105 \arcsin (c x)^2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-68 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+105 \arcsin (c x)^2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+105 \pi \arcsin (c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-210 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+210 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+210 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-210 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )+\frac {24-204 c x \arcsin (c x)+204 \arcsin (c x)^2-105 c x \arcsin (c x)^3+\left (20+658 \arcsin (c x)^2\right ) \cos (2 \arcsin (c x))-4 \left (6+35 \arcsin (c x)^2\right ) \cos (4 \arcsin (c x))-20 \cos (6 \arcsin (c x))-210 \arcsin (c x)^2 \cos (6 \arcsin (c x))-456 c x \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+456 c x \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+540 \arcsin (c x) \sin (2 \arcsin (c x))-204 \arcsin (c x) \sin (3 \arcsin (c x))-105 \arcsin (c x)^3 \sin (3 \arcsin (c x))-456 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))+456 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))+32 \arcsin (c x) \sin (4 \arcsin (c x))+68 \arcsin (c x) \sin (5 \arcsin (c x))+35 \arcsin (c x)^3 \sin (5 \arcsin (c x))+152 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (5 \arcsin (c x))-152 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (5 \arcsin (c x))-116 \arcsin (c x) \sin (6 \arcsin (c x))+68 \arcsin (c x) \sin (7 \arcsin (c x))+35 \arcsin (c x)^3 \sin (7 \arcsin (c x))+152 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (7 \arcsin (c x))-152 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (7 \arcsin (c x))}{1536 c^3 x^3 \left (1-c^2 x^2\right )^2}\right )}{d^3} \]

[In]

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

[Out]

-1/3*a^2/(d^3*x^3) - (3*a^2*c^2)/(d^3*x) + (a^2*c^4*x)/(4*d^3*(-1 + c^2*x^2)^2) - (11*a^2*c^4*x)/(8*d^3*(-1 +
c^2*x^2)) - (35*a^2*c^3*Log[1 - c*x])/(16*d^3) + (35*a^2*c^3*Log[1 + c*x])/(16*d^3) - (2*a*b*((c^3*((2 - c*x)*
Sqrt[1 - c^2*x^2] - 3*ArcSin[c*x]))/(48*(-1 + c*x)^2) - (11*c^3*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(16*(-1 + c
*x)) + (11*c^4*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(16*(c + c^2*x)) + (c^3*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*Arc
Sin[c*x]))/(48*(1 + c*x)^2) - 3*c^2*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^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*x^3) + (35*c^4*((((3*I)/2)*Pi*ArcSin[c*x])/c - ((I/2
)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c - (Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[
c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]
])/c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c))/16 - (35*c^4*(((I/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c
*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1
- I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*
I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/c))/16))/d^3 - (b^2*c^3*(((-19*I)/3)*PolyLog[2, -E^(I*ArcSin[c*x])] + ((19
*I)/3)*PolyLog[2, E^(I*ArcSin[c*x])] + (68*ArcSin[c*x] + 35*ArcSin[c*x]^3 - 105*ArcSin[c*x]^2*Log[1 - I*E^(I*A
rcSin[c*x])] - 105*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] + 105*
ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 105*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((
I/2)*ArcSin[c*x])] - 105*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])]
- 105*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] + 105*Pi*ArcSin[c*x]*
Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 68*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 105*ArcSin[c*x]^2*Log[Co
s[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 68*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 105*ArcSin[c*x]^2*L
og[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 105*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (210*I)*Ar
cSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (210*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + 210*PolyL
og[3, (-I)*E^(I*ArcSin[c*x])] - 210*PolyLog[3, I*E^(I*ArcSin[c*x])])/24 + (24 - 204*c*x*ArcSin[c*x] + 204*ArcS
in[c*x]^2 - 105*c*x*ArcSin[c*x]^3 + (20 + 658*ArcSin[c*x]^2)*Cos[2*ArcSin[c*x]] - 4*(6 + 35*ArcSin[c*x]^2)*Cos
[4*ArcSin[c*x]] - 20*Cos[6*ArcSin[c*x]] - 210*ArcSin[c*x]^2*Cos[6*ArcSin[c*x]] - 456*c*x*ArcSin[c*x]*Log[1 - E
^(I*ArcSin[c*x])] + 456*c*x*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + 540*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - 204*
ArcSin[c*x]*Sin[3*ArcSin[c*x]] - 105*ArcSin[c*x]^3*Sin[3*ArcSin[c*x]] - 456*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*
x])]*Sin[3*ArcSin[c*x]] + 456*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[3*ArcSin[c*x]] + 32*ArcSin[c*x]*Sin[4
*ArcSin[c*x]] + 68*ArcSin[c*x]*Sin[5*ArcSin[c*x]] + 35*ArcSin[c*x]^3*Sin[5*ArcSin[c*x]] + 152*ArcSin[c*x]*Log[
1 - E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]] - 152*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]] - 11
6*ArcSin[c*x]*Sin[6*ArcSin[c*x]] + 68*ArcSin[c*x]*Sin[7*ArcSin[c*x]] + 35*ArcSin[c*x]^3*Sin[7*ArcSin[c*x]] + 1
52*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])]*Sin[7*ArcSin[c*x]] - 152*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[
7*ArcSin[c*x]])/(1536*c^3*x^3*(1 - c^2*x^2)^2)))/d^3

