3.3.25 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {ArcSin}(c x))^2}{x^4} \, dx\) [225]

Optimal. Leaf size=400 \[ -\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{x}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b \sqrt {1-c^2 x^2}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {4 i b^2 c^3 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {1-c^2 x^2}} \]

[Out]

-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3-1/3*b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)/x+c^2*d*(a+b*arcsin(c*x))
^2*(-c^2*d*x^2+d)^(1/2)/x-1/3*b^2*c^3*d*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+4/3*I*c^3*d*(a+b*a
rcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/3*c^3*d*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/(-c
^2*x^2+1)^(1/2)-8/3*b*c^3*d*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^
2+1)^(1/2)+4/3*I*b^2*c^3*d*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/3
*b*c*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/x^2

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Rubi [A]
time = 0.39, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {4785, 4781, 4721, 3798, 2221, 2317, 2438, 4737, 4775, 283, 222} \begin {gather*} \frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{x}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 x^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{3 b \sqrt {1-c^2 x^2}}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 \sqrt {1-c^2 x^2}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {4 i b^2 c^3 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {b^2 c^3 d \text {ArcSin}(c x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^4,x]

[Out]

-1/3*(b^2*c^2*d*Sqrt[d - c^2*d*x^2])/x - (b^2*c^3*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3*Sqrt[1 - c^2*x^2]) - (
b*c*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*A
rcSin[c*x])^2)/x + (((4*I)/3)*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - ((d - c^2*d
*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*x^3) + (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*Sqrt[1 - c
^2*x^2]) - (8*b*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/(3*Sqrt[1 - c^2*
x^2]) + (((4*I)/3)*b^2*c^3*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 283

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 4737

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

Rule 4775

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^
(m + 1)*(d + e*x^2)^p*((a + b*ArcSin[c*x])/(f*(m + 1))), x] + (-Dist[b*c*(d^p/(f*(m + 1))), Int[(f*x)^(m + 1)*
(1 - c^2*x^2)^(p - 1/2), x], x] - Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*Arc
Sin[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

Rule 4781

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

Rule 4785

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*((a + b*ArcSin[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^
(m + 2)*(d + e*x^2)^(p - 1)*(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]) /; FreeQ[{a, b, c,
d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\left (c^2 d\right ) \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2}}{x^2} \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}+\frac {\left (4 i b c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (4 i b c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (i b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (i b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {4 i c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}-\frac {8 b c^3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {4 i b^2 c^3 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.25, size = 493, normalized size = 1.23 \begin {gather*} \frac {-a b c d x \sqrt {d-c^2 d x^2}-a^2 d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+4 a^2 c^2 d x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}-b^2 c^2 d x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}+b d \sqrt {d-c^2 d x^2} \left (3 a c^3 x^3+b \left (4 i c^3 x^3-\sqrt {1-c^2 x^2}+4 c^2 x^2 \sqrt {1-c^2 x^2}\right )\right ) \text {ArcSin}(c x)^2+b^2 c^3 d x^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^3-3 a^2 c^3 d^{3/2} x^3 \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-b d \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \left (b c x+2 a \left (1-4 c^2 x^2\right ) \sqrt {1-c^2 x^2}+8 b c^3 x^3 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )-8 a b c^3 d x^3 \sqrt {d-c^2 d x^2} \log (c x)+4 i b^2 c^3 d x^3 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 x^3 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^4,x]

[Out]

