\(\int \frac {(a+b \arcsin (c x))^2}{x^2 (d-c^2 d x^2)^2} \, dx\) [198]

Optimal result

Integrand size = 27, antiderivative size = 324 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \]

[Out]

-(a+b*arcsin(c*x))^2/d^2/x/(-c^2*x^2+1)+3/2*c^2*x*(a+b*arcsin(c*x))^2/d^2/(-c^2*x^2+1)-3*I*c*(a+b*arcsin(c*x))
^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d^2-4*b*c*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/d^2+b^2*c*ar
ctanh(c*x)/d^2+2*I*b^2*c*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d^2+3*I*b*c*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*
x+(-c^2*x^2+1)^(1/2)))/d^2-3*I*b*c*(a+b*arcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2-2*I*b^2*c*pol
ylog(2,I*c*x+(-c^2*x^2+1)^(1/2))/d^2-3*b^2*c*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2+3*b^2*c*polylog(3,I*
(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2-b*c*(a+b*arcsin(c*x))/d^2/(-c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {4789, 4747, 4749, 4266, 2611, 2320, 6724, 4767, 212, 4793, 4803, 4268, 2317, 2438} \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {3 i c \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d^2}-\frac {4 b c \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2} \]

[In]

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

[Out]

-((b*c*(a + b*ArcSin[c*x]))/(d^2*Sqrt[1 - c^2*x^2])) - (a + b*ArcSin[c*x])^2/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*
(a + b*ArcSin[c*x])^2)/(2*d^2*(1 - c^2*x^2)) - ((3*I)*c*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/d^2 -
(4*b*c*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/d^2 + (b^2*c*ArcTanh[c*x])/d^2 + ((2*I)*b^2*c*PolyLog[
2, -E^(I*ArcSin[c*x])])/d^2 + ((3*I)*b*c*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^2 - ((3*I)*
b*c*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^2 - ((2*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])])/d^2
- (3*b^2*c*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/d^2 + (3*b^2*c*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 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}{d^2 x \left (1-c^2 x^2\right )}+\left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2} \\ & = \frac {2 b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{d^2}-\frac {\left (2 b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2}-\frac {\left (3 b c^3\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 b^2 c \text {arctanh}(c x)}{d^2}+\frac {(3 c) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arcsin (c x)\right )}{2 d^2}+\frac {(2 b c) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{d^2}+\frac {\left (3 b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}-\frac {(3 b c) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {(3 b c) \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 (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (3 b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (3 b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 b^2 c \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. \(1161\) vs. \(2(324)=648\).

Time = 10.44 (sec) , antiderivative size = 1161, normalized size of antiderivative = 3.58 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {a^2}{d^2 x}-\frac {a^2 c^2 x}{2 d^2 \left (-1+c^2 x^2\right )}-\frac {3 a^2 c \log (1-c x)}{4 d^2}+\frac {3 a^2 c \log (1+c x)}{4 d^2}+\frac {2 a b c \left (\frac {\sqrt {1-c^2 x^2}-\arcsin (c x)}{4 (-1+c x)}-\frac {\arcsin (c x)}{c x}-\frac {\sqrt {1-c^2 x^2}+\arcsin (c x)}{4 (1+c x)}-\text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {3}{4} \left (\frac {3}{2} i \pi \arcsin (c x)-\frac {1}{2} i \arcsin (c x)^2+2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )-\pi \log \left (1+i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )+\frac {3}{4} \left (\frac {1}{2} i \pi \arcsin (c x)-\frac {1}{2} i \arcsin (c x)^2+2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )+\pi \log \left (1-i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )\right )}{d^2}+\frac {b^2 c \left (-4 \arcsin (c x)-2 \arcsin (c x)^2 \cot \left (\frac {1}{2} \arcsin (c x)\right )+8 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+6 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )+6 \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 )-6 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )-6 \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 )+6 \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 )-8 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+6 \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 )-6 \pi \arcsin (c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+6 \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 )+4 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-6 \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 )-6 \pi \arcsin (c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+8 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+12 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-12 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-8 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-12 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )+12 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )+\frac {\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 {4 \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 {\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 {4 \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 )}-2 \arcsin (c x)^2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{4 d^2} \]

