3.182 \(\int \frac{(d-c^2 d x^2)^3 (a+b \sin ^{-1}(c x))^2}{x^4} \, dx\)

Optimal. Leaf size=348 \[ -\frac{17}{3} i b^2 c^3 d^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+\frac{17}{3} i b^2 c^3 d^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+5 b c^3 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{27} b^2 c^6 d^3 x^3-\frac{50}{9} b^2 c^4 d^3 x-\frac{b^2 c^2 d^3}{3 x} \]

[Out]

-(b^2*c^2*d^3)/(3*x) - (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 + 5*b*c^3*d^3*Sqrt[1 - c^2*x^2]*(a + b*Ar
cSin[c*x]) - (b*c^3*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/9 - (b*c*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSi
n[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSin[c*x])^2)/3 + (8*c^4*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/
3 + (2*c^2*d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(3*x^3)
+ (34*b*c^3*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, -E^(I*Ar
cSin[c*x])] + ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, E^(I*ArcSin[c*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.981233, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4695, 4649, 4619, 4677, 8, 4699, 4697, 4709, 4183, 2279, 2391, 270} \[ -\frac{17}{3} i b^2 c^3 d^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+\frac{17}{3} i b^2 c^3 d^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+5 b c^3 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{27} b^2 c^6 d^3 x^3-\frac{50}{9} b^2 c^4 d^3 x-\frac{b^2 c^2 d^3}{3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*c^2*d^3)/(3*x) - (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 + 5*b*c^3*d^3*Sqrt[1 - c^2*x^2]*(a + b*Ar
cSin[c*x]) - (b*c^3*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/9 - (b*c*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSi
n[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSin[c*x])^2)/3 + (8*c^4*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/
3 + (2*c^2*d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(3*x^3)
+ (34*b*c^3*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, -E^(I*Ar
cSin[c*x])] + ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, E^(I*ArcSin[c*x])]

Rule 4695

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/
(f*(m + 1)*(1 - c^2*x^2)^FracPart[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]

Rule 4649

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*(
a + b*ArcSin[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n,
x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[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] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 4619

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

Rule 4677

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4699

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 + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

Rule 4697

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

Rule 4709

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

Rule 4183

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 2279

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 2391

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\left (2 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{1}{3} \left (2 b c d^3\right ) \int \frac{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\left (8 c^4 d^2\right ) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac{1}{3} \left (b^2 c^2 d^3\right ) \int \frac{\left (1-c^2 x^2\right )^2}{x^2} \, dx-\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\left (4 b c^3 d^3\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx\\ &=-\frac{17}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d^3\right ) \int \left (-2 c^2+\frac{1}{x^2}+c^4 x^2\right ) \, dx-\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\left (4 b c^3 d^3\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac{1}{3} \left (16 c^4 d^3\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac{1}{9} \left (5 b^2 c^4 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx+\frac{1}{3} \left (4 b^2 c^4 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx-\frac{1}{3} \left (16 b c^5 d^3\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{11}{9} b^2 c^4 d^3 x-\frac{14}{27} b^2 c^6 d^3 x^3-\frac{17}{3} b c^3 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx-\left (4 b c^3 d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx+\frac{1}{3} \left (5 b^2 c^4 d^3\right ) \int 1 \, dx-\frac{1}{9} \left (16 b^2 c^4 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx+\left (4 b^2 c^4 d^3\right ) \int 1 \, dx-\frac{1}{3} \left (32 b c^5 d^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{46}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{3} \left (5 b c^3 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\left (4 b c^3 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (32 b^2 c^4 d^3\right ) \int 1 \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{34}{3} b c^3 d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{34}{3} b c^3 d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac{1}{3} \left (5 i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\frac{1}{3} \left (5 i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\left (4 i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\left (4 i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{34}{3} b c^3 d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac{17}{3} i b^2 c^3 d^3 \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+\frac{17}{3} i b^2 c^3 d^3 \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.998796, size = 480, normalized size = 1.38 \[ -\frac{d^3 \left (153 i b^2 c^3 x^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-153 i b^2 c^3 x^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+9 a^2 c^6 x^6-81 a^2 c^4 x^4-81 a^2 c^2 x^2+9 a^2+6 a b c^5 x^5 \sqrt{1-c^2 x^2}-150 a b c^3 x^3 \sqrt{1-c^2 x^2}+9 a b c x \sqrt{1-c^2 x^2}+18 a b c^6 x^6 \sin ^{-1}(c x)-162 a b c^4 x^4 \sin ^{-1}(c x)-162 a b c^2 x^2 \sin ^{-1}(c x)-153 a b c^3 x^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+18 a b \sin ^{-1}(c x)-2 b^2 c^6 x^6+150 b^2 c^4 x^4+9 b^2 c^2 x^2+9 b^2 c^6 x^6 \sin ^{-1}(c x)^2+6 b^2 c^5 x^5 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-81 b^2 c^4 x^4 \sin ^{-1}(c x)^2-150 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-81 b^2 c^2 x^2 \sin ^{-1}(c x)^2+9 b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+153 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-153 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )+9 b^2 \sin ^{-1}(c x)^2\right )}{27 x^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(d^3*(9*a^2 - 81*a^2*c^2*x^2 + 9*b^2*c^2*x^2 - 81*a^2*c^4*x^4 + 150*b^2*c^4*x^4 + 9*a^2*c^6*x^6 - 2*b^2*c^6*x
^6 + 9*a*b*c*x*Sqrt[1 - c^2*x^2] - 150*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 6*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 18*a*
b*ArcSin[c*x] - 162*a*b*c^2*x^2*ArcSin[c*x] - 162*a*b*c^4*x^4*ArcSin[c*x] + 18*a*b*c^6*x^6*ArcSin[c*x] + 9*b^2
*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 150*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b^2*c^5*x^5*Sqrt[1 - c^
2*x^2]*ArcSin[c*x] + 9*b^2*ArcSin[c*x]^2 - 81*b^2*c^2*x^2*ArcSin[c*x]^2 - 81*b^2*c^4*x^4*ArcSin[c*x]^2 + 9*b^2
*c^6*x^6*ArcSin[c*x]^2 - 153*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] + 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 - E^(I
*ArcSin[c*x])] - 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (153*I)*b^2*c^3*x^3*PolyLog[2, -E^(I
*ArcSin[c*x])] - (153*I)*b^2*c^3*x^3*PolyLog[2, E^(I*ArcSin[c*x])]))/(27*x^3)