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.44

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a^{2} \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {105 \arcsin \left (c x \right )^{2} c^{6} x^{6}-58 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}-175 \arcsin \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}+54 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+56 \arcsin \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +8 \arcsin \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {35 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {19 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {19 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {19 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\)	\(821\)
default	\(c^{3} \left (-\frac {a^{2} \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {105 \arcsin \left (c x \right )^{2} c^{6} x^{6}-58 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}-175 \arcsin \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}+54 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+56 \arcsin \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +8 \arcsin \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {35 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {19 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {19 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {19 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\)	\(821\)
parts	\(-\frac {a^{2} \left (\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}-\frac {c^{3}}{16 \left (c x -1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x -1\right )}+\frac {35 c^{3} \ln \left (c x -1\right )}{16}+\frac {c^{3}}{16 \left (c x +1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x +1\right )}-\frac {35 c^{3} \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} c^{3} \left (\frac {105 \arcsin \left (c x \right )^{2} c^{6} x^{6}-58 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}-175 \arcsin \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}+54 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+56 \arcsin \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +8 \arcsin \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {35 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {19 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {19 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {19 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \,c^{3} \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\)	\(838\)

[In]

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

[Out]

c^3*(-a^2/d^3*(1/3/c^3/x^3+3/c/x-1/16/(c*x-1)^2+11/16/(c*x-1)+35/16*ln(c*x-1)+1/16/(c*x+1)^2+11/16/(c*x+1)-35/
16*ln(c*x+1))-b^2/d^3*(1/24*(105*arcsin(c*x)^2*c^6*x^6-58*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^5*x^5-175*arcsin(c*
x)^2*x^4*c^4+10*c^6*x^6+54*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3*x^3+56*arcsin(c*x)^2*x^2*c^2-18*c^4*x^4+8*(-c^2*
x^2+1)^(1/2)*arcsin(c*x)*x*c+8*arcsin(c*x)^2+8*c^2*x^2)/(c^4*x^4-2*c^2*x^2+1)/c^3/x^3+35/8*arcsin(c*x)^2*ln(1+
I*(I*c*x+(-c^2*x^2+1)^(1/2)))-35/4*I*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/4*polylog(3,-I*(I
*c*x+(-c^2*x^2+1)^(1/2)))-35/8*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/4*I*arcsin(c*x)*polylog(2,I
*(I*c*x+(-c^2*x^2+1)^(1/2)))-35/4*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-19/3*I*dilog(I*c*x+(-c^2*x^2+1)^(1/2
))-19/3*I*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+17/3*I*arctan(I*c*x+(-c^2*x^2+1)^(1/2))+19/3*arcsin(c*x)*ln(1+I*c*
x+(-c^2*x^2+1)^(1/2)))-2*a*b/d^3*(1/24*(105*arcsin(c*x)*c^6*x^6-29*c^5*x^5*(-c^2*x^2+1)^(1/2)-175*c^4*x^4*arcs
in(c*x)+27*c^3*x^3*(-c^2*x^2+1)^(1/2)+56*c^2*x^2*arcsin(c*x)+4*c*x*(-c^2*x^2+1)^(1/2)+8*arcsin(c*x))/(c^4*x^4-
2*c^2*x^2+1)/c^3/x^3+35/8*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-19/6*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-3
5/8*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+19/6*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-35/8*I*dilog(1+I*(I*c*x
+(-c^2*x^2+1)^(1/2)))+35/8*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \]

[In]

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

[Out]

integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^
4), x)

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx}{d^{3}} \]

[In]

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

[Out]

-(Integral(a**2/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4), x) + Integral(b**2*asin(c*x)**2/(c**6*x**10 -
3*c**4*x**8 + 3*c**2*x**6 - x**4), x) + Integral(2*a*b*asin(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x*
*4), x))/d**3

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \]

[In]

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

[Out]

1/48*a^2*(105*c^3*log(c*x + 1)/d^3 - 105*c^3*log(c*x - 1)/d^3 - 2*(105*c^6*x^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)
/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)) + 1/48*(105*(b^2*c^7*x^7 - 2*b^2*c^5*x^5 + b^2*c^3*x^3)*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 105*(b^2*c^7*x^7 - 2*b^2*c^5*x^5 + b^2*c^3*x^3)*arctan2(c*x, s
qrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(105*b^2*c^6*x^6 - 175*b^2*c^4*x^4 + 56*b^2*c^2*x^2 + 8*b^2)*
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 48*(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)*integrate(-1/24*(48*
a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (105*(b^2*c^8*x^8 - 2*b^2*c^6*x^6 + b^2*c^4*x^4)*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 105*(b^2*c^8*x^8 - 2*b^2*c^6*x^6 + b^2*c^4*x^4)*arctan2(c*x, sqr
t(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(105*b^2*c^7*x^7 - 175*b^2*c^5*x^5 + 56*b^2*c^3*x^3 + 8*b^2*c*x)*
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^
2*d^3*x^6 - d^3*x^4), x))/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]

[In]

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

[Out]

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