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

Optimal. Leaf size=319 \[ -\frac{2 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}-\frac{2 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+\frac{4 b c^3 \sqrt{1-c^2 x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt{d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.390316, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {4701, 4681, 4625, 3717, 2190, 2279, 2391, 4627, 264} \[ -\frac{2 i b^2 c^3 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{d-c^2 d x^2}}-\frac{2 i c^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+\frac{4 b c^3 \sqrt{1-c^2 x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*c^2*(1 - c^2*x^2))/(3*x*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*x^2*Sqrt[d
- c^2*d*x^2]) - (((2*I)/3)*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2] - (Sqrt[d - c^2*d
*x^2]*(a + b*ArcSin[c*x])^2)/(3*d*x^3) - (2*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*d*x) + (4*b*c^3*
Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/(3*Sqrt[d - c^2*d*x^2]) - (((2*I)/3)*b^2
*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/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 4681

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[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[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*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && 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 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 264

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]

Rubi steps

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

Mathematica [A]  time = 0.671191, size = 269, normalized size = 0.84 \[ -\frac{\sqrt{1-c^2 x^2} \left (2 i b^2 c^3 x^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+2 a^2 c^2 x^2 \sqrt{1-c^2 x^2}+a^2 \sqrt{1-c^2 x^2}-4 a b c^3 x^3 \log (c x)-b \sin ^{-1}(c x) \left (-2 a \sqrt{1-c^2 x^2} \left (2 c^2 x^2+1\right )+4 b c^3 x^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b c x\right )+a b c x+b^2 c^2 x^2 \sqrt{1-c^2 x^2}+b^2 \left (2 i c^3 x^3+2 c^2 x^2 \sqrt{1-c^2 x^2}+\sqrt{1-c^2 x^2}\right ) \sin ^{-1}(c x)^2\right )}{3 x^3 \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.336, size = 2320, normalized size = 7.3 \begin{align*} \text{result 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)^(1/2),x)

[Out]

-4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*arcsin(c*x)*(-c^2*x^2+1)*c^6-2*I*b^2*(-d*(c^2*
x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^2*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^5+a*b*(-d*(c^2*x^2-1))^(1/2)/(3
*c^4*x^4-2*c^2*x^2-1)/d*c^3*(-c^2*x^2+1)^(1/2)+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x^3*ar
csin(c*x)-4*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5+2/3*
a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*arcsin(c*x)*c^4+8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x
^4-2*c^2*x^2-1)/d/x*arcsin(c*x)*c^2-4/3*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*ln((I*c*x+
(-c^2*x^2+1)^(1/2))^2-1)*c^3-4/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^5*c^8+2/3*I*a*b*(-d*
(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*c^6-4*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^
3*arcsin(c*x)*c^6-2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*
c^3-4/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^5*arcsin(c*x)*c^8+2/3*I*b^2*(-d*(c^2*x^2-1))^
(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*arcsin(c*x)*c^6-I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^2
*(-c^2*x^2+1)^(1/2)*c^5+2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*arcsin(c*x)*c^4+4/3*I*b^2
*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*c^3*arcsin(c*x)^2+4/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c
^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*c^3*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+4/3*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*
x^2-1))^(1/2)/d/(c^2*x^2-1)*c^3*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-4/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1)
)^(1/2)/d/(c^2*x^2-1)*c^3*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-4/3*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1
))^(1/2)/d/(c^2*x^2-1)*c^3*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^
4-2*c^2*x^2-1)/d/x^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c+2/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)
/d*x*c^4+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x^2*(-c^2*x^2+1)^(1/2)*c-2/3*b^2*(-d*(c^2*x^
2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+1)*c^6-2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1
)/d*x^3*arcsin(c*x)^2*c^6+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*arcsin(c*x)^2*c^4+4/3*b^2
*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x*arcsin(c*x)^2*c^2+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*
c^2*x^2-1)/d*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3-1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*(-c
^2*x^2+1)^(1/2)*c^3-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*c^6+2/3*b^2*(-d*(c^2*x^2-1))^
(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x*c^2+1/3*b^2*(
-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x^3*arcsin(c*x)^2-2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-
2*c^2*x^2-1)/d*x*arcsin(c*x)*(-c^2*x^2+1)*c^4+8/3*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1
)*arcsin(c*x)*c^3-4/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+1)*c^6-2/3*I*a*b*(-
d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*(-c^2*x^2+1)*c^4-4/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-
2*c^2*x^2-1)/d*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3-2/3*a^2*c^2/d/x*(-c^2*d*x^2+d)^(1/2)-2/3*b^2*(-d*(c^2*x^2-1)
)^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^5*c^8-1/3*a^2/d/x^3*(-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^4/(-c^2*d*x^2+d)^(1/2),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^{2} d x^{6} - d 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)^(1/2),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^2*d*x^6 - d*x^4), x)

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

[Out]

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

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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}}{\sqrt{-c^{2} d x^{2} + d} 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)^(1/2),x, algorithm="giac")

[Out]

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