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

Optimal. Leaf size=329 \[ 2 i b^2 c d^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{22}{5} b c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{14}{75} b^2 c^4 d^3 x^3+\frac{122}{25} b^2 c^2 d^3 x \]

[Out]

(122*b^2*c^2*d^3*x)/25 - (14*b^2*c^4*d^3*x^3)/75 + (2*b^2*c^6*d^3*x^5)/125 - (22*b*c*d^3*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(5/2)*(
a + b*ArcSin[c*x]))/25 - (16*c^2*d^3*x*(a + b*ArcSin[c*x])^2)/5 - (8*c^2*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x
])^2)/5 - (6*c^2*d^3*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/5 - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/
x - 4*b*c*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d^3*PolyLog[2, -E^(I*ArcSin[c*x])]
- (2*I)*b^2*c*d^3*PolyLog[2, E^(I*ArcSin[c*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.706098, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 24, 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, 194, 4699, 4697, 4709, 4183, 2279, 2391} \[ 2 i b^2 c d^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{22}{5} b c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{14}{75} b^2 c^4 d^3 x^3+\frac{122}{25} b^2 c^2 d^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(122*b^2*c^2*d^3*x)/25 - (14*b^2*c^4*d^3*x^3)/75 + (2*b^2*c^6*d^3*x^5)/125 - (22*b*c*d^3*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(5/2)*(
a + b*ArcSin[c*x]))/25 - (16*c^2*d^3*x*(a + b*ArcSin[c*x])^2)/5 - (8*c^2*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x
])^2)/5 - (6*c^2*d^3*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/5 - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/
x - 4*b*c*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d^3*PolyLog[2, -E^(I*ArcSin[c*x])]
- (2*I)*b^2*c*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 194

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

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]

