\(\int \frac {(a+b \arcsin (c x))^2}{x^4 (d-c^2 d x^2)^2} \, dx\) [200]

Optimal result

Integrand size = 27, antiderivative size = 439 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \]

[Out]

-1/3*b^2*c^2/d^2/x-1/3*(a+b*arcsin(c*x))^2/d^2/x^3/(-c^2*x^2+1)-5/3*c^2*(a+b*arcsin(c*x))^2/d^2/x/(-c^2*x^2+1)
+5/2*c^4*x*(a+b*arcsin(c*x))^2/d^2/(-c^2*x^2+1)-5*I*c^3*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d
^2-26/3*b*c^3*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/d^2+b^2*c^3*arctanh(c*x)/d^2+13/3*I*b^2*c^3*
polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d^2+5*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/
d^2-5*I*b*c^3*(a+b*arcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2-13/3*I*b^2*c^3*polylog(2,I*c*x+(-c
^2*x^2+1)^(1/2))/d^2-5*b^2*c^3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2+5*b^2*c^3*polylog(3,I*(I*c*x+(-c^2
*x^2+1)^(1/2)))/d^2-2/3*b*c^3*(a+b*arcsin(c*x))/d^2/(-c^2*x^2+1)^(1/2)-1/3*b*c*(a+b*arcsin(c*x))/d^2/x^2/(-c^2
*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4789, 4747, 4749, 4266, 2611, 2320, 6724, 4767, 212, 4793, 4803, 4268, 2317, 2438, 331} \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {5 i c^3 \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d^2}-\frac {26 b c^3 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 d^2}+\frac {5 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {5 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}-\frac {b^2 c^2}{3 d^2 x} \]

[In]

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

[Out]

-1/3*(b^2*c^2)/(d^2*x) - (2*b*c^3*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(
3*d^2*x^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(3*d^2*x^3*(1 - c^2*x^2)) - (5*c^2*(a + b*ArcSin[c*x])^2)
/(3*d^2*x*(1 - c^2*x^2)) + (5*c^4*x*(a + b*ArcSin[c*x])^2)/(2*d^2*(1 - c^2*x^2)) - ((5*I)*c^3*(a + b*ArcSin[c*
x])^2*ArcTan[E^(I*ArcSin[c*x])])/d^2 - (26*b*c^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(3*d^2) + (b^
2*c^3*ArcTanh[c*x])/d^2 + (((13*I)/3)*b^2*c^3*PolyLog[2, -E^(I*ArcSin[c*x])])/d^2 + ((5*I)*b*c^3*(a + b*ArcSin
[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^2 - ((5*I)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x
])])/d^2 - (((13*I)/3)*b^2*c^3*PolyLog[2, E^(I*ArcSin[c*x])])/d^2 - (5*b^2*c^3*PolyLog[3, (-I)*E^(I*ArcSin[c*x
])])/d^2 + (5*b^2*c^3*PolyLog[3, I*E^(I*ArcSin[c*x])])/d^2

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 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^2 x^3 \left (1-c^2 x^2\right )}+\frac {1}{3} \left (5 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x^3 \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\left (5 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{3 d^2}+\frac {\left (b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\left (10 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}+\frac {13 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{d^2}+\frac {\left (10 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^2}+\frac {\left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^2}-\frac {\left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2}-\frac {\left (10 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^2}-\frac {\left (5 b c^5\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\left (5 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {4 b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arcsin (c x)\right )}{2 d^2}+\frac {\left (b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{d^2}+\frac {\left (10 b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{3 d^2}+\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}-\frac {\left (5 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 )}{d^2}+\frac {\left (5 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 )}{d^2}-\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}-\frac {\left (10 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^2}+\frac {\left (10 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (10 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^2}-\frac {\left (10 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^2}-\frac {\left (5 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 )}{d^2}+\frac {\left (5 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 )}{d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \\ \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. \(1390\) vs. \(2(439)=878\).

Time = 12.42 (sec) , antiderivative size = 1390, normalized size of antiderivative = 3.17 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {a^2}{3 d^2 x^3}-\frac {2 a^2 c^2}{d^2 x}-\frac {a^2 c^4 x}{2 d^2 \left (-1+c^2 x^2\right )}-\frac {5 a^2 c^3 \log (1-c x)}{4 d^2}+\frac {5 a^2 c^3 \log (1+c x)}{4 d^2}+\frac {2 a b \left (\frac {c^3 \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{4 (-1+c x)}-\frac {c^4 \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{4 \left (c+c^2 x\right )}+2 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 {5}{4} 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 {5}{4} 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^2}+\frac {b^2 c^3 \left (-24 \arcsin (c x)-\frac {6 \arcsin (c x)^2}{-1+c x}-4 \cot \left (\frac {1}{2} \arcsin (c x)\right )-26 \arcsin (c x)^2 \cot \left (\frac {1}{2} \arcsin (c x)\right )-2 \arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-\frac {1}{2} c x \arcsin (c x)^2 \csc ^4\left (\frac {1}{2} \arcsin (c x)\right )+104 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+60 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )+60 \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 )-60 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )-60 \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 )+60 \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 )-104 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+60 \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 )-60 \pi \arcsin (c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-24 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+60 \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 )+24 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-60 \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 )-60 \pi \arcsin (c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+104 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+120 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-120 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-104 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-120 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )+120 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-\frac {24 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )}-\frac {8 \arcsin (c x)^2 \sin ^4\left (\frac {1}{2} \arcsin (c x)\right )}{c^3 x^3}-\frac {6 \arcsin (c x)^2}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}+\frac {24 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}-4 \tan \left (\frac {1}{2} \arcsin (c x)\right )-26 \arcsin (c x)^2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{24 d^2} \]