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Maple [A]  time = 0.526, size = 547, normalized size = 1.6 \begin{align*}{\frac{17\,{c}^{3}{d}^{3}ab}{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) }-{\frac{17\,i}{3}}{b}^{2}{c}^{3}{d}^{3}{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +{\frac{17\,i}{3}}{b}^{2}{c}^{3}{d}^{3}{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,{\frac{{c}^{2}{d}^{3}ab\arcsin \left ( cx \right ) }{x}}-{\frac{2\,{c}^{5}{d}^{3}{b}^{2}\arcsin \left ( cx \right ){x}^{2}}{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{5}{d}^{3}ab{x}^{2}}{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{c{d}^{3}ab}{3\,{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{6}{d}^{3}ab{x}^{3}\arcsin \left ( cx \right ) }{3}}+6\,{c}^{4}{d}^{3}abx\arcsin \left ( cx \right ) -{\frac{{d}^{3}{b}^{2}c\arcsin \left ( cx \right ) }{3\,{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{d}^{3}{b}^{2}{c}^{2}}{3\,x}}-{\frac{50\,{b}^{2}{c}^{4}{d}^{3}x}{9}}+{\frac{2\,{b}^{2}{c}^{6}{d}^{3}{x}^{3}}{27}}-{\frac{{d}^{3}{a}^{2}}{3\,{x}^{3}}}+3\,{\frac{{c}^{2}{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{{c}^{6}{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{x}^{3}}{3}}+{\frac{50\,{c}^{3}{d}^{3}ab}{9}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{50\,{d}^{3}{b}^{2}{c}^{3}\arcsin \left ( cx \right ) }{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{17\,{d}^{3}{b}^{2}{c}^{3}\arcsin \left ( cx \right ) }{3}\ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{17\,{d}^{3}{b}^{2}{c}^{3}\arcsin \left ( cx \right ) }{3}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+3\,{c}^{4}{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}x-{\frac{2\,{d}^{3}ab\arcsin \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{{c}^{6}{d}^{3}{a}^{2}{x}^{3}}{3}}+3\,{c}^{4}{d}^{3}{a}^{2}x+3\,{\frac{{c}^{2}{d}^{3}{a}^{2}}{x}}-{\frac{{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-17/3*I*b^2*c^3*d^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+17/3*I*b^2*c^3*d^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)
)+6*c^2*d^3*a*b/x*arcsin(c*x)-2/9*c^5*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2-2/9*c^5*d^3*a*b*x^2*(-c^2*x^2
+1)^(1/2)-1/3*c*d^3*a*b/x^2*(-c^2*x^2+1)^(1/2)-2/3*c^6*d^3*a*b*x^3*arcsin(c*x)+6*c^4*d^3*a*b*x*arcsin(c*x)-1/3
*c*d^3*b^2/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/3*b^2*c^2*d^3/x-50/9*b^2*c^4*d^3*x+2/27*b^2*c^6*d^3*x^3-1/3*d^
3*a^2/x^3+3*c^2*d^3*b^2/x*arcsin(c*x)^2-1/3*c^6*d^3*b^2*arcsin(c*x)^2*x^3+50/9*c^3*d^3*a*b*(-c^2*x^2+1)^(1/2)+
17/3*c^3*d^3*a*b*arctanh(1/(-c^2*x^2+1)^(1/2))+50/9*c^3*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-17/3*c^3*d^3*b^
2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+17/3*c^3*d^3*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+3*c^4
*d^3*b^2*arcsin(c*x)^2*x-2/3*d^3*a*b*arcsin(c*x)/x^3-1/3*c^6*d^3*a^2*x^3+3*c^4*d^3*a^2*x+3*c^2*d^3*a^2/x-1/3*d
