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

Optimal. Leaf size=130 \[ -\frac{8 \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a \sqrt{1-a^2 x^2}}+\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}} \]

[Out]

(-2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2])/(3*a*ArcSin[a*x]^(3/2)) + (8*x*Sqrt[c - a^2*c*x^2])/(3*Sqrt[ArcSin[
a*x]]) - (8*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*a*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0733973, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4659, 4631, 3304, 3352} \[ -\frac{8 \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a \sqrt{1-a^2 x^2}}+\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2])/(3*a*ArcSin[a*x]^(3/2)) + (8*x*Sqrt[c - a^2*c*x^2])/(3*Sqrt[ArcSin[
a*x]]) - (8*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*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 4631

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

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(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 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[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)^{5/2}} \, dx &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{\left (4 a \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sin ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{\left (8 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{\left (16 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a \sqrt{1-a^2 x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{8 \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.470017, size = 142, normalized size = 1.09 \[ \frac{2 \sqrt{c-a^2 c x^2} \left (-\sqrt{2} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )+\frac{\sqrt{2} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )}{\sqrt{i \sin ^{-1}(a x)}}+a^2 x^2+4 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-1\right )}{3 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{-{a}^{2}c{x}^{2}+c} \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{5}{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)^(5/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(5/2),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)^(5/2),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)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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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{5}{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)^(5/2),x, algorithm="giac")

[Out]

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