Integrand size = 24, antiderivative size = 98 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}-\frac {2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{a \sqrt {1-a^2 x^2}} \] Output:
-2*(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/arcsin(a*x)^(1/2)-2*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)
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=-\frac {\sqrt {c \left (1-a^2 x^2\right )} \left (1+\cos (2 \arcsin (a x))+2 \sqrt {\pi } \sqrt {\arcsin (a x)} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )\right )}{a \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \] Input:
Integrate[Sqrt[c - a^2*c*x^2]/ArcSin[a*x]^(3/2),x]
Output:
-((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]]))
Time = 0.50 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5166, 5146, 4906, 27, 3042, 3786, 3832}
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.
\(\displaystyle \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 5166 |
\(\displaystyle -\frac {4 a \sqrt {c-a^2 c x^2} \int \frac {x}{\sqrt {\arcsin (a x)}}dx}{\sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}\) |
\(\Big \downarrow \) 5146 |
\(\displaystyle -\frac {4 \sqrt {c-a^2 c x^2} \int \frac {a x \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {4 \sqrt {c-a^2 c x^2} \int \frac {\sin (2 \arcsin (a x))}{2 \sqrt {\arcsin (a x)}}d\arcsin (a x)}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 \sqrt {c-a^2 c x^2} \int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \sqrt {c-a^2 c x^2} \int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle -\frac {4 \sqrt {c-a^2 c x^2} \int \sin (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -\frac {2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{a \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}{a \sqrt {\arcsin (a x)}}\) |
Input:
Int[Sqrt[c - a^2*c*x^2]/ArcSin[a*x]^(3/2),x]
Output:
(-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])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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]
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]
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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[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]
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]
\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\arcsin \left (a x \right )^{\frac {3}{2}}}d x\]
Input:
int((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x)
Output:
int((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x)
Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:
integrate((-a**2*c*x**2+c)**(1/2)/asin(a*x)**(3/2),x)
Output:
Integral(sqrt(-c*(a*x - 1)*(a*x + 1))/asin(a*x)**(3/2), x)
Exception generated. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c}}{\arcsin \left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(-a^2*c*x^2 + c)/arcsin(a*x)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \] Input:
int((c - a^2*c*x^2)^(1/2)/asin(a*x)^(3/2),x)
Output:
int((c - a^2*c*x^2)^(1/2)/asin(a*x)^(3/2), x)
\[ \int \frac {\sqrt {c-a^2 c x^2}}{\arcsin (a x)^{3/2}} \, dx=\sqrt {c}\, \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}}{\mathit {asin} \left (a x \right )^{2}}d x \right ) \] Input:
int((-a^2*c*x^2+c)^(1/2)/asin(a*x)^(3/2),x)
Output:
sqrt(c)*int((sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x)))/asin(a*x)**2,x)