\(\int \frac {x^2}{\sqrt {a+b \arcsin (c x)}} \, dx\) [156]

Optimal result

Integrand size = 16, antiderivative size = 223 \[ \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 \sqrt {b} c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 \sqrt {b} c^3}-\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{2 \sqrt {b} c^3} \] Output:

1/4*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^ 
(1/2)/b^(1/2))/b^(1/2)/c^3-1/12*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelC(6^(1/ 
2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(1/2)/c^3+1/4*2^(1/2)*Pi^(1 
/2)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^ 
(1/2)/c^3-1/12*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x) 
)^(1/2)/b^(1/2))*sin(3*a/b)/b^(1/2)/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.02 \[ \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}} \, dx=-\frac {i e^{-\frac {3 i a}{b}} \left (3 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )-3 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )+\sqrt {3} \left (-\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{24 c^3 \sqrt {a+b \arcsin (c x)}} \] Input:

Integrate[x^2/Sqrt[a + b*ArcSin[c*x]],x]
 

Output:

((-1/24*I)*(3*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2 
, ((-I)*(a + b*ArcSin[c*x]))/b] - 3*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[ 
c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*(-(Sqrt[((-I)*(a 
 + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b]) + E^(((6 
*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[ 
c*x]))/b])))/(c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5146, 4906, 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.

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

\(\Big \downarrow \) 5146

\(\displaystyle \frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b c^3}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {\int \left (\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b c^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b c^3}\)

Input:

Int[x^2/Sqrt[a + b*ArcSin[c*x]],x]
 

Output:

((Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]] 
)/Sqrt[b]])/2 - (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt 
[a + b*ArcSin[c*x]])/Sqrt[b]])/2 + (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi 
]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*Fres 
nelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2)/(b*c^3 
)
 

Defintions of rubi rules used

rule 2009

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

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 = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {\sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {3}{b}}\, \left (\cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {3}{b}}\, b -\sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {3}{b}}\, b +\cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right )-\sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right )\right )}{12 c^{3}}\)	\(196\)

Input:

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

Output:

-1/12/c^3*Pi^(1/2)*2^(1/2)*(-3/b)^(1/2)*(cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2 
)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)*(-3/b)^(1/2)*b-sin( 
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)*(-3/b)^(1/2)*b+cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/ 
2)*(a+b*arcsin(c*x))^(1/2)/b)-sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b 
)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b))
 

Fricas [F(-2)]

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

integrate(x^2/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.42 \[ \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {b}} - \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (\frac {3 i \, a}{b}\right )}}{4 \, {\left (\sqrt {6} \sqrt {b} + \frac {i \, \sqrt {6} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} c^{3}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{4 \, c^{3} {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{4 \, c^{3} {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a}}{2 \, \sqrt {b}} + \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (-\frac {3 i \, a}{b}\right )}}{4 \, {\left (\sqrt {6} \sqrt {b} - \frac {i \, \sqrt {6} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} c^{3}} \] Input:

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

Output:

1/4*sqrt(pi)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt 
(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*sqrt(b) 
+ I*sqrt(6)*b^(3/2)/abs(b))*c^3) - 1/4*sqrt(pi)*erf(-1/2*I*sqrt(2)*sqrt(b* 
arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(a 
bs(b))/b)*e^(I*a/b)/(c^3*(I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) 
) - 1/4*sqrt(pi)*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 
1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(c^3*(-I*sq 
rt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b)))) + 1/4*sqrt(pi)*erf(-1/2*sqrt 
(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a 
)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b 
))*c^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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