(-(a*b*c*d*x*Sqrt[d - c^2*d*x^2]) - a^2*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 4*a^2*c^2*d*x^2*Sqrt[1 - c^2
*x^2]*Sqrt[d - c^2*d*x^2] - b^2*c^2*d*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + b*d*Sqrt[d - c^2*d*x^2]*(3*a
*c^3*x^3 + b*((4*I)*c^3*x^3 - Sqrt[1 - c^2*x^2] + 4*c^2*x^2*Sqrt[1 - c^2*x^2]))*ArcSin[c*x]^2 + b^2*c^3*d*x^3*
Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 3*a^2*c^3*d^(3/2)*x^3*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(
Sqrt[d]*(-1 + c^2*x^2))] - b*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(b*c*x + 2*a*(1 - 4*c^2*x^2)*Sqrt[1 - c^2*x^2]
+ 8*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])]) - 8*a*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*Log[c*x] + (4*I)*b^2*c^3*d
*x^3*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(3*x^3*Sqrt[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. 3280 vs. \(2 (372 ) = 744\).
time = 0.46, size = 3281, normalized size = 8.20

method	result	size
default	\(\text {Expression too large to display}\)	\(3281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

-16/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8+64*a*b*(-d*(c^2*x^2-1))^(1/2
)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^8-16*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2
)*arcsin(c*x)*d*c^3/(3*c^2*x^2-3)+20/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)
*c^6-104*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^6+8*a*b*(-d*(c^2*
x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5+4/3*I*b^2*(-d*(c^2*x^2-1))^(1/
2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)*c^4+32*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24
*c^4*x^4-9*c^2*x^2+1)*x^4/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^7-16/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d
/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)*c^6-12*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c
^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^5+8/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(2
4*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+4/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c
^4*x^4-9*c^2*x^2+1)*x/(c^2*x^2-1)*(-c^2*x^2+1)*c^4-16/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1
)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^3
*d*c^3-20/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8+29/3*b^2*(-d*(c^2*x^2-1)
)^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*c^6-10/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2
+1)*x/(c^2*x^2-1)*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x/(c^2*x^2-1)*c^2+1/3*b^2*(-d*
(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)^2-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(5/2)
+2/3*a^2*c^4*x*(-c^2*d*x^2+d)^(3/2)+2/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(5/2)+a^2*c^4*d*x*(-c^2*d*x^2+d)^(1/2)+a^2*
c^4*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-4/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*
x^4-9*c^2*x^2+1)*x/(c^2*x^2-1)*c^4+20/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^3/(c^2*x^2-1
)*arcsin(c*x)*c^6+3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)
*c^5+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*
c+8*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5-4
/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4-16/3*I*b^2*(-d*(c^2*x
^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^8+4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24
*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^3-8*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^
4*x^4-9*c^2*x^2+1)*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^7+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x
^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*ln((I*c*x+(-c^
2*x^2+1)^(1/2))^2-1)*d*c^3-a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^2*d*c^3-3*a*b
*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^3+8*b^2*(-c^2*x^2+1)^(1/2)
*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+32*b^2*(-d*(c^2*x^2-1))
^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)^2*c^8-52*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^
4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)^2*c^6+73/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/
(c^2*x^2-1)*arcsin(c*x)^2*c^4-14/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin(
c*x)^2*c^2-3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*
c^3+8*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1
/2))-8*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(3*c^2*x^2-3)*arcsin(c*x)^2-8*I*b^2*(-c^2*x^2+1)^
(1/2)*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(3*c^2*x^2-3)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-8*I*b^2*(-c^2*x^2+1)^(1/
2)*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(3*c^2*x^2-3)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/3*I*b^2*(-d*(c^2*x^2-1))^(
1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^3+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^
4-9*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6+146/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x/
(c^2*x^2-1)*arcsin(c*x)*c^4-28/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin(c*
x)*c^2-24*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/
2)*c^5+64*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/
2)*c^7+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(24*c^4*x^4-9*c^2*x^2+1)/x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c

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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((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="maxima")

[Out]

1/3*(3*sqrt(-c^2*d*x^2 + d)*c^4*d*x + 3*c^3*d^(3/2)*arcsin(c*x) + 2*(-c^2*d*x^2 + d)^(3/2)*c^2/x - (-c^2*d*x^2
+ d)^(5/2)/(d*x^3))*a^2 - sqrt(d)*integrate(((b^2*c^2*d*x^2 - b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1
))^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/x^4,
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((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2/x**4, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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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\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^4,x)

[Out]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2))/x^4, x)

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