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^2} \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\left (6 c^2 d\right ) \int \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\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} \, dx\\ &=\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac{1}{5} \left (24 c^2 d^2\right ) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\left (2 b c 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{1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \, dx+\frac{1}{5} \left (12 b c^3 d^3\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2}{3} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac{1}{5} \left (16 c^2 d^3\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx+\frac{1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right )^2 \, dx-\frac{1}{3} \left (2 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx+\frac{1}{5} \left (16 b c^3 d^3\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{16}{15} b^2 c^2 d^3 x+\frac{22}{45} b^2 c^4 d^3 x^3-\frac{2}{25} b^2 c^6 d^3 x^5+2 b c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx+\frac{1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx+\frac{1}{15} \left (16 b^2 c^2 d^3\right ) \int \left (1-c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^3\right ) \int 1 \, dx+\frac{1}{5} \left (32 b c^3 d^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{38}{25} b^2 c^2 d^3 x-\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac{1}{5} \left (32 b^2 c^2 d^3\right ) \int 1 \, dx\\ &=\frac{122}{25} b^2 c^2 d^3 x-\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (2 b^2 c d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\left (2 b^2 c d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac{122}{25} b^2 c^2 d^3 x-\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\left (2 i b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\left (2 i b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )\\ &=\frac{122}{25} b^2 c^2 d^3 x-\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{16}{5} c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )^2-\frac{8}{5} c^2 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{6}{5} c^2 d^3 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+2 i b^2 c d^3 \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d^3 \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 1.25003, size = 483, normalized size = 1.47 \[ \frac{1}{720} d^3 \left (1440 i b^2 c \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-1440 i b^2 c \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-144 a^2 c^6 x^5+720 a^2 c^4 x^3-2160 a^2 c^2 x-\frac{720 a^2}{x}-\frac{288}{5} a b c^5 x^4 \sqrt{1-c^2 x^2}+\frac{2016}{5} a b c^3 x^2 \sqrt{1-c^2 x^2}-\frac{17568}{5} a b c \sqrt{1-c^2 x^2}-288 a b c^6 x^5 \sin ^{-1}(c x)+1440 a b c^4 x^3 \sin ^{-1}(c x)-1440 a b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-4320 a b c^2 x \sin ^{-1}(c x)-\frac{1440 a b \sin ^{-1}(c x)}{x}-3420 b^2 c \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+3460 b^2 c^2 x-1890 b^2 c^2 x \sin ^{-1}(c x)^2-360 b^2 c^2 x \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )+80 b^2 c^2 x \cos \left (2 \sin ^{-1}(c x)\right )-10 b^2 c \sin \left (3 \sin ^{-1}(c x)\right )+45 b^2 c \sin ^{-1}(c x)^2 \sin \left (3 \sin ^{-1}(c x)\right )+\frac{18}{25} b^2 c \sin \left (5 \sin ^{-1}(c x)\right )-9 b^2 c \sin ^{-1}(c x)^2 \sin \left (5 \sin ^{-1}(c x)\right )-\frac{720 b^2 \sin ^{-1}(c x)^2}{x}+1440 b^2 c \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-1440 b^2 c \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-90 b^2 c \sin ^{-1}(c x) \cos \left (3 \sin ^{-1}(c x)\right )-\frac{18}{5} b^2 c \sin ^{-1}(c x) \cos \left (5 \sin ^{-1}(c x)\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^3*((-720*a^2)/x - 2160*a^2*c^2*x + 3460*b^2*c^2*x + 720*a^2*c^4*x^3 - 144*a^2*c^6*x^5 - (17568*a*b*c*Sqrt[1
- c^2*x^2])/5 + (2016*a*b*c^3*x^2*Sqrt[1 - c^2*x^2])/5 - (288*a*b*c^5*x^4*Sqrt[1 - c^2*x^2])/5 - (1440*a*b*Ar
cSin[c*x])/x - 4320*a*b*c^2*x*ArcSin[c*x] + 1440*a*b*c^4*x^3*ArcSin[c*x] - 288*a*b*c^6*x^5*ArcSin[c*x] - 3420*
b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - (720*b^2*ArcSin[c*x]^2)/x - 1890*b^2*c^2*x*ArcSin[c*x]^2 - 1440*a*b*c*Ar
cTanh[Sqrt[1 - c^2*x^2]] + 80*b^2*c^2*x*Cos[2*ArcSin[c*x]] - 360*b^2*c^2*x*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] -
90*b^2*c*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (18*b^2*c*ArcSin[c*x]*Cos[5*ArcSin[c*x]])/5 + 1440*b^2*c*ArcSin[c*x]
*Log[1 - E^(I*ArcSin[c*x])] - 1440*b^2*c*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (1440*I)*b^2*c*PolyLog[2, -E
^(I*ArcSin[c*x])] - (1440*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])] - 10*b^2*c*Sin[3*ArcSin[c*x]] + 45*b^2*c*ArcS
in[c*x]^2*Sin[3*ArcSin[c*x]] + (18*b^2*c*Sin[5*ArcSin[c*x]])/25 - 9*b^2*c*ArcSin[c*x]^2*Sin[5*ArcSin[c*x]]))/7
20

________________________________________________________________________________________

Maple [A]  time = 0.336, size = 535, normalized size = 1.6 \begin{align*} -2\,c{d}^{3}ab{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) +2\,i{b}^{2}c{d}^{3}{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,i{b}^{2}c{d}^{3}{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +{\frac{122\,{b}^{2}{c}^{2}{d}^{3}x}{25}}-{\frac{14\,{b}^{2}{c}^{4}{d}^{3}{x}^{3}}{75}}+{\frac{2\,{b}^{2}{c}^{6}{d}^{3}{x}^{5}}{125}}-{\frac{{d}^{3}{a}^{2}}{x}}-{\frac{{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{6}{x}^{5}}{5}}+{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{4}{x}^{3}-3\,{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{2}x-{\frac{122\,c{d}^{3}ab}{25}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{122\,{d}^{3}{b}^{2}c\arcsin \left ( cx \right ) }{25}\sqrt{-{c}^{2}{x}^{2}+1}}+2\,c{d}^{3}{b}^{2}\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,c{d}^{3}{b}^{2}\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,{\frac{{d}^{3}ab\arcsin \left ( cx \right ) }{x}}-{\frac{2\,{d}^{3}ab\arcsin \left ( cx \right ){c}^{6}{x}^{5}}{5}}+{\frac{14\,{d}^{3}{b}^{2}\arcsin \left ( cx \right ){c}^{3}{x}^{2}}{25}\sqrt{-{c}^{2}{x}^{2}+1}}+2\,{d}^{3}ab{c}^{4}{x}^{3}\arcsin \left ( cx \right ) -6\,{d}^{3}ab{c}^{2}x\arcsin \left ( cx \right ) -{\frac{2\,{d}^{3}ab{c}^{5}{x}^{4}}{25}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{14\,{d}^{3}ab{c}^{3}{x}^{2}}{25}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2\,{d}^{3}{b}^{2}\arcsin \left ( cx \right ){c}^{5}{x}^{4}}{25}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{d}^{3}{a}^{2}{c}^{6}{x}^{5}}{5}}+{d}^{3}{a}^{2}{c}^{4}{x}^{3}-3\,{d}^{3}{a}^{2}{c}^{2}x-{\frac{{d}^{3}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{x}} \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^2,x)

