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

Optimal. Leaf size=99 \[ \frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{1-a^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.121941, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4663, 4661, 3312, 3304, 3352} \[ \frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4663

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(d^(p - 1/2)*Sqrt[
d + e*x^2])/Sqrt[1 - c^2*x^2], Int[(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] &&  !(IntegerQ[p] || GtQ[d, 0])

Rule 4661

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[(
a + b*x)^n*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && I
GtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 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}}{\sqrt{\sin ^{-1}(a x)}} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{\sin ^{-1}(a x)}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{2 a \sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a \sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{1-a^2 x^2}}+\frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.128655, size = 118, normalized size = 1.19 \[ \frac{\sqrt{c \left (1-a^2 x^2\right )} \left (-i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )+i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )+8 \sin ^{-1}(a x)\right )}{8 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[c*(1 - a^2*x^2)]*(8*ArcSin[a*x] - I*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] + I*Sq
rt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]]))/(8*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])

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Maple [F]  time = 0.235, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{-{a}^{2}c{x}^{2}+c}{\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}}\, 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)^(1/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(1/2),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)^(1/2),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)^(1/2),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 )}}{\sqrt{\operatorname{asin}{\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)**(1/2),x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))/sqrt(asin(a*x)), 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}}{\sqrt{\arcsin \left (a x\right )}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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