3.3.62 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x^3 (d-c^2 d x^2)^{5/2}} \, dx\) [262]

Optimal. Leaf size=752 \[ \frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c (a+b \text {ArcSin}(c x))}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {ArcSin}(c x))^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {ArcSin}(c x))^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

5/6*c^2*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)-1/2*(a+b*arcsin(c*x))^2/d/x^2/(-c^2*d*x^2+d)^(3/2)+1/3*b^2*
c^2/d^2/(-c^2*d*x^2+d)^(1/2)+5/2*c^2*(a+b*arcsin(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)-b*c*(a+b*arcsin(c*x))/d^2/x/
(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+2/3*b*c^3*x*(a+b*arcsin(c*x))/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1
/2)+26/3*I*b*c^2*(a+b*arcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2
)-5*c^2*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-b^2*
c^2*arctanh((-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+5*I*b*c^2*(a+b*arcsin(c*x))*polylo
g(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-13/3*I*b^2*c^2*polylog(2,-I*(I*c*x+
(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+13/3*I*b^2*c^2*polylog(2,I*(I*c*x+(-c^2*x^2+1
)^(1/2)))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-5*I*b*c^2*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)
^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-5*b^2*c^2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1
)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+5*b^2*c^2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*
x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.79, antiderivative size = 752, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 17, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {4789, 4793, 4803, 4268, 2611, 2320, 6724, 4749, 4266, 2317, 2438, 4747, 267, 272, 53, 65, 214} \begin {gather*} \frac {26 i b c^2 \sqrt {1-c^2 x^2} \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {ArcSin}(c x))^2}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c (a+b \text {ArcSin}(c x))}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {ArcSin}(c x))^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b c^3 x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(b^2*c^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(d^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]
) + (2*b*c^3*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcSin[c*x])
^2)/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcSin[c*x])^2/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcSin
[c*x])^2)/(2*d^2*Sqrt[d - c^2*d*x^2]) + (((26*I)/3)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*Ar
cSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*
x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/(d^2*Sqrt[d - c^2*d*x
^2]) + ((5*I)*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*
x^2]) - (((13*I)/3)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) +
(((13*I)/3)*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - ((5*I)*b*c^
2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*b^2*c^2*
Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (5*b^2*c^2*Sqrt[1 - c^2*x^2]*Pol
yLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 267

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

Rule 272

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

Rule 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 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*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {1}{2} \left (5 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (5 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{2 d}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {d-c^2 d x^2}} \, dx}{2 d^2}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^4 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^4 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 x \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {2 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {26 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 9.13, size = 1090, normalized size = 1.45 \begin {gather*} \sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a^2}{2 d^3 x^2}+\frac {a^2 c^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {2 a^2 c^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {5 a^2 c^2 \log (x)}{2 d^{5/2}}-\frac {5 a^2 c^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{2 d^{5/2}}+\frac {a b c^2 \sqrt {1-c^2 x^2} \left (-\frac {2 (-1+\text {ArcSin}(c x))}{-1+c x}+52 \text {ArcSin}(c x)-6 \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )-3 \text {ArcSin}(c x) \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+60 \text {ArcSin}(c x) \left (\log \left (1-e^{i \text {ArcSin}(c x)}\right )-\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )+52 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-52 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+60 i \left (\text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )+3 \text {ArcSin}(c x) \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+\frac {4 \text {ArcSin}(c x) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3}+\frac {52 \text {ArcSin}(c x) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}-\frac {4 \text {ArcSin}(c x) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3}+\frac {2 (1+\text {ArcSin}(c x))}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}-\frac {52 \text {ArcSin}(c x) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}-6 \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{12 d^2 \sqrt {d \left (1-c^2 x^2\right )}}+\frac {b^2 c^2 \sqrt {1-c^2 x^2} \left (8-\frac {2 (-2+\text {ArcSin}(c x)) \text {ArcSin}(c x)}{-1+c x}+52 \text {ArcSin}(c x)^2-12 \text {ArcSin}(c x) \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )-3 \text {ArcSin}(c x)^2 \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+24 \log \left (\tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-104 \left (\text {ArcSin}(c x) \left (\log \left (1-i e^{i \text {ArcSin}(c x)}\right )-\log \left (1+i e^{i \text {ArcSin}(c x)}\right )\right )+i \left (\text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )\right )\right )+60 \left (\text {ArcSin}(c x)^2 \left (\log \left (1-e^{i \text {ArcSin}(c x)}\right )-\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )+2 i \text {ArcSin}(c x) \left (\text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )+2 \left (-\text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )+\text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )\right )\right )+3 \text {ArcSin}(c x)^2 \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+\frac {4 \text {ArcSin}(c x)^2 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3}+\frac {4 \left (2+13 \text {ArcSin}(c x)^2\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}-\frac {4 \text {ArcSin}(c x)^2 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3}+\frac {2 \text {ArcSin}(c x) (2+\text {ArcSin}(c x))}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}-\frac {4 \left (2+13 \text {ArcSin}(c x)^2\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}-12 \text {ArcSin}(c x) \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{24 d^2 \sqrt {d \left (1-c^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a^2/(d^3*x^2) + (a^2*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a^2*c^2)/(d^3*(-1 + c^
2*x^2))) + (5*a^2*c^2*Log[x])/(2*d^(5/2)) - (5*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2))
+ (a*b*c^2*Sqrt[1 - c^2*x^2]*((-2*(-1 + ArcSin[c*x]))/(-1 + c*x) + 52*ArcSin[c*x] - 6*Cot[ArcSin[c*x]/2] - 3*
ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 60*ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) +
52*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (60*I)*(Po
lyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin
[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(C
os[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[
c*x]/2])^3 + (2*(1 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (52*ArcSin[c*x]*Sin[ArcSin[c*
x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 6*Tan[ArcSin[c*x]/2]))/(12*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b^
2*c^2*Sqrt[1 - c^2*x^2]*(8 - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) + 52*ArcSin[c*x]^2 - 12*ArcSin[c*x]
*Cot[ArcSin[c*x]/2] - 3*ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 + 24*Log[Tan[ArcSin[c*x]/2]] - 104*(ArcSin[c*x]*(Lo
g[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - PolyLog[2
, I*E^(I*ArcSin[c*x])])) + 60*(ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + (2*I)
*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcSin[c*
x])] + PolyLog[3, E^(I*ArcSin[c*x])])) + 3*ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]^2*Sin[ArcSin[c*
x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (4*(2 + 13*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin
[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]
)^3 + (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (4*(2 + 13*ArcSin[c*x]^2
)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 12*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(24*d^2*
Sqrt[d*(1 - c^2*x^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. 1875 vs. \(2 (739 ) = 1478\).
time = 0.46, size = 1876, normalized size = 2.49

