3.3.53 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x^4 (d-c^2 d x^2)^{3/2}} \, dx\) [253]

Optimal. Leaf size=483 \[ -\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {ArcSin}(c x))^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}} \]

[Out]

-1/3*b^2*c^2*(-c^2*x^2+1)/d/x/(-c^2*d*x^2+d)^(1/2)-1/3*(a+b*arcsin(c*x))^2/d/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*c^2*
(a+b*arcsin(c*x))^2/d/x/(-c^2*d*x^2+d)^(1/2)+8/3*c^4*x*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)-1/3*b*c*(a+b
*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/d/x^2/(-c^2*d*x^2+d)^(1/2)-8/3*I*c^3*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/d
/(-c^2*d*x^2+d)^(1/2)-20/3*b*c^3*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d/
(-c^2*d*x^2+d)^(1/2)+16/3*b*c^3*(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/(-c^
2*d*x^2+d)^(1/2)-I*b^2*c^3*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-
5/3*I*b^2*c^3*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.58, antiderivative size = 483, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {4789, 4745, 4765, 3800, 2221, 2317, 2438, 4769, 4504, 4268, 270} \begin {gather*} -\frac {4 c^2 (a+b \text {ArcSin}(c x))^2}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

-1/3*(b^2*c^2*(1 - c^2*x^2))/(d*x*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*d*x^2*
Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(3*d*x^3*Sqrt[d - c^2*d*x^2]) - (4*c^2*(a + b*ArcSin[c*x])^2)/(3*
d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*c^3*Sqrt[1 -
c^2*x^2]*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (20*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Arc
Tanh[E^((2*I)*ArcSin[c*x])])/(3*d*Sqrt[d - c^2*d*x^2]) + (16*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1
+ E^((2*I)*ArcSin[c*x])])/(3*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSi
n[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (((5*I)/3)*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(d
*Sqrt[d - c^2*d*x^2])

Rule 270

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

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]

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 )^{3/2}} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {1}{3} \left (4 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, 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 )} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {1}{3} \left (8 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+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 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 \sqrt {d-c^2 d x^2}}+\frac {\left (8 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 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \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 )}{3 d \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 )}{1-c^2 x^2} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 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 \sqrt {d-c^2 d x^2}}+\frac {\left (32 i b c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 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 \sqrt {d-c^2 d x^2}}+\frac {16 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 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 i b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 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 \sqrt {d-c^2 d x^2}}+\frac {16 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 \sqrt {d-c^2 d x^2}}+\frac {5 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 \sqrt {d-c^2 d x^2}}-\frac {5 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 \sqrt {d-c^2 d x^2}}+\frac {\left (8 i b^2 c^3 \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 )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 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 \sqrt {d-c^2 d x^2}}+\frac {16 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 \sqrt {d-c^2 d x^2}}-\frac {i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {5 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 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 462, normalized size = 0.96 \begin {gather*} \frac {-a^2-4 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4+b^2 c^4 x^4-a b c x \sqrt {1-c^2 x^2}-2 a b \text {ArcSin}(c x)-8 a b c^2 x^2 \text {ArcSin}(c x)+16 a b c^4 x^4 \text {ArcSin}(c x)-b^2 c x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)-b^2 \text {ArcSin}(c x)^2-4 b^2 c^2 x^2 \text {ArcSin}(c x)^2+8 b^2 c^4 x^4 \text {ArcSin}(c x)^2-8 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2+10 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+6 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )+10 a b c^3 x^3 \sqrt {1-c^2 x^2} \log (c x)+3 a b c^3 x^3 \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )-3 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-5 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 d x^3 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

(-a^2 - 4*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 + b^2*c^4*x^4 - a*b*c*x*Sqrt[1 - c^2*x^2] - 2*a*b*ArcSin[c
*x] - 8*a*b*c^2*x^2*ArcSin[c*x] + 16*a*b*c^4*x^4*ArcSin[c*x] - b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - b^2*Arc
Sin[c*x]^2 - 4*b^2*c^2*x^2*ArcSin[c*x]^2 + 8*b^2*c^4*x^4*ArcSin[c*x]^2 - (8*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*A
rcSin[c*x]^2 + 10*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 6*b^2*c^3*x^3*Sqr
t[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + 10*a*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[c*x] + 3*a*b*
c^3*x^3*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - (3*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c
*x])] - (5*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(3*d*x^3*Sqrt[d - c^2*d*x^2])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2843 vs. \(2 (470 ) = 940\).
time = 0.58, size = 2844, normalized size = 5.89