[In]

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

[Out]

-1/3*a^2/(d^2*x^3) - (2*a^2*c^2)/(d^2*x) - (a^2*c^4*x)/(2*d^2*(-1 + c^2*x^2)) - (5*a^2*c^3*Log[1 - c*x])/(4*d^
2) + (5*a^2*c^3*Log[1 + c*x])/(4*d^2) + (2*a*b*((c^3*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(4*(-1 + c*x)) - (c^4*
(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(4*(c + c^2*x)) + 2*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) - (5*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))/4 + (5*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))/4))/d^2 + (b^2*c^3*(-24*ArcSin[c*x] - (6*ArcSin[c*
x]^2)/(-1 + c*x) - 4*Cot[ArcSin[c*x]/2] - 26*ArcSin[c*x]^2*Cot[ArcSin[c*x]/2] - 2*ArcSin[c*x]*Csc[ArcSin[c*x]/
2]^2 - (c*x*ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^4)/2 + 104*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 60*ArcSin[c*x
]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 60*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*Ar
cSin[c*x]))] - 60*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] - 60*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*Ar
cSin[c*x])))/E^((I/2)*ArcSin[c*x])] + 60*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2
)*ArcSin[c*x])] - 104*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + 60*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*Ar
cSin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] - 60*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 24*Log[Cos[ArcS
in[c*x]/2] - Sin[ArcSin[c*x]/2]] + 60*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 24*Log[Cos[
ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 60*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 60*Pi*A
rcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (104*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (120*I)*ArcSin[c*x]*Pol
yLog[2, (-I)*E^(I*ArcSin[c*x])] - (120*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] - (104*I)*PolyLog[2, E^(
I*ArcSin[c*x])] - 120*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 120*PolyLog[3, I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]
*Sec[ArcSin[c*x]/2]^2 - (24*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (8*Arc
Sin[c*x]^2*Sin[ArcSin[c*x]/2]^4)/(c^3*x^3) - (6*ArcSin[c*x]^2)/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 + (
24*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 4*Tan[ArcSin[c*x]/2] - 26*ArcSi
n[c*x]^2*Tan[ArcSin[c*x]/2]))/(24*d^2)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.61

method	result	size
derivativedivides	\(c^{3} \left (\frac {a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{2}}+\frac {2 a b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(707\)
default	\(c^{3} \left (\frac {a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{2}}+\frac {2 a b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(707\)
parts	\(\frac {a^{2} \left (-\frac {1}{3 x^{3}}-\frac {2 c^{2}}{x}-\frac {c^{3}}{4 \left (c x -1\right )}-\frac {5 c^{3} \ln \left (c x -1\right )}{4}-\frac {c^{3}}{4 \left (c x +1\right )}+\frac {5 c^{3} \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} c^{3} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{2}}+\frac {2 a b \,c^{3} \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\)	\(718\)

[In]

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

[Out]

c^3*(a^2/d^2*(-1/3/c^3/x^3-2/c/x-1/4/(c*x-1)-5/4*ln(c*x-1)-1/4/(c*x+1)+5/4*ln(c*x+1))+b^2/d^2*(-1/6*(15*arcsin
(c*x)^2*x^4*c^4-4*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3*x^3-10*arcsin(c*x)^2*x^2*c^2+2*c^4*x^4-2*(-c^2*x^2+1)^(1/
2)*arcsin(c*x)*x*c-2*arcsin(c*x)^2-2*c^2*x^2)/c^3/x^3/(c^2*x^2-1)-5/2*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1)
^(1/2)))+5*I*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-5*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5
/2*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-5*I*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+5*
polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+13/3*I*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+13/3*I*dilog(1+I*c*x+(-c^2*x^2+
1)^(1/2))-2*I*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-13/3*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-1/
6*(15*c^4*x^4*arcsin(c*x)-2*c^3*x^3*(-c^2*x^2+1)^(1/2)-10*c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-2*arcsin(
c*x))/(c^2*x^2-1)/c^3/x^3-5/2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+13/6*ln(I*c*x+(-c^2*x^2+1)^(1/2)-
1)+5/2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-13/6*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+5/2*I*dilog(1+I*(I*c
*x+(-c^2*x^2+1)^(1/2)))-5/2*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/12*(15*c^3*log(c*x + 1)/d^2 - 15*c^3*log(c*x - 1)/d^2 - 2*(15*c^4*x^4 - 10*c^2*x^2 - 2)/(c^2*d^2*x^5 - d^2*x
^3))*a^2 + 1/12*(15*(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) - 15
*(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) - 2*(15*b^2*c^4*x^4 -
10*b^2*c^2*x^2 - 2*b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 12*(c^2*d^2*x^5 - d^2*x^3)*integrate(1/
6*(12*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (15*(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) - 15*(b^2*c^6*x^6 - b^2*c^4*x^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*l
og(-c*x + 1) - 2*(15*b^2*c^5*x^5 - 10*b^2*c^3*x^3 - 2*b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqr
t(c*x + 1)*sqrt(-c*x + 1))/(c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2*x^4), x))/(c^2*d^2*x^5 - d^2*x^3)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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