3.261 \(\int \frac{(a+b \sin ^{-1}(c x))^2}{x^2 (d-c^2 d x^2)^{5/2}} \, dx\)
Optimal. Leaf size=452 \[ -\frac{5 i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 b c \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2 c^2 x}{3 d^2 \sqrt{d-c^2 d x^2}} \]
[Out]
(b^2*c^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^
2]) - (a + b*ArcSin[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2)) + (4*c^2*x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)
^(3/2)) + (8*c^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*c*Sqrt[1 - c^2*x^2]*(a + b*
ArcSin[c*x])^2)/(d^2*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcS
in[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (16*b*c*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]) - (((5*I)/3)*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(d
^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*c*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 = 0.616969, antiderivative size = 452, normalized size of antiderivative = 1.,
number of steps used = 19, number of rules used = 14, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.483, Rules used
= {4701, 4655, 4653, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4705, 4679, 4419, 4183}
\[ -\frac{5 i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 b c \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2 c^2 x}{3 d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)),x]
[Out]
(b^2*c^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^
2]) - (a + b*ArcSin[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2)) + (4*c^2*x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)
^(3/2)) + (8*c^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*c*Sqrt[1 - c^2*x^2]*(a + b*
ArcSin[c*x])^2)/(d^2*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^((2*I)*ArcS
in[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (16*b*c*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]) - (((5*I)/3)*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(d
^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*c*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]
Rubi steps
\begin{align*} \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 (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\left (4 c^2\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 (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 )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \left (a+b \sin ^{-1}(c x)\right )}{d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{\left (8 c^2\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 (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^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c^2 \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 (8 b c^3 \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 x}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \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{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c^2 \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 (16 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}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 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 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^2 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 \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 (16 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 )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 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 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i c \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{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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (32 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 )}{3 d^2 \sqrt{d-c^2 d x^2}}-\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^2 \sqrt{d-c^2 d x^2}}+\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^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 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 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i c \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{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^2 \sqrt{d-c^2 d x^2}}+\frac{16 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 )}{3 d^2 \sqrt{d-c^2 d x^2}}+\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^2 \sqrt{d-c^2 d x^2}}-\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^2 \sqrt{d-c^2 d x^2}}-\frac{\left (16 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 )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 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 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i c \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{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^2 \sqrt{d-c^2 d x^2}}+\frac{16 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 )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 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 )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 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 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac{4 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{8 c^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i c \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{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^2 \sqrt{d-c^2 d x^2}}+\frac{16 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 )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 c \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{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 2.58508, size = 352, normalized size = 0.78 \[ -\frac{c \left (-b^2 \left (1-c^2 x^2\right )^{3/2} \left (-5 i \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )-3 i \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\frac{c x}{\sqrt{1-c^2 x^2}}-\frac{3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{c x}+\frac{5 c x \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\frac{c x \sin ^{-1}(c x)^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac{\sin ^{-1}(c x)}{c^2 x^2-1}-8 i \sin ^{-1}(c x)^2+6 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+10 \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )\right )+\frac{a^2 \left (8 c^4 x^4-12 c^2 x^2+3\right )}{c x}+a b \sqrt{1-c^2 x^2} \left (6 \left (c^2 x^2-1\right ) \log (c x)+5 \left (c^2 x^2-1\right ) \log \left (1-c^2 x^2\right )+1\right )+\frac{2 a b \left (8 c^4 x^4-12 c^2 x^2+3\right ) \sin ^{-1}(c x)}{c x}\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)),x]
[Out]
-(c*((a^2*(3 - 12*c^2*x^2 + 8*c^4*x^4))/(c*x) + (2*a*b*(3 - 12*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x])/(c*x) + a*b*S
