3.5.78 \(\int \frac {\sqrt {c-a^2 c x^2}}{\text {ArcSin}(a x)^{5/2}} \, dx\) [478]

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

[Out]

-8/3*FresnelC(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)-2/3*(-a^2*c*x^2
+c)^(1/2)*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(3/2)+8/3*x*(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, 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 = {4751, 4727, 3385, 3433} \begin {gather*} -\frac {8 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(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 {\text {ArcSin}(a x)}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \text {ArcSin}(a x)^{3/2}} \end {gather*}

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 3385

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 3433

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

Rule 4727

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^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.27, size = 142, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {c-a^2 c x^2} \left (-1+a^2 x^2+4 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)-\sqrt {2} (-i \text {ArcSin}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcSin}(a x)\right )+\frac {\sqrt {2} \text {ArcSin}(a x)^2 \text {Gamma}\left (\frac {1}{2},2 i \text {ArcSin}(a x)\right )}{\sqrt {i \text {ArcSin}(a x)}}\right )}{3 a \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\arcsin \left (a x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(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 = 0.00, 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)^(5/2),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)^(5/2),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 {5}{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)**(5/2),x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))/asin(a*x)**(5/2), 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)^(5/2),x, algorithm="giac")

[Out]

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

[Out]

int((c - a^2*c*x^2)^(1/2)/asin(a*x)^(5/2), x)

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