3.5.73 \(\int \frac {\sqrt {c-a^2 c x^2}}{\text {ArcSin}(a x)^{3/2}} \, dx\) [473]

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

[Out]

-2*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-2*(-a^2*c*x^2+c)^
(1/2)*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4751, 4731, 4491, 12, 3386, 3432} \begin {gather*} -\frac {2 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(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 {\text {ArcSin}(a x)}} \end {gather*}

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 12

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

Rule 3386

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 3432

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

Rule 4491

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 4731

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

Rule 4751

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)/(b*(n + 1)))*Simp[(d + e*x^2)
^p/(1 - c^2*x^2)^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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.11, size = 83, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {c \left (1-a^2 x^2\right )} \left (1+\cos (2 \text {ArcSin}(a x))+2 \sqrt {\pi } \sqrt {\text {ArcSin}(a x)} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )\right )}{a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)}} \end {gather*}

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 = 0.33, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\arcsin \left (a x \right )^{\frac {3}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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