\(\int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx\) [55]

Optimal result

Integrand size = 14, antiderivative size = 197 \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=-\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}-\frac {x}{b^2 c^2 (a+b \arcsin (c x))}+\frac {3 x^3}{2 b^2 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{8 b^3 c^3}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{8 b^3 c^3} \] Output:

-1/2*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))^2-x/b^2/c^2/(a+b*arcsin( 
c*x))+3/2*x^3/b^2/(a+b*arcsin(c*x))-1/8*cos(a/b)*Ci((a+b*arcsin(c*x))/b)/b 
^3/c^3+9/8*cos(3*a/b)*Ci(3*(a+b*arcsin(c*x))/b)/b^3/c^3-1/8*sin(a/b)*Si((a 
+b*arcsin(c*x))/b)/b^3/c^3+9/8*sin(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^3/c^ 
3
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=-\frac {\frac {4 b^2 x^2 \sqrt {1-c^2 x^2}}{c (a+b \arcsin (c x))^2}+\frac {8 b x}{c^2 (a+b \arcsin (c x))}-\frac {12 b x^3}{a+b \arcsin (c x)}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{c^3}-\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{c^3}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^3}}{8 b^3} \] Input:

Integrate[x^2/(a + b*ArcSin[c*x])^3,x]
 

Output:

-1/8*((4*b^2*x^2*Sqrt[1 - c^2*x^2])/(c*(a + b*ArcSin[c*x])^2) + (8*b*x)/(c 
^2*(a + b*ArcSin[c*x])) - (12*b*x^3)/(a + b*ArcSin[c*x]) + (Cos[a/b]*CosIn 
tegral[a/b + ArcSin[c*x]])/c^3 - (9*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcS 
in[c*x])])/c^3 + (Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/c^3 - (9*Sin[(3 
*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/c^3)/b^3
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.26, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5144, 5222, 5134, 3042, 3784, 25, 3042, 3780, 3783, 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.

\(\displaystyle \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx\)

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^2/(a + b*ArcSin[c*x])^3,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 5134

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   Su 
bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, 
c, n}, x]
 

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 = 0.05 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{8 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}+\frac {\cos \left (3 \arcsin \left (c x \right )\right )}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}+\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}-\frac {3 \sin \left (3 \arcsin \left (c x \right )\right ) b}{8}}{\left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)
default	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{8 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}+\frac {\cos \left (3 \arcsin \left (c x \right )\right )}{8 \left (a +b \arcsin \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}+\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}-\frac {3 \sin \left (3 \arcsin \left (c x \right )\right ) b}{8}}{\left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)

Input:

int(x^2/(a+b*arcsin(c*x))^3,x,method=_RETURNVERBOSE)
 

Output:

1/c^3*(-1/8*(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^2/b-1/8*(arcsin(c*x)*Si(a 
rcsin(c*x)+a/b)*sin(a/b)*b+arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+Si(a 
rcsin(c*x)+a/b)*sin(a/b)*a+Ci(arcsin(c*x)+a/b)*cos(a/b)*a-x*b*c)/(a+b*arcs 
in(c*x))/b^3+1/8*cos(3*arcsin(c*x))/(a+b*arcsin(c*x))^2/b+3/8*(3*arcsin(c* 
x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b+3*arcsin(c*x)*Si(3*arcsin(c*x)+3*a 
/b)*sin(3*a/b)*b+3*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+3*Si(3*arcsin(c*x) 
+3*a/b)*sin(3*a/b)*a-sin(3*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^3)
 

Fricas [F]

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

integrate(x^2/(a+b*arcsin(c*x))^3,x, algorithm="fricas")
 

Output:

integral(x^2/(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]

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

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

Output:

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

Maxima [F]

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

integrate(x^2/(a+b*arcsin(c*x))^3,x, algorithm="maxima")
 

Output:

1/2*(3*a*c^2*x^3 - sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x^2 - 2*a*x + (3*b*c^2 
*x^3 - 2*b*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - 2*(b^4*c^2*arct 
an2(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(1/2*(9*c^2*x^2 - 2)/(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, sq 
rt(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. 1539 vs. \(2 (183) = 366\).

Time = 0.22 (sec) , antiderivative size = 1539, normalized size of antiderivative = 7.81 \[ \int \frac {x^2}{(a+b \arcsin (c x))^3} \, dx=\text {Too large to display} \] Input:

integrate(x^2/(a+b*arcsin(c*x))^3,x, algorithm="giac")
 

Output:

9/2*b^2*arcsin(c*x)^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5* 
c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9/2*b^2*arcsi 
n(c*x)^2*cos(a/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3* 
arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9*a*b*arcsin(c*x) 
*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2 
*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9*a*b*arcsin(c*x)*cos(a/b)^2*sin(a 
/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c 
^3*arcsin(c*x) + a^2*b^3*c^3) + 3/2*(c^2*x^2 - 1)*b^2*c*x*arcsin(c*x)/(b^5 
*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) - 27/8*b^2*arc 
sin(c*x)^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c* 
x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9/2*a^2*cos(a/b)^3*cos_int 
egral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c 
*x) + a^2*b^3*c^3) - 1/8*b^2*arcsin(c*x)^2*cos(a/b)*cos_integral(a/b + arc 
sin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) 
- 9/8*b^2*arcsin(c*x)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5* 
c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) + 9/2*a^2*cos(a 
/b)^2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 
+ 2*a*b^4*c^3*arcsin(c*x) + a^2*b^3*c^3) - 1/8*b^2*arcsin(c*x)^2*sin(a/b)* 
sin_integral(a/b + arcsin(c*x))/(b^5*c^3*arcsin(c*x)^2 + 2*a*b^4*c^3*arcsi 
n(c*x) + a^2*b^3*c^3) + 3/2*(c^2*x^2 - 1)*a*b*c*x/(b^5*c^3*arcsin(c*x)^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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