\(\int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx\) [157]

Optimal result

Integrand size = 14, antiderivative size = 99 \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} c^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.24 \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=-\frac {e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{4 \sqrt {2} c^2 \sqrt {a+b \arcsin (c x)}} \] Input:

Integrate[x/Sqrt[a + b*ArcSin[c*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5146, 25, 4906, 27, 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.

\(\displaystyle \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx\)

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

Input:

Int[x/Sqrt[a + b*ArcSin[c*x]],x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 5146

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]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+\sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )\right )}{2 c^{2}}\)	\(91\)

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x/sqrt(b*arcsin(c*x) + a), x)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, c^{2} {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, \sqrt {b} c^{2} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \] Input:

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

Output:

1/4*I*sqrt(pi)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) 
 + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(c^2*(sqrt(b) - I*b^(3/2)/abs(b))) - 1/ 
4*I*sqrt(pi)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + 
 a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(sqrt(b)*c^2*(I*b/abs(b) + 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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