[Out]

2*I*b^2*c*d^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b^2*c*d^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+122/25*b^
2*c^2*d^3*x-14/75*b^2*c^4*d^3*x^3+2/125*b^2*c^6*d^3*x^5-d^3*a^2/x-1/5*d^3*b^2*arcsin(c*x)^2*c^6*x^5+d^3*b^2*ar
csin(c*x)^2*c^4*x^3-3*d^3*b^2*arcsin(c*x)^2*c^2*x-122/25*c*d^3*a*b*(-c^2*x^2+1)^(1/2)-2*c*d^3*a*b*arctanh(1/(-
c^2*x^2+1)^(1/2))-122/25*c*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+2*c*d^3*b^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2
+1)^(1/2))-2*c*d^3*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*d^3*a*b/x*arcsin(c*x)-2/5*d^3*a*b*arcsin(c
*x)*c^6*x^5+14/25*d^3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3*x^2+2*d^3*a*b*c^4*x^3*arcsin(c*x)-6*d^3*a*b*c^2*x
*arcsin(c*x)-2/25*d^3*a*b*c^5*x^4*(-c^2*x^2+1)^(1/2)+14/25*d^3*a*b*c^3*x^2*(-c^2*x^2+1)^(1/2)-2/25*d^3*b^2*arc
sin(c*x)*(-c^2*x^2+1)^(1/2)*c^5*x^4-1/5*d^3*a^2*c^6*x^5+d^3*a^2*c^4*x^3-3*d^3*a^2*c^2*x-d^3*b^2/x*arcsin(c*x)^
2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{5} \, a^{2} c^{6} d^{3} x^{5} - \frac{2}{75} \,{\left (15 \, x^{5} \arcsin \left (c x\right ) +{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{6} d^{3} + a^{2} c^{4} d^{3} x^{3} + \frac{2}{3} \,{\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^{4} d^{3} - 3 \, b^{2} c^{2} d^{3} x \arcsin \left (c x\right )^{2} + 6 \, b^{2} c^{2} d^{3}{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} - 3 \, a^{2} c^{2} d^{3} x - 6 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b c d^{3} - 2 \,{\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 d^{3} - \frac{a^{2} d^{3}}{x} - \frac{{\left (b^{2} c^{6} d^{3} x^{6} - 5 \, b^{2} c^{4} d^{3} x^{4} + 5 \, b^{2} d^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \, x \int \frac{{\left (b^{2} c^{7} d^{3} x^{6} - 5 \, b^{2} c^{5} d^{3} x^{4} + 5 \, 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^{3} - x}\,{d x}}{5 \, 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^2,x, algorithm="maxima")

[Out]

-1/5*a^2*c^6*d^3*x^5 - 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4
+ 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*c^6*d^3 + a^2*c^4*d^3*x^3 + 2/3*(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^4*d^3 - 3*b^2*c^2*d^3*x*arcsin(c*x)^2 + 6*b^2*c^2*d^3*(x - sqrt(-c
^2*x^2 + 1)*arcsin(c*x)/c) - 3*a^2*c^2*d^3*x - 6*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*c*d^3 - 2*(c*log(2
*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*a*b*d^3 - a^2*d^3/x - 1/5*((b^2*c^6*d^3*x^6 - 5*b^2*c^
4*d^3*x^4 + 5*b^2*d^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 5*x*integrate(2/5*(b^2*c^7*d^3*x^6 - 5*b
^2*c^5*d^3*x^4 + 5*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^3
- x), x))/x

________________________________________________________________________________________

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^{2}}, 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^2,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^2, x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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^2,x, algorithm="giac")

[Out]

sage0*x