3.473 \(\int \frac{\sqrt{c-a^2 c x^2}}{\sin ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{2 \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[ArcSin[a*x]]) - (2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2
*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(a*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0799091, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4659, 4635, 4406, 12, 3305, 3351} \[ -\frac{2 \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]/ArcSin[a*x]^(3/2),x]

[Out]

(-2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[ArcSin[a*x]]) - (2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2
*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(a*Sqrt[1 - a^2*x^2])

Rule 4659

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(Sqrt[1 - c^2*x^2]*
(d + e*x^2)^p*(a + b*ArcSin[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^Frac
Part[p])/(b*(n + 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, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{c-a^2 c x^2}}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (4 a \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sin ^{-1}(a x)}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (4 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (4 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (2 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\left (4 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.170107, size = 83, normalized size = 0.85 \[ -\frac{\sqrt{c \left (1-a^2 x^2\right )} \left (2 \sqrt{\pi } \sqrt{\sin ^{-1}(a x)} S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )+\cos \left (2 \sin ^{-1}(a x)\right )+1\right )}{a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a^2*c*x^2]/ArcSin[a*x]^(3/2),x]

[Out]

-((Sqrt[c*(1 - a^2*x^2)]*(1 + Cos[2*ArcSin[a*x]] + 2*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])
/Sqrt[Pi]]))/(a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]]))

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Maple [F]  time = 0.234, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{-{a}^{2}c{x}^{2}+c} \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(1/2)/asin(a*x)**(3/2),x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))/asin(a*x)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)/arcsin(a*x)^(3/2), x)