\(\int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx\) [163]

Optimal result

Integrand size = 12, antiderivative size = 137 \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c} \] Output:

-2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))^(1/2)-2*2^(1/2)*Pi^(1/2)*cos(a 
/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(3/2)/c+2 
*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2 
))*sin(a/b)/b^(3/2)/c
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {e^{-\frac {i (a+b \arcsin (c x))}{b}} \left (e^{i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+e^{\frac {i a}{b}} \left (-1-e^{2 i \arcsin (c x)}+e^{\frac {i (a+b \arcsin (c x))}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )\right )}{b c \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

(E^(I*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a 
+ b*ArcSin[c*x]))/b] + E^((I*a)/b)*(-1 - E^((2*I)*ArcSin[c*x]) + E^((I*(a 
+ b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b* 
ArcSin[c*x]))/b]))/(b*c*E^((I*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c* 
x]])
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5132, 5224, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 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.

\(\displaystyle \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx\)

\(\Big \downarrow \) 5132

\(\displaystyle -\frac {2 c \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

\(\displaystyle -\frac {2 \left (-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

\(\displaystyle -\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 3786

\(\displaystyle -\frac {2 \left (2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\)

Input:

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

Output:

(-2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (2*(Sqrt[b]*Sqrt[2* 
Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] - Sqrt 
[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[ 
a/b]))/(b^2*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3786

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]
 

rule 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
 

rule 3832

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 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 5132

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2 
*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + Simp[c/(b*(n + 1)) 
  Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a 
, b, c}, x] && 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 [A] (verified)

Time = 0.00 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {2 \left (-\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+\cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )\right )}{c b \sqrt {a +b \arcsin \left (c x \right )}}\)	\(158\)

Input:

int(1/(a+b*arcsin(c*x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2/c/b/(a+b*arcsin(c*x))^(1/2)*(-(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin 
(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c 
*x))^(1/2)/b)-(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/ 
b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)+cos(- 
(a+b*arcsin(c*x))/b+a/b))
                                                                                    
                                                                                    
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(1/(a+b*asin(c*x))**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate(1/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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