\(\int \frac {x^3}{(a+b \arcsin (c x))^3} \, dx\) [54]

Optimal result

Integrand size = 14, antiderivative size = 198 \[ \int \frac {x^3}{(a+b \arcsin (c x))^3} \, dx=-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}-\frac {3 x^2}{2 b^2 c^2 (a+b \arcsin (c x))}+\frac {2 x^4}{b^2 (a+b \arcsin (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b^3 c^4}-\frac {\operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{b^3 c^4}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b^3 c^4}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{b^3 c^4} \] Output:

-1/2*x^3*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))^2-3/2*x^2/b^2/c^2/(a+b*a 
rcsin(c*x))+2*x^4/b^2/(a+b*arcsin(c*x))+1/2*Ci(2*(a+b*arcsin(c*x))/b)*sin( 
2*a/b)/b^3/c^4-Ci(4*(a+b*arcsin(c*x))/b)*sin(4*a/b)/b^3/c^4-1/2*cos(2*a/b) 
*Si(2*(a+b*arcsin(c*x))/b)/b^3/c^4+cos(4*a/b)*Si(4*(a+b*arcsin(c*x))/b)/b^ 
3/c^4
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.81 \[ \int \frac {x^3}{(a+b \arcsin (c x))^3} \, dx=\frac {-\frac {b^2 c^3 x^3 \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2}+\frac {b c^2 x^2 \left (-3+4 c^2 x^2\right )}{a+b \arcsin (c x)}+\operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )-2 \operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {4 a}{b}\right )-\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+2 \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{2 b^3 c^4} \] Input:

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

Output:

(-((b^2*c^3*x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2) + (b*c^2*x^2*(-3 
 + 4*c^2*x^2))/(a + b*ArcSin[c*x]) + CosIntegral[2*(a/b + ArcSin[c*x])]*Si 
n[(2*a)/b] - 2*CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] - Cos[(2*a) 
/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 2*Cos[(4*a)/b]*SinIntegral[4*(a/b 
 + ArcSin[c*x])])/(2*b^3*c^4)
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.31, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5144, 5222, 5146, 25, 4906, 27, 2009, 3042, 3784, 25, 3042, 3780, 3783}

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.

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

\(\Big \downarrow \) 5144

\(\displaystyle \frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}dx}{2 b c}-\frac {2 c \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}dx}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 5222

\(\displaystyle \frac {3 \left (\frac {2 \int \frac {x}{a+b \arcsin (c x)}dx}{b c}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {2 c \left (\frac {4 \int \frac {x^3}{a+b \arcsin (c x)}dx}{b c}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 5146

\(\displaystyle -\frac {2 c \left (\frac {4 \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}+\frac {3 \left (\frac {2 \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 c \left (-\frac {4 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}+\frac {3 \left (-\frac {2 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {2 c \left (-\frac {4 \int \left (\frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{4 (a+b \arcsin (c x))}-\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}+\frac {3 \left (-\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 (a+b \arcsin (c x))}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 c \left (-\frac {4 \int \left (\frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{4 (a+b \arcsin (c x))}-\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}+\frac {3 \left (-\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (-\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {2 c \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {2 c \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {3 \left (\frac {-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))-\cos \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {2 c \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {2 c \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {2 c \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {2 c \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

\(\Big \downarrow \) 3783

\(\displaystyle -\frac {2 c \left (\frac {4 \left (-\frac {1}{4} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^5}-\frac {x^4}{b c (a+b \arcsin (c x))}\right )}{b}+\frac {3 \left (\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^3}-\frac {x^2}{b c (a+b \arcsin (c x))}\right )}{2 b c}-\frac {x^3 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}\)

Input:

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

Output:

-1/2*(x^3*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])^2) + (3*(-(x^2/(b*c* 
(a + b*ArcSin[c*x]))) + (-(CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a 
)/b]) + Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^3)))/( 
2*b*c) - (2*c*(-(x^4/(b*c*(a + b*ArcSin[c*x]))) + (4*(-1/4*(CosIntegral[(2 
*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b]) + (CosIntegral[(4*(a + b*ArcSin[c*x 
]))/b]*Sin[(4*a)/b])/8 + (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x])) 
/b])/4 - (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/8))/(b^2*c^ 
5)))/b
 

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

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

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

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

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 5144

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] + (Sim 
p[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt 
[1 - c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1))   Int[x^(m - 1)*((a + b*ArcSi 
n[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[ 
m, 0] && LtQ[n, -2]
 

