\(\int \frac {x}{(a+b \arcsin (c x))^3} \, dx\) [56]

Optimal result

Integrand size = 12, antiderivative size = 130 \[ \int \frac {x}{(a+b \arcsin (c x))^3} \, dx=-\frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arcsin (c x))}+\frac {x^2}{b^2 (a+b \arcsin (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 c^2}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^3 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.83 \[ \int \frac {x}{(a+b \arcsin (c x))^3} \, dx=\frac {-\frac {b^2 c x \sqrt {1-c^2 x^2}}{(a+b \arcsin (c x))^2}+\frac {b \left (-1+2 c^2 x^2\right )}{a+b \arcsin (c x)}+2 \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )-2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{2 b^3 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5144, 5152, 5222, 5146, 25, 4906, 27, 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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

Input:

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

Output:

-1/2*(x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])^2) - 1/(2*b^2*c^2*(a + 
 b*ArcSin[c*x])) - (c*(-(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)))/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 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 5152

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

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 = 0.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{4 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {2 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -2 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -2 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +\cos \left (2 \arcsin \left (c x \right )\right ) b}{2 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{2}}\)	\(157\)
default	\(\frac {-\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{4 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {2 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -2 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -2 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +\cos \left (2 \arcsin \left (c x \right )\right ) b}{2 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{2}}\)	\(157\)

Input:

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

Output:

1/c^2*(-1/4*sin(2*arcsin(c*x))/(a+b*arcsin(c*x))^2/b-1/2*(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)*s 
in(2*a/b)*b+2*Si(2*arcsin(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]

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

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

Output:

integral(x/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(2*a*c^2*x^2 - sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x + (2*b*c^2*x^2 - b)* 
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - 4*(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)*integrate(x/(b^3*arctan2(c*x, sqrt(c*x + 1)*sqrt( 
-c*x + 1)) + a*b^2), x) - a)/(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. 864 vs. \(2 (124) = 248\).

Time = 0.18 (sec) , antiderivative size = 864, normalized size of antiderivative = 6.65 \[ \int \frac {x}{(a+b \arcsin (c x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

2*b^2*arcsin(c*x)^2*cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a/b)/ 
(b^5*c^2*arcsin(c*x)^2 + 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) - 2*b^2*ar 
csin(c*x)^2*cos(a/b)^2*sin_integral(2*a/b + 2*arcsin(c*x))/(b^5*c^2*arcsin 
(c*x)^2 + 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) + 4*a*b*arcsin(c*x)*cos(a 
/b)*cos_integral(2*a/b + 2*arcsin(c*x))*sin(a/b)/(b^5*c^2*arcsin(c*x)^2 + 
2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) - 4*a*b*arcsin(c*x)*cos(a/b)^2*sin_ 
integral(2*a/b + 2*arcsin(c*x))/(b^5*c^2*arcsin(c*x)^2 + 2*a*b^4*c^2*arcsi 
n(c*x) + a^2*b^3*c^2) + 2*a^2*cos(a/b)*cos_integral(2*a/b + 2*arcsin(c*x)) 
*sin(a/b)/(b^5*c^2*arcsin(c*x)^2 + 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) 
+ b^2*arcsin(c*x)^2*sin_integral(2*a/b + 2*arcsin(c*x))/(b^5*c^2*arcsin(c* 
x)^2 + 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) - 2*a^2*cos(a/b)^2*sin_integ 
ral(2*a/b + 2*arcsin(c*x))/(b^5*c^2*arcsin(c*x)^2 + 2*a*b^4*c^2*arcsin(c*x 
) + a^2*b^3*c^2) - 1/2*sqrt(-c^2*x^2 + 1)*b^2*c*x/(b^5*c^2*arcsin(c*x)^2 + 
 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) + (c^2*x^2 - 1)*b^2*arcsin(c*x)/(b 
^5*c^2*arcsin(c*x)^2 + 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) + 2*a*b*arcs 
in(c*x)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^5*c^2*arcsin(c*x)^2 + 2*a*b 
^4*c^2*arcsin(c*x) + a^2*b^3*c^2) + (c^2*x^2 - 1)*a*b/(b^5*c^2*arcsin(c*x) 
^2 + 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) + 1/2*b^2*arcsin(c*x)/(b^5*c^2 
*arcsin(c*x)^2 + 2*a*b^4*c^2*arcsin(c*x) + a^2*b^3*c^2) + a^2*sin_integral 
(2*a/b + 2*arcsin(c*x))/(b^5*c^2*arcsin(c*x)^2 + 2*a*b^4*c^2*arcsin(c*x...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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