method	result	size
default	\(\text {Expression too large to display}\)	\(2844\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-8/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*arcsin(c*x)*(-c^2*x^2+1)*c^4-40/3*b^2*(-d*(c^2
*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^5*c^8+7/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*
x*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x
^4-7*c^2*x^2-1)/d^2/x^3*arcsin(c*x)^2+32/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^7*c^10+64/
3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^5*arcsin(c*x)*(-c^2*x^2+1)*c^8-32/3*I*b^2*(-d*(c^
2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*arcsin(c*x)*(-c^2*x^2+1)*c^6-64/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)
/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^2*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^5+I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1)
)^(1/2)/d^2/(c^2*x^2-1)*c^3*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-10/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-
1))^(1/2)/d^2/(c^2*x^2-1)*c^3*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2
-1))^(1/2)/d^2/(c^2*x^2-1)*c^3*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-10/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*
(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*c^3*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+16/3*I*b^2*(-c^2*x^2+1)^(1/2
)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*c^3*arcsin(c*x)^2+10/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2
)/d^2/(c^2*x^2-1)*c^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+10/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2
)/d^2/(c^2*x^2-1)*c^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-32*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2
-1)/d^2*x^5*arcsin(c*x)*c^8+8*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*arcsin(c*x)*c^6-8/3
*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^2*(-c^2*x^2+1)^(1/2)*c^5+8/3*I*b^2*(-d*(c^2*x^2-1)
)^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*arcsin(c*x)*c^4-8/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)
/d^2*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^3+64/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^7*ar
csin(c*x)*c^10+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c
-64/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*arcsin(c*x)^2*c^6+8*b^2*(-d*(c^2*x^2-1))^(1/2
)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*arcsin(c*x)^2*c^4+4*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x*a
rcsin(c*x)^2*c^2-1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*(-c^2*x^2+1)^(1/2)*c^3-8/3*b^2*(
-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*(-c^2*x^2+1)*c^6+32/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*
x^4-7*c^2*x^2-1)/d^2*x^5*(-c^2*x^2+1)*c^8+8/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*arcsin(c*
x)*(-c^2*x^2+1)^(1/2)*c^3+8/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*c^4+64/3*I*a*b*(-d*(c
^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^7*c^10-128/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)
/d^2*x^3*arcsin(c*x)*c^6-32*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^5*c^8+8*I*a*b*(-d*(c^2*
x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*c^6+16*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*
arcsin(c*x)*c^4+8*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x*arcsin(c*x)*c^2+1/3*a*b*(-d*(c^2*x^
2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x^2*(-c^2*x^2+1)^(1/2)*c-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/
2)/d^2/(c^2*x^2-1)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*c^3-10/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d
^2/(c^2*x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c^3+a^2*(-1/3/d/x^3/(-c^2*d*x^2+d)^(1/2)+4/3*c^2*(-1/d/x/(-c
^2*d*x^2+d)^(1/2)+2*c^2/d*x/(-c^2*d*x^2+d)^(1/2)))-128/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/
d^2*x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5+32/3*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1
)*arcsin(c*x)*c^3-8/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*(-c^2*x^2+1)*c^4-32/3*I*a*b*(
-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*(-c^2*x^2+1)*c^6-16/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^
4*x^4-7*c^2*x^2-1)/d^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+64/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x
^2-1)/d^2*x^5*(-c^2*x^2+1)*c^8+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*(-c^2*x^2+1)^(1/2)*c
^3+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x^3*arcsin(c*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^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

1/3*(8*c^4*x/(sqrt(-c^2*d*x^2 + d)*d) - 4*c^2/(sqrt(-c^2*d*x^2 + d)*d*x) - 1/(sqrt(-c^2*d*x^2 + d)*d*x^3))*a^2
+ sqrt(d)*integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt
(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2*x^4), x)

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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^4/(-c^2*d*x^2+d)^(3/2),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^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2
*x^4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

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)**(3/2),x)

[Out]

Integral((a + b*asin(c*x))**2/(x**4*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)

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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^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2/((-c^2*d*x^2 + d)^(3/2)*x^4), 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^4\,{\left (d-c^2\,d\,x^2\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^4*(d - c^2*d*x^2)^(3/2)),x)

[Out]

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

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