3.3.42 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x^3 \sqrt {d-c^2 d x^2}} \, dx\) [242]

Optimal. Leaf size=402 \[ -\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 d x^2}-\frac {c^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b c^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b c^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.29, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {4789, 4803, 4268, 2611, 2320, 6724, 4723, 272, 65, 214} \begin {gather*} \frac {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))}{\sqrt {d-c^2 d x^2}}-\frac {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))}{\sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 d x^2}-\frac {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}{\sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \text {ArcSin}(c x)}\right )}{\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 )}{\sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -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 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 \sqrt {d-c^2 d x^2}} \, dx &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {1}{2} c^2 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^2} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {\left (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 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {\left (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 \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 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {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 )}{\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 )}{\sqrt {d-c^2 d x^2}}-\frac {\left (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 )}{\sqrt {d-c^2 d x^2}}+\frac {\left (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 )}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {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 )}{\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 )}{\sqrt {d-c^2 d x^2}}+\frac {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 )}{\sqrt {d-c^2 d x^2}}-\frac {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 )}{\sqrt {d-c^2 d x^2}}-\frac {\left (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 )}{\sqrt {d-c^2 d x^2}}+\frac {\left (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 )}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {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 )}{\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 )}{\sqrt {d-c^2 d x^2}}+\frac {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 )}{\sqrt {d-c^2 d x^2}}-\frac {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 )}{\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 {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\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 {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {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 )}{\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 )}{\sqrt {d-c^2 d x^2}}+\frac {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 )}{\sqrt {d-c^2 d x^2}}-\frac {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 )}{\sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 3.73, size = 487, normalized size = 1.21 \begin {gather*} \frac {-\frac {4 a^2 \sqrt {d-c^2 d x^2}}{x^2}+4 a^2 c^2 \sqrt {d} \log (x)-4 a^2 c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b c^2 d^2 \left (1-c^2 x^2\right )^{3/2} \left (-2 \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\text {ArcSin}(c x) \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+4 \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-4 \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )+4 i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-4 i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )+\text {ArcSin}(c x) \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )-2 \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{\left (d-c^2 d x^2\right )^{3/2}}+\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right )^{3/2} \left (-4 \text {ArcSin}(c x) \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\text {ArcSin}(c x)^2 \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+4 \text {ArcSin}(c x)^2 \log \left (1-e^{i \text {ArcSin}(c x)}\right )-4 \text {ArcSin}(c x)^2 \log \left (1+e^{i \text {ArcSin}(c x)}\right )+8 \log \left (\tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+8 i \text {ArcSin}(c x) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-8 i \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )-8 \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )+8 \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )+\text {ArcSin}(c x)^2 \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )-4 \text {ArcSin}(c x) \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{\left (d-c^2 d x^2\right )^{3/2}}}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a^2*Sqrt[d - c^2*d*x^2])/x^2 + 4*a^2*c^2*Sqrt[d]*Log[x] - 4*a^2*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*
d*x^2]] + (2*a*b*c^2*d^2*(1 - c^2*x^2)^(3/2)*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 4*Arc
Sin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, -E^(I*ArcSin
[c*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(d -
c^2*d*x^2)^(3/2) + (b^2*c^2*d^2*(1 - c^2*x^2)^(3/2)*(-4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[Ar
cSin[c*x]/2]^2 + 4*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + 8*L
og[Tan[ArcSin[c*x]/2]] + (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (8*I)*ArcSin[c*x]*PolyLog[2, E^(I*
ArcSin[c*x])] - 8*PolyLog[3, -E^(I*ArcSin[c*x])] + 8*PolyLog[3, E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Sec[ArcSin[
c*x]/2]^2 - 4*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(d - c^2*d*x^2)^(3/2))/(8*d)

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1106 vs. \(2 (406 ) = 812\).
time = 0.33, size = 1107, normalized size = 2.75

method	result	size
default	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{2 d \,x^{2}}-\frac {a^{2} c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}-\frac {b^{2} \arcsin \left (c x \right )^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2}}{2 d \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{x d \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} d \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{2 d \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{2 d \left (c^{2} x^{2}-1\right )}-\frac {i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arctanh \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{2}}{d \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{x d \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{x^{2} d \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {i b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}+\frac {i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d \left (c^{2} x^{2}-1\right )}\)	\(1107\)

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)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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)^(1/2),x, algorithm="maxima")

[Out]

-1/2*(c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d) + sqrt(-c^2*d*x^2 + d)/(d*x^2))*a^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^2*d*x^5 - d*x^3), x)

________________________________________________________________________________________

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)^(1/2),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^2*d*x^5 - d*x^3), x)

________________________________________________________________________________________

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} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, 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)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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)^(1/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

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\,\sqrt {d-c^2\,d\,x^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)^(1/2)),x)

[Out]

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

________________________________________________________________________________________