qrt[1 - c^2*x^2]*(1 + 6*(-1 + c^2*x^2)*Log[c*x] + 5*(-1 + c^2*x^2)*Log[1 - c^2*x^2]) - b^2*(1 - c^2*x^2)^(3/2)
*((c*x)/Sqrt[1 - c^2*x^2] + ArcSin[c*x]/(-1 + c^2*x^2) - (8*I)*ArcSin[c*x]^2 + (c*x*ArcSin[c*x]^2)/(1 - c^2*x^
2)^(3/2) + (5*c*x*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - (3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*x) + 6*ArcSin[c*x]
*Log[1 - E^((2*I)*ArcSin[c*x])] + 10*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (5*I)*PolyLog[2, -E^((2*I)*A
rcSin[c*x])] - (3*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])])))/(3*d*(d - c^2*d*x^2)^(3/2))
________________________________________________________________________________________
Maple [B] time = 0.316, size = 3777, normalized size = 8.4 \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^2/(-c^2*d*x^2+d)^(5/2),x)
[Out]
136/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^2*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2
)*c^3+56*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*arcsin(c*x)^2*c^4-44*b^2*(-d*(
c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*arcsin(c*x)^2*c^2+3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*
c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c-88/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x
^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*(-c^2*x^2+1)*c^6+32/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*
c^2*x^2-9)/d^3*x^7*(-c^2*x^2+1)*c^8-8*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*(-c
^2*x^2+1)*c^2+80/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*(-c^2*x^2+1)*c^4-3*I
*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*(-c^2*x^2+1)^(1/2)*c+18*a*b*(-d*(c^2*x^2-1
))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3/x*arcsin(c*x)+3*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*
x^4+26*c^2*x^2-9)/d^3*(-c^2*x^2+1)^(1/2)*c-128/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2
-9)/d^3*x^4*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5+272/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2
*x^2-9)/d^3*x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+8*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2
*x^2-9)/d^3*x*c^2-128/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*arcsin(c*x)*c^6
+112*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*arcsin(c*x)*c^4-8/3*a*b*(-d*(c^2*x
^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^2*(-c^2*x^2+1)^(1/2)*c^3-88*a*b*(-d*(c^2*x^2-1))^(1/2)/
(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*arcsin(c*x)*c^2-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/
(c^2*x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c-10/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/(c^2*x
^2-1)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*c+64/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-
9)/d^3*x^9*c^10-224/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^7*c^8+280/3*I*a*b
*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*c^6-48*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6
*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*c^4-8/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)
/d^3*x^4*(-c^2*x^2+1)^(1/2)*c^5-48*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*ar
csin(c*x)*c^4-24*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*arcsin(c*x)^2*(-c^2*x^2+
1)^(1/2)*c-224/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^7*arcsin(c*x)*c^8+17/3
*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^2*(-c^2*x^2+1)^(1/2)*c^3+8*I*b^2*(-d*(
c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*arcsin(c*x)*c^2+16/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*
(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c*arcsin(c*x)^2+2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*
x^2-1)*c*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+5/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2
-1)*c*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-
1)*c*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+64/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)
/d^3*x^9*arcsin(c*x)*c^10+280/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*arcsi
n(c*x)*c^6-2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+
1)^(1/2))-8/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^2*arcsin(c*x)*(-c^2*x^2+1)^
(1/2)*c^3-10/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x
^2+1)^(1/2))^2)-2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*c*arcsin(c*x)*ln(1-I*c*x-(-c^2
*x^2+1)^(1/2))-a^2/d/x/(-c^2*d*x^2+d)^(3/2)+32/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9
)/d^3*x^9*c^10-40*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^7*c^8+160/3*b^2*(-d*(c^
2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*c^6-29*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^
4*x^4+26*c^2*x^2-9)/d^3*x^3*c^4+5*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*c^2+9*b
^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3/x*arcsin(c*x)^2+4/3*a^2*c^2/d*x/(-c^2*d*x^2+
d)^(3/2)-8*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*arcsin(c*x)*(-c^2*x^2+1)*c^2
+40*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*(-c^2*x^2+1)*c^4-8*I*a*b*(-d*(c^2
*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x*(-c^2*x^2+1)*c^2+32/3*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c
^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)*arcsin(c*x)*c-48*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^
2-9)/d^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c+64/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9
)/d^3*x^7*(-c^2*x^2+1)*c^8-160/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*(-c^
2*x^2+1)*c^6+40*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^3*arcsin(c*x)*(-c^2*x^2
+1)*c^4-64/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^4*arcsin(c*x)^2*(-c^2*x^2+
1)^(1/2)*c^5-160/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*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^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^7*arcsin(c*x)*(-c^2*x^2
+1)*c^8-64/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/d^3*x^5*arcsin(c*x)^2*c^6+8/3*a^2*
c^2/d^2*x/(-c^2*d*x^2+d)^(1/2)
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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^2/(-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^{8} - 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} - d^{3} 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^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^8 - 3*c^4*d^3*x^6 + 3*
c^2*d^3*x^4 - d^3*x^2), 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**2/(-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^{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^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^2), x)