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

Optimal. Leaf size=268 \[ -\frac{11}{3} i b^2 c^3 d^2 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+\frac{11}{3} i b^2 c^3 d^2 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+\frac{5}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{22}{3} b c^3 d^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-2 b^2 c^4 d^2 x-\frac{b^2 c^2 d^2}{3 x} \]

[Out]

-(b^2*c^2*d^2)/(3*x) - 2*b^2*c^4*d^2*x + (5*b*c^3*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/3 - (b*c*d^2*(1 -
c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*x^2) + (8*c^4*d^2*x*(a + b*ArcSin[c*x])^2)/3 + (4*c^2*d^2*(1 - c^2*x^2
)*(a + b*ArcSin[c*x])^2)/(3*x) - (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*x^3) + (22*b*c^3*d^2*(a + b*Ar
cSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((11*I)/3)*b^2*c^3*d^2*PolyLog[2, -E^(I*ArcSin[c*x])] + ((11*I)/3)*
b^2*c^3*d^2*PolyLog[2, E^(I*ArcSin[c*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.675485, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {4695, 4619, 4677, 8, 4697, 4709, 4183, 2279, 2391, 14} \[ -\frac{11}{3} i b^2 c^3 d^2 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+\frac{11}{3} i b^2 c^3 d^2 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+\frac{5}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{8}{3} c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{22}{3} b c^3 d^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-2 b^2 c^4 d^2 x-\frac{b^2 c^2 d^2}{3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*c^2*d^2)/(3*x) - 2*b^2*c^4*d^2*x + (5*b*c^3*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/3 - (b*c*d^2*(1 -
c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*x^2) + (8*c^4*d^2*x*(a + b*ArcSin[c*x])^2)/3 + (4*c^2*d^2*(1 - c^2*x^2
)*(a + b*ArcSin[c*x])^2)/(3*x) - (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*x^3) + (22*b*c^3*d^2*(a + b*Ar
cSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((11*I)/3)*b^2*c^3*d^2*PolyLog[2, -E^(I*ArcSin[c*x])] + ((11*I)/3)*
b^2*c^3*d^2*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 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 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{3} \left (4 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{1}{3} \left (2 b c d^2\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d^2\right ) \int \frac{1-c^2 x^2}{x^2} \, dx-\left (b c^3 d^2\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac{1}{3} \left (8 b c^3 d^2\right ) \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac{1}{3} \left (8 c^4 d^2\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=-\frac{11}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d^2\right ) \int \left (-c^2+\frac{1}{x^2}\right ) \, dx-\left (b c^3 d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx-\frac{1}{3} \left (8 b c^3 d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx+\left (b^2 c^4 d^2\right ) \int 1 \, dx+\frac{1}{3} \left (8 b^2 c^4 d^2\right ) \int 1 \, dx-\frac{1}{3} \left (16 b c^5 d^2\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^2}{3 x}+\frac{10}{3} b^2 c^4 d^2 x+\frac{5}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}-\left (b c^3 d^2\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (8 b c^3 d^2\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (16 b^2 c^4 d^2\right ) \int 1 \, dx\\ &=-\frac{b^2 c^2 d^2}{3 x}-2 b^2 c^4 d^2 x+\frac{5}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{22}{3} b c^3 d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )+\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\left (b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )+\frac{1}{3} \left (8 b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{3} \left (8 b^2 c^3 d^2\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^2}{3 x}-2 b^2 c^4 d^2 x+\frac{5}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{22}{3} b c^3 d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\left (i b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\left (i b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )-\frac{1}{3} \left (8 i b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )+\frac{1}{3} \left (8 i b^2 c^3 d^2\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^2}{3 x}-2 b^2 c^4 d^2 x+\frac{5}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{4 c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 x^3}+\frac{22}{3} b c^3 d^2 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )-\frac{11}{3} i b^2 c^3 d^2 \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )+\frac{11}{3} i b^2 c^3 d^2 \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.82151, size = 374, normalized size = 1.4 \[ \frac{d^2 \left (-11 i b^2 c^3 x^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )+11 i b^2 c^3 x^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+3 a^2 c^4 x^4+6 a^2 c^2 x^2-a^2+6 a b c^3 x^3 \sqrt{1-c^2 x^2}-a b c x \sqrt{1-c^2 x^2}+6 a b c^4 x^4 \sin ^{-1}(c x)+12 a b c^2 x^2 \sin ^{-1}(c x)+11 a b c^3 x^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-2 a b \sin ^{-1}(c x)-6 b^2 c^4 x^4-b^2 c^2 x^2+3 b^2 c^4 x^4 \sin ^{-1}(c x)^2+6 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+6 b^2 c^2 x^2 \sin ^{-1}(c x)^2-b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-11 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )+11 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-b^2 \sin ^{-1}(c x)^2\right )}{3 x^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(-a^2 + 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 3*a^2*c^4*x^4 - 6*b^2*c^4*x^4 - a*b*c*x*Sqrt[1 - c^2*x^2] + 6*a*b*c
^3*x^3*Sqrt[1 - c^2*x^2] - 2*a*b*ArcSin[c*x] + 12*a*b*c^2*x^2*ArcSin[c*x] + 6*a*b*c^4*x^4*ArcSin[c*x] - b^2*c*
x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - b^2*ArcSin[c*x]^2 + 6*b^2*c^2*
x^2*ArcSin[c*x]^2 + 3*b^2*c^4*x^4*ArcSin[c*x]^2 + 11*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 11*b^2*c^3*x^3*A
rcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 11*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (11*I)*b^2*c^3
*x^3*PolyLog[2, -E^(I*ArcSin[c*x])] + (11*I)*b^2*c^3*x^3*PolyLog[2, E^(I*ArcSin[c*x])]))/(3*x^3)

