3.2.99 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x^3 (d-c^2 d x^2)^2} \, dx\) [199]

Optimal. Leaf size=270 \[ -\frac {b c (a+b \text {ArcSin}(c x))}{d^2 x \sqrt {1-c^2 x^2}}+\frac {c^2 (a+b \text {ArcSin}(c x))^2}{d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {2 i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{d^2}-\frac {2 i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{d^2}-\frac {b^2 c^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right )}{d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 15, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4789, 4793, 4769, 4504, 4268, 2611, 2320, 6724, 4745, 266, 277, 197, 4779, 457, 78} \begin {gather*} \frac {2 i b c^2 \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2}-\frac {2 i b c^2 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2}+\frac {c^2 (a+b \text {ArcSin}(c x))^2}{d^2 \left (1-c^2 x^2\right )}-\frac {b c (a+b \text {ArcSin}(c x))}{d^2 x \sqrt {1-c^2 x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 i \text {ArcSin}(c x)}\right )}{d^2}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {b^2 c^2 \log (x)}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 277

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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 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 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 4504

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

Rule 4745

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

Rule 4769

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

Rule 4779

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

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

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Mathematica [A]
time = 1.10, size = 430, normalized size = 1.59 \begin {gather*} \frac {-\frac {a^2}{x^2}+\frac {a^2 c^2}{1-c^2 x^2}+4 a^2 c^2 \log (x)-2 a^2 c^2 \log \left (1-c^2 x^2\right )+2 a b \left (-\frac {c^3 x}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {1-c^2 x^2}}{x}-\frac {\text {ArcSin}(c x)}{x^2}+\frac {c^2 \text {ArcSin}(c x)}{1-c^2 x^2}+2 c^2 \left (2 \text {ArcSin}(c x) \left (\log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-\log \left (1+e^{2 i \text {ArcSin}(c x)}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )\right )\right )+b^2 c^2 \left (-\frac {2 c x \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}-\frac {2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c x}-\frac {\text {ArcSin}(c x)^2}{c^2 x^2}+\frac {\text {ArcSin}(c x)^2}{1-c^2 x^2}+4 \text {ArcSin}(c x)^2 \left (\log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-\log \left (1+e^{2 i \text {ArcSin}(c x)}\right )\right )+2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+4 i \text {ArcSin}(c x) \left (\text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )+2 \left (-\text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )+\text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right )\right )\right )}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2/x^2) + (a^2*c^2)/(1 - c^2*x^2) + 4*a^2*c^2*Log[x] - 2*a^2*c^2*Log[1 - c^2*x^2] + 2*a*b*(-((c^3*x)/Sqrt[
1 - c^2*x^2]) - (c*Sqrt[1 - c^2*x^2])/x - ArcSin[c*x]/x^2 + (c^2*ArcSin[c*x])/(1 - c^2*x^2) + 2*c^2*(2*ArcSin[
c*x]*(Log[1 - E^((2*I)*ArcSin[c*x])] - Log[1 + E^((2*I)*ArcSin[c*x])]) + I*(PolyLog[2, -E^((2*I)*ArcSin[c*x])]
- PolyLog[2, E^((2*I)*ArcSin[c*x])]))) + b^2*c^2*((-2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (2*Sqrt[1 - c^2*x^
2]*ArcSin[c*x])/(c*x) - ArcSin[c*x]^2/(c^2*x^2) + ArcSin[c*x]^2/(1 - c^2*x^2) + 4*ArcSin[c*x]^2*(Log[1 - E^((2
*I)*ArcSin[c*x])] - Log[1 + E^((2*I)*ArcSin[c*x])]) + 2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + (4*I)*ArcSin[c*x]*(Poly
Log[2, -E^((2*I)*ArcSin[c*x])] - PolyLog[2, E^((2*I)*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^((2*I)*ArcSin[c*x])] +
PolyLog[3, E^((2*I)*ArcSin[c*x])])))/(2*d^2)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 844 vs. \(2 (312 ) = 624\).
time = 0.28, size = 845, normalized size = 3.13

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a^{2}}{4 d^{2} \left (c x -1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2} \ln \left (c x +1\right )}{d^{2}}-\frac {a^{2} \ln \left (c x -1\right )}{d^{2}}+\frac {a^{2}}{4 d^{2} \left (c x +1\right )}+\frac {2 a^{2} \ln \left (c x \right )}{d^{2}}+\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2}}{2 d^{2} c^{2} x^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d^{2}}+\frac {b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c x \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}\right )\)	\(845\)
default	\(c^{2} \left (-\frac {a^{2}}{4 d^{2} \left (c x -1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}-\frac {b^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2} \ln \left (c x +1\right )}{d^{2}}-\frac {a^{2} \ln \left (c x -1\right )}{d^{2}}+\frac {a^{2}}{4 d^{2} \left (c x +1\right )}+\frac {2 a^{2} \ln \left (c x \right )}{d^{2}}+\frac {2 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}+\frac {2 i a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {a^{2}}{2 d^{2} c^{2} x^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d^{2}}+\frac {b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c x \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 i b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {4 a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}\right )\)	\(845\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*(2*c^2*log(c*x + 1)/d^2 + 2*c^2*log(c*x - 1)/d^2 - 4*c^2*log(x)/d^2 + (2*c^2*x^2 - 1)/(c^2*d^2*x^4 -
d^2*x^2)) + integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqr
t(-c*x + 1)))/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/x^3/(-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^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(b**2*asin(c*x)**2/(c**4*x**7 - 2*c**2*x**5 + x*
*3), x) + Integral(2*a*b*asin(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3), x))/d**2

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/x^3/(-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^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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