\(\int \frac {(c-a^2 c x^2)^{3/2}}{\arcsin (a x)^{3/2}} \, dx\) [134]

Optimal result

Integrand size = 24, antiderivative size = 164 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\arcsin (a x)^{3/2}} \, dx=-\frac {2 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a \sqrt {1-a^2 x^2}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{a \sqrt {1-a^2 x^2}} \] Output:

-2*c*(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/arcsin(a*x)^(1/2)-1/2*c*2^( 
1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a*x) 
^(1/2))/a/(-a^2*x^2+1)^(1/2)-2*c*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*FresnelS(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 = 0.32 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.29 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\arcsin (a x)^{3/2}} \, dx=-\frac {c e^{-4 i \arcsin (a x)} \sqrt {c-a^2 c x^2} \left (1+6 e^{4 i \arcsin (a x)}+e^{8 i \arcsin (a x)}+8 e^{4 i \arcsin (a x)} \cos (2 \arcsin (a x))+16 e^{4 i \arcsin (a x)} \sqrt {\pi } \sqrt {\arcsin (a x)} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-2 e^{4 i \arcsin (a x)} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {1}{2},-4 i \arcsin (a x)\right )-2 e^{4 i \arcsin (a x)} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {1}{2},4 i \arcsin (a x)\right )\right )}{8 a \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \] Input:

Integrate[(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(3/2),x]
 

Output:

-1/8*(c*Sqrt[c - a^2*c*x^2]*(1 + 6*E^((4*I)*ArcSin[a*x]) + E^((8*I)*ArcSin 
[a*x]) + 8*E^((4*I)*ArcSin[a*x])*Cos[2*ArcSin[a*x]] + 16*E^((4*I)*ArcSin[a 
*x])*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] - 
 2*E^((4*I)*ArcSin[a*x])*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a 
*x]] - 2*E^((4*I)*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (4*I)*ArcSin 
[a*x]]))/(a*E^((4*I)*ArcSin[a*x])*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5166, 5224, 4906, 2009}

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.

\(\displaystyle \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\arcsin (a x)^{3/2}} \, dx\)

\(\Big \downarrow \) 5166

\(\displaystyle -\frac {8 a c \sqrt {c-a^2 c x^2} \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx}{\sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\arcsin (a x)}}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {8 c \sqrt {c-a^2 c x^2} \int \frac {a x \left (1-a^2 x^2\right )^{3/2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\arcsin (a x)}}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {8 c \sqrt {c-a^2 c x^2} \int \left (\frac {\sin (2 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}+\frac {\sin (4 \arcsin (a x))}{8 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\arcsin (a x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8 c \sqrt {c-a^2 c x^2} \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )\right )}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\arcsin (a x)}}\)

Input:

Int[(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(3/2),x]
 

Output:

(-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSin[a*x]]) - (8*c* 
Sqrt[c - a^2*c*x^2]*((Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]]) 
/8 + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/4))/(a*Sqrt[1 - a 
^2*x^2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG 
tQ[p, 0]
 

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]
 

rule 5224

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x 
^2)^p]   Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, 
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [F]

\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arcsin \left (a x \right )^{\frac {3}{2}}}d x\]

Input:

int((-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(3/2),x)
 

Output:

int((-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((-a**2*c*x**2+c)**(3/2)/asin(a*x)**(3/2),x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((-a^2*c*x^2+c)^(3/2)/arcsin(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((-a^2*c*x^2 + c)^(3/2)/arcsin(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\arcsin (a x)^{3/2}} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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