\(\int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx\) [140]

Optimal result

Integrand size = 24, antiderivative size = 130 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \arcsin (a x)^{3/2}}+\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\arcsin (a x)}}-\frac {8 \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{3 a \sqrt {1-a^2 x^2}} \] Output:

-2/3*(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/arcsin(a*x)^(3/2)+8/3*x*(-a 
^2*c*x^2+c)^(1/2)/arcsin(a*x)^(1/2)-8/3*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*Fres 
nelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))/a/(-a^2*x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=\frac {2 \sqrt {c-a^2 c x^2} \left (-1+a^2 x^2+4 a x \sqrt {1-a^2 x^2} \arcsin (a x)-\sqrt {2} (-i \arcsin (a x))^{3/2} \Gamma \left (\frac {1}{2},-2 i \arcsin (a x)\right )+\frac {\sqrt {2} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},2 i \arcsin (a x)\right )}{\sqrt {i \arcsin (a x)}}\right )}{3 a \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}} \] Input:

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

Output:

(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))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5166, 5142, 3042, 3785, 3833}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx\)

\(\Big \downarrow \) 5166

\(\displaystyle -\frac {4 a \sqrt {c-a^2 c x^2} \int \frac {x}{\arcsin (a x)^{3/2}}dx}{3 \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 5142

\(\displaystyle -\frac {4 a \sqrt {c-a^2 c x^2} \left (\frac {2 \int \frac {\cos (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 a \sqrt {c-a^2 c x^2} \left (\frac {2 \int \frac {\sin \left (2 \arcsin (a x)+\frac {\pi }{2}\right )}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 3785

\(\displaystyle -\frac {4 a \sqrt {c-a^2 c x^2} \left (\frac {4 \int \cos (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \arcsin (a x)^{3/2}}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {4 a \sqrt {c-a^2 c x^2} \left (\frac {2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{a^2}-\frac {2 x \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{3 a \arcsin (a x)^{3/2}}\)

Input:

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

Output:

(-2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2])/(3*a*ArcSin[a*x]^(3/2)) - (4*a* 
Sqrt[c - a^2*c*x^2]*((-2*x*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) + (2*S 
qrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/a^2))/(3*Sqrt[1 - a^2*x^ 
2])
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[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 3833

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 5142

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] - Simp 
[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 5166

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] + Simp[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]
 

Maple [F]

\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\arcsin \left (a x \right )^{\frac {5}{2}}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\arcsin \left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{5/2}} \, dx=\sqrt {c}\, \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}}{\mathit {asin} \left (a x \right )^{3}}d x \right ) \] Input:

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

Output:

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