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

Optimal. Leaf size=314 \[ \frac{i b^2 c^3 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}+\frac{i c^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{2 b c^3 \sqrt{d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{b^2 c^2 \sqrt{d-c^2 d x^2}}{3 x}-\frac{b^2 c^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt{1-c^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.271609, antiderivative size = 314, 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 = {4681, 4685, 277, 216, 4625, 3717, 2190, 2279, 2391} \[ \frac{i b^2 c^3 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt{1-c^2 x^2}}+\frac{i c^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{2 b c^3 \sqrt{d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{b^2 c^2 \sqrt{d-c^2 d x^2}}{3 x}-\frac{b^2 c^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4685

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((f*x)
^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c*d^p)/(f*(m + 1)), Int[(f*x)^(m + 1)*
(1 - c^2*x^2)^(p - 1/2), x], x] - Dist[(2*e*p)/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*Arc
Sin[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

Rule 277

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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]

Rubi steps

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

Mathematica [A]  time = 1.18935, size = 248, normalized size = 0.79 \[ \frac{\sqrt{d-c^2 d x^2} \left (2 i b^2 c^3 x^3 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-2 \left (a^2 \left (1-c^2 x^2\right )^{3/2}+2 a b c^3 x^3 \log (c x)+a b c x+b^2 c^2 x^2 \sqrt{1-c^2 x^2}\right )-b \sin ^{-1}(c x) \left (3 a \sqrt{1-c^2 x^2}+a \cos \left (3 \sin ^{-1}(c x)\right )+4 b c^3 x^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+2 b c x\right )+2 b^2 \left (i c^3 x^3+c^2 x^2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2}\right ) \sin ^{-1}(c x)^2\right )}{6 x^3 \sqrt{1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(2*b^2*(I*c^3*x^3 - Sqrt[1 - c^2*x^2] + c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - b*ArcS
in[c*x]*(2*b*c*x + 3*a*Sqrt[1 - c^2*x^2] + a*Cos[3*ArcSin[c*x]] + 4*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])])
- 2*(a*b*c*x + b^2*c^2*x^2*Sqrt[1 - c^2*x^2] + a^2*(1 - c^2*x^2)^(3/2) + 2*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])]))/(6*x^3*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.361, size = 3017, normalized size = 9.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^4,x)

[Out]

2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^6-1/3*I*b^2*(-d*(c^2*x^
2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^8+1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^
4-3*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)^2*(-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^4-3*
c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4+I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*
(-c^2*x^2+1)^(1/2)*c^5+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^
2+1)^(1/2)*c^5+2*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^8-1/3*I*a*b*
(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8+2/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4
-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*c^6-1/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*c^4+1
/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c-6*a*b*(
-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^6+a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^
4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5+20/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2
+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4-10/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin
(c*x)*c^2+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c-I*b^2*(-
d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^7-4*I*a*b*(-d*(c^2*x^2-1))^(
1/2)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3/(3*c^2*x^2-3)-5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x
/(c^2*x^2-1)*arcsin(c*x)^2*c^2-1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+
1)^(1/2)*c^3-2*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^3/(3*c^2*x^2-3)*arcsin(c*x)^2-2*I*b^2*(-d*(c^
2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^3/(3*c^2*x^2-3)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b^2*(-d*(c^2*x^2
-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c^3/(3*c^2*x^2-3)*polylog(2,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^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*
c^3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*c
^3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)
/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^
2-1)*arcsin(c*x)^2*c^8-3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)^2*c^6+
10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)^2*c^4-a*b*(-d*(c^2*x^2-1))^(
1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^3+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^
2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*ln((I*c*x+(
-c^2*x^2+1)^(1/2))^2-1)*c^3+1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)
*(-c^2*x^2+1)*c^4-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8+5/3*b^2*(-d*(c^2*
x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*c^6-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+
1)*x/(c^2*x^2-1)*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x/(c^2*x^2-1)*c^2+1/3*b^2*(-d*(c^2
*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)^2+2*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4
-3*c^2*x^2+1)*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^7-2*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c
^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5-1/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2
*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6+1/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1
)*(-c^2*x^2+1)*c^4+2/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+
1)^(1/2)*c^3+I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(
1/2)*c^7-1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)*c^6
-I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^5-1/3
*a^2/d/x^3*(-c^2*d*x^2+d)^(3/2)

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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((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^4,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 )}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^4,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)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/x^4,x, algorithm="giac")

[Out]

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