[In]

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

[Out]

-(a^2/(d^2*x)) - (a^2*c^2*x)/(2*d^2*(-1 + c^2*x^2)) - (3*a^2*c*Log[1 - c*x])/(4*d^2) + (3*a^2*c*Log[1 + c*x])/
(4*d^2) + (2*a*b*c*((Sqrt[1 - c^2*x^2] - ArcSin[c*x])/(4*(-1 + c*x)) - ArcSin[c*x]/(c*x) - (Sqrt[1 - c^2*x^2]
+ ArcSin[c*x])/(4*(1 + c*x)) - ArcTanh[Sqrt[1 - c^2*x^2]] - (3*(((3*I)/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2
+ 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[
c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] + Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, (-I)*E^(I*ArcS
in[c*x])]))/4 + (3*((I/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + Pi*Log[1
- I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] - Pi*Log[S
in[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]))/4))/d^2 + (b^2*c*(-4*ArcSin[c*x] - 2*Arc
Sin[c*x]^2*Cot[ArcSin[c*x]/2] + 8*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcS
in[c*x])] + 6*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] - 6*ArcSin[
c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcS
in[c*x])] + 6*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 8*ArcSin[
c*x]*Log[1 + E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin
[c*x]))] - 6*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]
+ 6*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2
]] - 6*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 6*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*
x])/4]] + (8*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (12*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (12*I
)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] - (8*I)*PolyLog[2, E^(I*ArcSin[c*x])] - 12*PolyLog[3, (-I)*E^(I*
ArcSin[c*x])] + 12*PolyLog[3, I*E^(I*ArcSin[c*x])] + ArcSin[c*x]^2/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2
- (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - ArcSin[c*x]^2/(Cos[ArcSin[c*
x]/2] + Sin[ArcSin[c*x]/2])^2 + (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) -
2*ArcSin[c*x]^2*Tan[ArcSin[c*x]/2]))/(4*d^2)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.87

method	result	size
derivativedivides	\(c \left (\frac {a^{2} \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(605\)
default	\(c \left (\frac {a^{2} \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(605\)
parts	\(\frac {a^{2} \left (-\frac {1}{x}-\frac {c}{4 \left (c x -1\right )}-\frac {3 c \ln \left (c x -1\right )}{4}-\frac {c}{4 \left (c x +1\right )}+\frac {3 c \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} c \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b c \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\)	\(606\)

[In]

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

[Out]

c*(a^2/d^2*(-1/c/x-1/4/(c*x-1)-3/4*ln(c*x-1)-1/4/(c*x+1)+3/4*ln(c*x+1))+b^2/d^2*(-1/2/c/x/(c^2*x^2-1)*(3*c^2*x
^2*arcsin(c*x)-2*c*x*(-c^2*x^2+1)^(1/2)-2*arcsin(c*x))*arcsin(c*x)-3/2*arcsin(c*x)^2*ln(1+I*(I*c*x+(-c^2*x^2+1
)^(1/2)))+3*I*arcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+
3/2*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3*I*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3
*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*I*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+2*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))-2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-1/2*(3*c^2
*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-2*arcsin(c*x))/c/x/(c^2*x^2-1)-3/2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2
+1)^(1/2)))+ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+3/2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-ln(1+I*c*x+(-c^2
*x^2+1)^(1/2))+3/2*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/2*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/4*a^2*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 3*c*log(c*x - 1)/d^2) + 1/4*(3*(b^2
*c^3*x^3 - b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 3*(b^2*c^3*x^3 - b^2*c*x)*arct
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(3*b^2*c^2*x^2 - 2*b^2)*arctan2(c*x, sqrt(c*x + 1)*
sqrt(-c*x + 1))^2 + 4*(c^2*d^2*x^3 - d^2*x)*integrate(1/2*(4*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) +
(3*(b^2*c^4*x^4 - b^2*c^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 3*(b^2*c^4*x^4 - b^2*
c^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(3*b^2*c^3*x^3 - 2*b^2*c*x)*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x))/(c^
2*d^2*x^3 - d^2*x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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