3.274 \(\int \frac{\sin ^{-1}(a x)^2}{(c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=283 \[ -\frac{2 i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{x}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{3 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.216651, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {4655, 4653, 4675, 3719, 2190, 2279, 2391, 4677, 191} \[ -\frac{2 i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{x}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{3 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2),x]

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.586031, size = 149, normalized size = 0.53 \[ \frac{-2 i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )+\left (a x \left (\frac{1}{1-a^2 x^2}+2\right )-2 i \sqrt{1-a^2 x^2}\right ) \sin ^{-1}(a x)^2+\frac{\sin ^{-1}(a x) \left (-1+\left (4-4 a^2 x^2\right ) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )\right )}{\sqrt{1-a^2 x^2}}+a x}{3 a c^2 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.167, size = 365, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,{c}^{3} \left ( 3\,{a}^{6}{x}^{6}-10\,{a}^{4}{x}^{4}+11\,{a}^{2}{x}^{2}-4 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}+2\,{a}^{3}{x}^{3}-2\,i\sqrt{-{a}^{2}{x}^{2}+1}-3\,ax \right ) \left ( -2\,i\arcsin \left ( ax \right ){x}^{4}{a}^{4}-2\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}+i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}-{a}^{4}{x}^{4}+3\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+4\,i\arcsin \left ( ax \right ){x}^{2}{a}^{2}+3\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa-i\sqrt{-{a}^{2}{x}^{2}+1}xa+3\,{a}^{2}{x}^{2}-4\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}-2\,i\arcsin \left ( ax \right ) -2 \right ) }+{\frac{{\frac{2\,i}{3}}}{a{c}^{3} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,i\arcsin \left ( ax \right ) \ln \left ( 1+ \left ( iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) +2\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}+{\it polylog} \left ( 2,- \left ( iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x)

[Out]

-1/3*(-c*(a^2*x^2-1))^(1/2)*(2*I*(-a^2*x^2+1)^(1/2)*x^2*a^2+2*a^3*x^3-2*I*(-a^2*x^2+1)^(1/2)-3*a*x)*(-2*I*arcs
in(a*x)*x^4*a^4-2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x^3*a^3+I*(-a^2*x^2+1)^(1/2)*x^3*a^3-a^4*x^4+3*arcsin(a*x)^2*
x^2*a^2+4*I*arcsin(a*x)*x^2*a^2+3*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x*a-I*(-a^2*x^2+1)^(1/2)*x*a+3*a^2*x^2-4*arcs
in(a*x)^2-2*I*arcsin(a*x)-2)/c^3/(3*a^6*x^6-10*a^4*x^4+11*a^2*x^2-4)/a+2/3*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1
))^(1/2)*(2*I*arcsin(a*x)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)+2*arcsin(a*x)^2+polylog(2,-(I*a*x+(-a^2*x^2+1)^(1
/2))^2))/a/c^3/(a^2*x^2-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{2}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^2/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**2/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(asin(a*x)**2/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(5/2), x)