method	result	size
default	\(\text {Expression too large to display}\)	\(1876\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-5/2*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^2*arcsin(c*x)^2*c^4+b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^
2*x^2-1))^(1/2)*c^2/d^3/(c^2*x^2-1)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/
2)*c^2/d^3/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+5*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/
(c^2*x^2-1)*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))-5*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*
x^2-1)*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))+5*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*x^2-
1)*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-5*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(
c^2*x^2-1)*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*
x^2+1)*x*(-c^2*x^2+1)^(1/2)*c^3-a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x*(-c^2*x^2+1)^(1/2)*c-5*
a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^2*arcsin(c*x)*c^4+5/6*a^2*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/
2*a^2*c^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a^2*c^2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-1/2*a^2/d/x^
2/(-c^2*d*x^2+d)^(3/2)+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*c^2-1/3*b^2*(-d*(c^2*x^2-1))^(
1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*x^2*c^4+10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)^2
*c^2-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*arcsin(c*x)^2+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)
/d^3/(c^4*x^4-2*c^2*x^2+1)*x*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3-b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*
x^2+1)/x*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c+5/2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*x^2-1
)*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-5/2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*
x^2-1)*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+13/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d
^3/(c^2*x^2-1)*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-13/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^
3/(c^2*x^2-1)*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+13/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2
)*c^2/d^3/(c^2*x^2-1)*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-13/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^
2-1))^(1/2)*c^2/d^3/(c^2*x^2-1)*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+20/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4
*x^4-2*c^2*x^2+1)*arcsin(c*x)*c^2-a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*arcsin(c*x)+5*a*b*(
-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*x^2-1)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-26/3*I
*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*x^2-1)*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-5*I*a*b*(-
c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*x^2-1)*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-5*I*a*b*(-c^2*x^
2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^2/d^3/(c^2*x^2-1)*dilog(I*c*x+(-c^2*x^2+1)^(1/2))

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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)^(5/2),x, algorithm="maxima")

[Out]

-1/6*a^2*(15*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2) - 15*c^2/(sqrt(-c^2*d*x^2 + d
)*d^2) - 5*c^2/((-c^2*d*x^2 + d)^(3/2)*d) + 3/((-c^2*d*x^2 + d)^(3/2)*d*x^2)) - sqrt(d)*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)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-
c*x + 1)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*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)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*
c^2*d^3*x^5 - d^3*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \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)**(5/2),x)

[Out]

Integral((a + b*asin(c*x))**2/(x**3*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)

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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)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2/((-c^2*d*x^2 + d)^(5/2)*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 )}^{5/2}} \,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)^(5/2)),x)

[Out]

int((a + b*asin(c*x))^2/(x^3*(d - c^2*d*x^2)^(5/2)), x)

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