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

Optimal. Leaf size=396 \[ -\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b c \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d e x \sqrt{c d x+d} \sqrt{e-c e x}} \]

[Out]

-((a + b*ArcSin[c*x])^2/(d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])) + (2*c^2*x*(a + b*ArcSin[c*x])^2)/(d*e*Sqrt[d
+ c*d*x]*Sqrt[e - c*e*x]) - ((2*I)*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c
*e*x]) - (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqr
t[e - c*e*x]) + (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d
*x]*Sqrt[e - c*e*x]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqr
t[e - c*e*x]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*
e*x])

________________________________________________________________________________________

Rubi [A]  time = 0.857531, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {4739, 4701, 4651, 4675, 3719, 2190, 2279, 2391, 4679, 4419, 4183} \[ -\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b c \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d e x \sqrt{c d x+d} \sqrt{e-c e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcSin[c*x])^2/(d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])) + (2*c^2*x*(a + b*ArcSin[c*x])^2)/(d*e*Sqrt[d
+ c*d*x]*Sqrt[e - c*e*x]) - ((2*I)*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c
*e*x]) - (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqr
t[e - c*e*x]) + (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d
*x]*Sqrt[e - c*e*x]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqr
t[e - c*e*x]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*
e*x])

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(
q_), x_Symbol] :> Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^F
racPart[q], Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]

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 4651

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

Rubi steps

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

Mathematica [A]  time = 2.47636, size = 564, normalized size = 1.42 \[ \frac{c \csc \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \sec \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (-2 i b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-2 i b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )-i b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+4 a^2 c^2 x^2-2 a^2+2 a b \log (c x) \sin \left (2 \sin ^{-1}(c x)\right )-4 a b \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )+2 a b \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+2 a b \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-2 i b^2 \sin ^{-1}(c x)^2 \sin \left (2 \sin ^{-1}(c x)\right )+2 i \pi b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )+4 \pi b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+2 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+\pi b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+2 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-\pi b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+2 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\pi b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-2 b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )-4 \pi b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+\pi b^2 \sin \left (2 \sin ^{-1}(c x)\right ) \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )\right )}{4 d e \sqrt{c d x+d} \sqrt{e-c e x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Csc[ArcSin[c*x]/2]*Sec[ArcSin[c*x]/2]*(-2*a^2 + 4*a^2*c^2*x^2 - 4*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 2*b^
2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + (2*I)*b^2*Pi*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - (2*I)*b^2*ArcSin[c*x]^2*Sin
[2*ArcSin[c*x]] + 4*b^2*Pi*Log[1 + E^((-I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + b^2*Pi*Log[1 - I*E^(I*ArcSin[c*x
])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - b^2*Pi*Log[1 + I*
E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*
b^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[c*x]*Sin[2*ArcSin[c*x]] - 4*b^2*
Pi*Log[Cos[ArcSin[c*x]/2]]*Sin[2*ArcSin[c*x]] + b^2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]]*Sin[2*ArcSin[c*x]] +
2*a*b*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[Cos[ArcSin[c*x]/2] + Sin[Arc
Sin[c*x]/2]]*Sin[2*ArcSin[c*x]] - b^2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]*Sin[2*ArcSin[c*x]] - (2*I)*b^2*PolyL
og[2, (-I)*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - (2*I)*b^2*PolyLog[2, I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]
] - I*b^2*PolyLog[2, E^((2*I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]]))/(4*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])

________________________________________________________________________________________

Maple [F]  time = 0.49, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{{x}^{2}} \left ( cdx+d \right ) ^{-{\frac{3}{2}}} \left ( -cex+e \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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**2/(c*d*x+d)**(3/2)/(-c*e*x+e)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

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 d x + d\right )}^{\frac{3}{2}}{\left (-c e x + e\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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