\(\int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx\) [162]

Optimal result

Integrand size = 14, antiderivative size = 130 \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} c^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.19 \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {i e^{-\frac {2 i a}{b}} \left (-\sqrt {2} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+\sqrt {2} e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )+2 i e^{\frac {2 i a}{b}} \sin (2 \arcsin (c x))\right )}{2 b c^2 \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

((I/2)*(-(Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a 
 + b*ArcSin[c*x]))/b]) + Sqrt[2]*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x 
]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] + (2*I)*E^(((2*I)*a)/b)*S 
in[2*ArcSin[c*x]]))/(b*c^2*E^(((2*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5142, 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.

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

\(\Big \downarrow \) 5142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

Input:

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

Output:

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

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 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]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+\sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )}{c^{2} b \sqrt {a +b \arcsin \left (c x \right )}}\)	\(156\)

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate(x/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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