^3*b^2/x^3*arcsin(c*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3} \, a^{2} c^{6} d^{3} x^{3} - \frac{2}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{6} d^{3} + 3 \, b^{2} c^{4} d^{3} x \arcsin \left (c x\right )^{2} - 6 \, b^{2} c^{4} d^{3}{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + 3 \, a^{2} c^{4} d^{3} x + 6 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b c^{3} d^{3} + 6 \,{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} a b c^{2} d^{3} - \frac{1}{3} \,{\left ({\left (c^{2} \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\sqrt{-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac{2 \, \arcsin \left (c x\right )}{x^{3}}\right )} a b d^{3} + \frac{3 \, a^{2} c^{2} d^{3}}{x} - \frac{a^{2} d^{3}}{3 \, x^{3}} - \frac{2 \, x^{3} \int \frac{{\left (b^{2} c^{7} d^{3} x^{6} - 9 \, b^{2} c^{3} d^{3} x^{2} + b^{2} c d^{3}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{c^{2} x^{5} - x^{3}}\,{d x} +{\left (b^{2} c^{6} d^{3} x^{6} - 9 \, b^{2} c^{2} d^{3} x^{2} + b^{2} d^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*c^6*d^3*x^3 - 2/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b
*c^6*d^3 + 3*b^2*c^4*d^3*x*arcsin(c*x)^2 - 6*b^2*c^4*d^3*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + 3*a^2*c^4*d^
3*x + 6*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c^3*d^3 + 6*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x))
+ arcsin(c*x)/x)*a*b*c^2*d^3 - 1/3*((c^2*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1)/x^2)
*c + 2*arcsin(c*x)/x^3)*a*b*d^3 + 3*a^2*c^2*d^3/x - 1/3*a^2*d^3/x^3 - 1/3*(3*x^3*integrate(2/3*(b^2*c^7*d^3*x^
6 - 9*b^2*c^3*d^3*x^2 + b^2*c*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^
2*x^5 - x^3), x) + (b^2*c^6*d^3*x^6 - 9*b^2*c^2*d^3*x^2 + b^2*d^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^
2)/x^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a^{2} c^{6} d^{3} x^{6} - 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} - a^{2} d^{3} +{\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arcsin \left (c x\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - d^{3} \left (\int - 3 a^{2} c^{4}\, dx + \int - \frac{a^{2}}{x^{4}}\, dx + \int \frac{3 a^{2} c^{2}}{x^{2}}\, dx + \int a^{2} c^{6} x^{2}\, dx + \int - 3 b^{2} c^{4} \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int - \frac{b^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int - 6 a b c^{4} \operatorname{asin}{\left (c x \right )}\, dx + \int - \frac{2 a b \operatorname{asin}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{3 b^{2} c^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{6} x^{2} \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int \frac{6 a b c^{2} \operatorname{asin}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{6} x^{2} \operatorname{asin}{\left (c x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d**3*(Integral(-3*a**2*c**4, x) + Integral(-a**2/x**4, x) + Integral(3*a**2*c**2/x**2, x) + Integral(a**2*c**
6*x**2, x) + Integral(-3*b**2*c**4*asin(c*x)**2, x) + Integral(-b**2*asin(c*x)**2/x**4, x) + Integral(-6*a*b*c
**4*asin(c*x), x) + Integral(-2*a*b*asin(c*x)/x**4, x) + Integral(3*b**2*c**2*asin(c*x)**2/x**2, x) + Integral
(b**2*c**6*x**2*asin(c*x)**2, x) + Integral(6*a*b*c**2*asin(c*x)/x**2, x) + Integral(2*a*b*c**6*x**2*asin(c*x)
, x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x