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]
 

rule 5222

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) 
+ (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^ 
2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Simp[f*(m/(b*c*(n 
 + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[(f*x)^(m - 1)*(a + b* 
ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2* 
d + e, 0] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{16 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {4 \arcsin \left (c x \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) b -4 \arcsin \left (c x \right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b +4 \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) a -4 \,\operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a -\cos \left (4 \arcsin \left (c x \right )\right ) b}{4 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}-\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {2 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -2 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -2 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +\cos \left (2 \arcsin \left (c x \right )\right ) b}{4 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{4}}\)	\(309\)
default	\(\frac {\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{16 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {4 \arcsin \left (c x \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) b -4 \arcsin \left (c x \right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b +4 \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) a -4 \,\operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a -\cos \left (4 \arcsin \left (c x \right )\right ) b}{4 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}-\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {2 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -2 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -2 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +\cos \left (2 \arcsin \left (c x \right )\right ) b}{4 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{4}}\)	\(309\)

Input:

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

Output:

1/c^4*(1/16*sin(4*arcsin(c*x))/(a+b*arcsin(c*x))^2/b-1/4*(4*arcsin(c*x)*si 
n(4*a/b)*Ci(4*arcsin(c*x)+4*a/b)*b-4*arcsin(c*x)*Si(4*arcsin(c*x)+4*a/b)*c 
os(4*a/b)*b+4*sin(4*a/b)*Ci(4*arcsin(c*x)+4*a/b)*a-4*Si(4*arcsin(c*x)+4*a/ 
b)*cos(4*a/b)*a-cos(4*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^3-1/8*sin(2*arcs 
in(c*x))/(a+b*arcsin(c*x))^2/b-1/4*(2*arcsin(c*x)*Si(2*arcsin(c*x)+2*a/b)* 
cos(2*a/b)*b-2*arcsin(c*x)*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*b+2*Si(2*arc 
sin(c*x)+2*a/b)*cos(2*a/b)*a-2*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*a+cos(2* 
arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^3)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(4*a*c^2*x^4 - sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x^3 - 3*a*x^2 + (4*b*c 
^2*x^4 - 3*b*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - 2*(b^4*c^2* 
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*arctan2(c*x, sq 
rt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2)*integrate((8*c^2*x^3 - 3*x)/(b^ 
3*c^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b^2*c^2), x))/(b^4*c^ 
2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*arctan2(c*x, 
sqrt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2075 vs. \(2 (188) = 376\).

Time = 0.22 (sec) , antiderivative size = 2075, normalized size of antiderivative = 10.48 \[ \int \frac {x^3}{(a+b \arcsin (c x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-8*b^2*arcsin(c*x)^2*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(c*x))*sin(a/ 
b)/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c^4) + 8*b^2 
*arcsin(c*x)^2*cos(a/b)^4*sin_integral(4*a/b + 4*arcsin(c*x))/(b^5*c^4*arc 
sin(c*x)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c^4) - 16*a*b*arcsin(c*x)*c 
os(a/b)^3*cos_integral(4*a/b + 4*arcsin(c*x))*sin(a/b)/(b^5*c^4*arcsin(c*x 
)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c^4) + 16*a*b*arcsin(c*x)*cos(a/b) 
^4*sin_integral(4*a/b + 4*arcsin(c*x))/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^ 
4*arcsin(c*x) + a^2*b^3*c^4) + 4*b^2*arcsin(c*x)^2*cos(a/b)*cos_integral(4 
*a/b + 4*arcsin(c*x))*sin(a/b)/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^4*arcsin 
(c*x) + a^2*b^3*c^4) - 8*a^2*cos(a/b)^3*cos_integral(4*a/b + 4*arcsin(c*x) 
)*sin(a/b)/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c^4) 
 + b^2*arcsin(c*x)^2*cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a/b) 
/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c^4) - 8*b^2*a 
rcsin(c*x)^2*cos(a/b)^2*sin_integral(4*a/b + 4*arcsin(c*x))/(b^5*c^4*arcsi 
n(c*x)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c^4) + 8*a^2*cos(a/b)^4*sin_i 
ntegral(4*a/b + 4*arcsin(c*x))/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^4*arcsin 
(c*x) + a^2*b^3*c^4) - b^2*arcsin(c*x)^2*cos(a/b)^2*sin_integral(2*a/b + 2 
*arcsin(c*x))/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c 
^4) + 8*a*b*arcsin(c*x)*cos(a/b)*cos_integral(4*a/b + 4*arcsin(c*x))*sin(a 
/b)/(b^5*c^4*arcsin(c*x)^2 + 2*a*b^4*c^4*arcsin(c*x) + a^2*b^3*c^4) + 2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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