3.2.64 \(\int \frac {x}{\sqrt {a+b \arcsin (c+d x)}} \, dx\) [164]

3.2.64.1 Optimal result

Integrand size = 16, antiderivative size = 211 \[ \int \frac {x}{\sqrt {a+b \arcsin (c+d x)}} \, dx=-\frac {c \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{\sqrt {b} d^2}+\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} d^2}-\frac {c \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{\sqrt {b} d^2}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} d^2} \]

output

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

3.2.64.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.47 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03 \[ \int \frac {x}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\frac {\frac {i c e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )-e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{\sqrt {a+b \arcsin (c+d x)}}+\frac {\sqrt {\pi } \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )\right )}{\sqrt {b}}}{2 d^2} \]

input

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

output

((I*c*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcS 
in[c + d*x]))/b] - E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gam 
ma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcSin[c 
+ d*x]]) + (Sqrt[Pi]*(Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]] 
)/(Sqrt[b]*Sqrt[Pi])] - FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]* 
Sqrt[Pi])]*Sin[(2*a)/b]))/Sqrt[b])/(2*d^2)
 

3.2.64.3 Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5304, 25, 27, 5246, 7267, 7292, 7293, 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.

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

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {x}{\sqrt {a+b \arcsin (c+d x)}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5246

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {\frac {\pi }{2}} \sqrt {b} c \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\sqrt {\frac {\pi }{2}} \sqrt {b} c \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b d^2}\)

input

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

output

(-2*(Sqrt[b]*c*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[ 
c + d*x]])/Sqrt[b]] - (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + 
b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/4 + Sqrt[b]*c*Sqrt[Pi/2]*FresnelS 
[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b] + (Sqrt[b]*Sqr 
t[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2* 
a)/b])/4))/(b*d^2)
 

3.2.64.3.1 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 2009

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

rule 5246

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

rule 5304

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.2.64.4 Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \left (2 \sqrt {2}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) c -2 \sqrt {2}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) c +\cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+\sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )\right )}{2 d^{2}}\)	\(179\)

input

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

output

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

3.2.64.5 Fricas [F(-2)]

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

input

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

output

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

3.2.64.6 Sympy [F]

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

input

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

output

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

3.2.64.7 Maxima [F]

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

input

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

output

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

3.2.64.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.45 \[ \int \frac {x}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\frac {\sqrt {\pi } c \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{d^{2} {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {\sqrt {\pi } c \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{d^{2} {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, d^{2} {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, \sqrt {b} d^{2} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]

input

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

output

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

3.2.64.9 Mupad [F(-1)]

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

input

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

output

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