________________________________________________________________________________________

Maple [A]  time = 0.421, size = 425, normalized size = 1.6 \begin{align*}{c}^{4}{d}^{2}{a}^{2}x+2\,{\frac{{c}^{2}{d}^{2}{a}^{2}}{x}}-{\frac{{d}^{2}{a}^{2}}{3\,{x}^{3}}}+2\,{c}^{3}{d}^{2}{b}^{2}\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1}+{c}^{4}{d}^{2}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}x-2\,{b}^{2}{c}^{4}{d}^{2}x+2\,{\frac{{c}^{2}{d}^{2}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{{d}^{2}{b}^{2}c\arcsin \left ( cx \right ) }{3\,{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{d}^{2}{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{{d}^{2}{b}^{2}{c}^{2}}{3\,x}}+{\frac{11\,{c}^{3}{d}^{2}{b}^{2}\arcsin \left ( cx \right ) }{3}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{11\,i}{3}}{b}^{2}{c}^{3}{d}^{2}{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{11\,{c}^{3}{d}^{2}{b}^{2}\arcsin \left ( cx \right ) }{3}\ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{11\,i}{3}}{b}^{2}{c}^{3}{d}^{2}{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,{c}^{4}{d}^{2}abx\arcsin \left ( cx \right ) +4\,{\frac{{c}^{2}{d}^{2}ab\arcsin \left ( cx \right ) }{x}}-{\frac{2\,{d}^{2}ab\arcsin \left ( cx \right ) }{3\,{x}^{3}}}+2\,{c}^{3}{d}^{2}ab\sqrt{-{c}^{2}{x}^{2}+1}+{\frac{11\,{c}^{3}{d}^{2}ab}{3}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) }-{\frac{c{d}^{2}ab}{3\,{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*d^2*a^2*x+2*c^2*d^2*a^2/x-1/3*d^2*a^2/x^3+2*c^3*d^2*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+c^4*d^2*b^2*arcsin(
c*x)^2*x-2*b^2*c^4*d^2*x+2*c^2*d^2*b^2/x*arcsin(c*x)^2-1/3*c*d^2*b^2/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-1/3*d^
2*b^2/x^3*arcsin(c*x)^2-1/3*b^2*c^2*d^2/x+11/3*c^3*d^2*b^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-11/3*I*b
^2*c^3*d^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-11/3*c^3*d^2*b^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+11
/3*I*b^2*c^3*d^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+2*c^4*d^2*a*b*x*arcsin(c*x)+4*c^2*d^2*a*b/x*arcsin(c*x)-2
/3*d^2*a*b*arcsin(c*x)/x^3+2*c^3*d^2*a*b*(-c^2*x^2+1)^(1/2)+11/3*c^3*d^2*a*b*arctanh(1/(-c^2*x^2+1)^(1/2))-1/3
*c*d^2*a*b/x^2*(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*(Integral(a**2*c**4, x) + Integral(a**2/x**4, x) + Integral(-2*a**2*c**2/x**2, x) + Integral(b**2*c**4*as
in(c*x)**2, x) + Integral(b**2*asin(c*x)**2/x**4, x) + Integral(2*a*b*c**4*asin(c*x), x) + Integral(2*a*b*asin
(c*x)/x**4, x) + Integral(-2*b**2*c**2*asin(c*x)**2/x**2, x) + Integral(-4*a*b*c**2*asin(c*x)/x**2, 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)^2*(a+b*arcsin(c*x))^2/x^4,x, algorithm="giac")

[Out]

sage0*x