3.6.95 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx\) [595]

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

[Out]

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

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Rubi [A]
time = 0.60, antiderivative size = 396, normalized size of antiderivative = 1.00, 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 = {4823, 4789, 4745, 4765, 3800, 2221, 2317, 2438, 4769, 4504, 4268} \begin {gather*} -\frac {2 i c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^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 \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{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 \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d e \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 c^2 x (a+b \text {ArcSin}(c x))^2}{d e \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {(a+b \text {ArcSin}(c x))^2}{d e x \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(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 {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}} \end {gather*}

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 3800

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 4268

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 4504

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 4745

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/d)*Simp[Sqrt[1 - c^2*x^2]/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 4765

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 4769

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 4789

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/(f*(m + 1)))*Simp[(d + e*x
^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4823

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)^Fr
acPart[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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.58, size = 564, normalized size = 1.42 \begin {gather*} \frac {c \csc \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \sec \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left (-2 a^2+4 a^2 c^2 x^2-4 a b \text {ArcSin}(c x) \cos (2 \text {ArcSin}(c x))-2 b^2 \text {ArcSin}(c x)^2 \cos (2 \text {ArcSin}(c x))+2 i b^2 \pi \text {ArcSin}(c x) \sin (2 \text {ArcSin}(c x))-2 i b^2 \text {ArcSin}(c x)^2 \sin (2 \text {ArcSin}(c x))+4 b^2 \pi \log \left (1+e^{-i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))+b^2 \pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))+2 b^2 \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))-b^2 \pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))+2 b^2 \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))+2 b^2 \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))+2 a b \log (c x) \sin (2 \text {ArcSin}(c x))-4 b^2 \pi \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \sin (2 \text {ArcSin}(c x))+b^2 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right ) \sin (2 \text {ArcSin}(c x))+2 a b \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \sin (2 \text {ArcSin}(c x))+2 a b \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \sin (2 \text {ArcSin}(c x))-b^2 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right ) \sin (2 \text {ArcSin}(c x))-2 i b^2 \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))-2 i b^2 \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))-i b^2 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right ) \sin (2 \text {ArcSin}(c x))\right )}{4 d e \sqrt {d+c d x} \sqrt {e-c e x}} \end {gather*}

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])

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Maple [F]
time = 1.32, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{x^{2} \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

a*b*c*(e^(-3/2)*log(c*x + 1)/d^(3/2) + e^(-3/2)*log(c*x - 1)/d^(3/2) + 2*e^(-3/2)*log(x)/d^(3/2)) + 2*(2*c^2*x
*e^(-1)/(sqrt(-c^2*d*x^2*e + d*e)*d) - e^(-1)/(sqrt(-c^2*d*x^2*e + d*e)*d*x))*a*b*arcsin(c*x) - b^2*e^(-1/2)*i
ntegrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/((c^2*d*x^4*e - d*x^2*e)*sqrt(c*x + 1)*sqrt(-c*x + 1)),
x)/sqrt(d) + (2*c^2*x*e^(-1)/(sqrt(-c^2*d*x^2*e + d*e)*d) - e^(-1)/(sqrt(-c^2*d*x^2*e + d*e)*d*x))*a^2

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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*x - 1)*e)*e^(-2)/(c^4*d^2*x^6
- 2*c^2*d^2*x^4 + d^2*x^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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