3.263 \(\int \frac{(a+b \sin ^{-1}(c x))^2}{x^4 (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=538 \[ -\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{32 b c^3 \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}} \]

[Out]

-(b^2*c^2)/(3*d^2*x*Sqrt[d - c^2*d*x^2]) + (2*b^2*c^4*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]
))/(3*d^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(3*d*x^3*(d - c^2*d*x^2)^(3/2)) -
(2*c^2*(a + b*ArcSin[c*x])^2)/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x
^2)^(3/2)) + (16*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((16*I)/3)*c^3*Sqrt[1 - c^2*x^2]*
(a + b*ArcSin[c*x])^2)/(d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(
(2*I)*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(
(2*I)*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*A
rcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x]
)])/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.05303, antiderivative size = 538, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 15, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.517, Rules used = {4701, 4655, 4653, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4705, 4679, 4419, 4183, 271} \[ -\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{32 b c^3 \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

-(b^2*c^2)/(3*d^2*x*Sqrt[d - c^2*d*x^2]) + (2*b^2*c^4*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]
))/(3*d^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(3*d*x^3*(d - c^2*d*x^2)^(3/2)) -
(2*c^2*(a + b*ArcSin[c*x])^2)/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x
^2)^(3/2)) + (16*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((16*I)/3)*c^3*Sqrt[1 - c^2*x^2]*
(a + b*ArcSin[c*x])^2)/(d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(
(2*I)*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(
(2*I)*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*A
rcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x]
)])/(d^2*Sqrt[d - c^2*d*x^2])

Rule 4701

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m
+ 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^F
racPart[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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && Inte
gerQ[m]

Rule 4655

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcSin[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(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] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 4653

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSin[c
*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 - c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSin[c*x
])^(n - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4675

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

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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 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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 4705

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[
((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*d*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d*(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)^Frac
Part[p])/(2*f*(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] && LtQ[p, -1] &&  !GtQ
[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rule 4679

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

Rule 4419

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[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 271

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\left (2 c^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (2 b c \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\left (8 c^4\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (b^2 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{8 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (16 c^4\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c^4 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (16 b c^5 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 c^4 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (32 b c^5 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (32 b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (64 i b c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{32 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (32 b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{32 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (16 i b^2 c^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{32 b c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 3.58719, size = 441, normalized size = 0.82 \[ \frac{b^2 c^3 \left (1-c^2 x^2\right )^{3/2} \left (-8 i \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )-8 i \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{\sqrt{1-c^2 x^2}}{c x}+\frac{c x}{\sqrt{1-c^2 x^2}}-\frac{8 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{c x}-\frac{\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{c^3 x^3}+\frac{8 c x \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\frac{c x \sin ^{-1}(c x)^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac{\sin ^{-1}(c x)}{c^2 x^2-1}-\frac{\sin ^{-1}(c x)}{c^2 x^2}-16 i \sin ^{-1}(c x)^2+16 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+16 \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )\right )-\frac{a^2 \left (16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right )}{x^3}-\frac{a b \left (c x \sqrt{1-c^2 x^2} \left (16 c^2 x^2 \left (c^2 x^2-1\right ) \log (c x)+8 c^2 x^2 \left (c^2 x^2-1\right ) \log \left (1-c^2 x^2\right )+1\right )+2 \left (16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right ) \sin ^{-1}(c x)\right )}{x^3}}{3 d \left (d-c^2 d x^2\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(-((a^2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))/x^3) - (a*b*(2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*Ar
cSin[c*x] + c*x*Sqrt[1 - c^2*x^2]*(1 + 16*c^2*x^2*(-1 + c^2*x^2)*Log[c*x] + 8*c^2*x^2*(-1 + c^2*x^2)*Log[1 - c
^2*x^2])))/x^3 + b^2*c^3*(1 - c^2*x^2)^(3/2)*((c*x)/Sqrt[1 - c^2*x^2] - Sqrt[1 - c^2*x^2]/(c*x) - ArcSin[c*x]/
(c^2*x^2) + ArcSin[c*x]/(-1 + c^2*x^2) - (16*I)*ArcSin[c*x]^2 + (c*x*ArcSin[c*x]^2)/(1 - c^2*x^2)^(3/2) + (8*c
*x*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - (Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c^3*x^3) - (8*Sqrt[1 - c^2*x^2]*ArcSi
n[c*x]^2)/(c*x) + 16*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 16*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])
] - (8*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (8*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(3*d*(d - c^2*d*x^2)^
(3/2))

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Maple [B]  time = 0.381, size = 5229, normalized size = 9.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{10} - 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} - d^{3} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3
*c^2*d^3*x^6 - d^3*x^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**2/x**4/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2/((-c^2*d*x^2 + d)^(5/2)*x^4), x)