3.1.62 \(\int \frac {x^2}{\arcsin (a x)^3} \, dx\) [62]

3.1.62.1 Optimal result

Integrand size = 10, antiderivative size = 82 \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{8 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a x))}{8 a^3} \]

output

-x/a^2/arcsin(a*x)+3/2*x^3/arcsin(a*x)-1/8*Ci(arcsin(a*x))/a^3+9/8*Ci(3*ar 
csin(a*x))/a^3-1/2*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^2
 

3.1.62.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\frac {\frac {4 a x \left (-a x \sqrt {1-a^2 x^2}+\left (-2+3 a^2 x^2\right ) \arcsin (a x)\right )}{\arcsin (a x)^2}-\operatorname {CosIntegral}(\arcsin (a x))+9 \operatorname {CosIntegral}(3 \arcsin (a x))}{8 a^3} \]

input

Integrate[x^2/ArcSin[a*x]^3,x]
 

output

((4*a*x*(-(a*x*Sqrt[1 - a^2*x^2]) + (-2 + 3*a^2*x^2)*ArcSin[a*x]))/ArcSin[ 
a*x]^2 - CosIntegral[ArcSin[a*x]] + 9*CosIntegral[3*ArcSin[a*x]])/(8*a^3)
 

3.1.62.3 Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5144, 5222, 5134, 3042, 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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3}{2} a \left (\frac {3 \left (\frac {1}{4} \operatorname {CosIntegral}(\arcsin (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arcsin (a x))\right )}{a^4}-\frac {x^3}{a \arcsin (a x)}\right )+\frac {\frac {\operatorname {CosIntegral}(\arcsin (a x))}{a^2}-\frac {x}{a \arcsin (a x)}}{a}-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}\)

input

Int[x^2/ArcSin[a*x]^3,x]
 

output

-1/2*(x^2*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^2) + (-(x/(a*ArcSin[a*x])) + C 
osIntegral[ArcSin[a*x]]/a^2)/a - (3*a*(-(x^3/(a*ArcSin[a*x])) + (3*(CosInt 
egral[ArcSin[a*x]]/4 - CosIntegral[3*ArcSin[a*x]]/4))/a^4))/2
 

3.1.62.3.1 Defintions of rubi rules used

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 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 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]
 

3.1.62.4 Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )^{2}}+\frac {a x}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\)	\(82\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )^{2}}+\frac {a x}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\)	\(82\)

input

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

output

1/a^3*(-1/8/arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+1/8*a*x/arcsin(a*x)-1/8*Ci(ar 
csin(a*x))+1/8/arcsin(a*x)^2*cos(3*arcsin(a*x))-3/8/arcsin(a*x)*sin(3*arcs 
in(a*x))+9/8*Ci(3*arcsin(a*x)))
 

3.1.62.5 Fricas [F]

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

input

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

output

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

3.1.62.6 Sympy [F]

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

input

integrate(x**2/asin(a*x)**3,x)
 

output

Integral(x**2/asin(a*x)**3, x)
 

3.1.62.7 Maxima [F]

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

input

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

output

-1/2*(sqrt(a*x + 1)*sqrt(-a*x + 1)*a*x^2 + arctan2(a*x, sqrt(a*x + 1)*sqrt 
(-a*x + 1))^2*integrate((9*a^2*x^2 - 2)/arctan2(a*x, sqrt(a*x + 1)*sqrt(-a 
*x + 1)), x) - (3*a^2*x^3 - 2*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1) 
))/(a^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)
 

3.1.62.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\frac {3 \, {\left (a^{2} x^{2} - 1\right )} x}{2 \, a^{2} \arcsin \left (a x\right )} + \frac {x}{2 \, a^{2} \arcsin \left (a x\right )} + \frac {9 \, \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{8 \, a^{3}} - \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a^{3}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, a^{3} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a^{3} \arcsin \left (a x\right )^{2}} \]

input

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

output

3/2*(a^2*x^2 - 1)*x/(a^2*arcsin(a*x)) + 1/2*x/(a^2*arcsin(a*x)) + 9/8*cos_ 
integral(3*arcsin(a*x))/a^3 - 1/8*cos_integral(arcsin(a*x))/a^3 + 1/2*(-a^ 
2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x)^2) - 1/2*sqrt(-a^2*x^2 + 1)/(a^3*arcsin( 
a*x)^2)
 

3.1.62.9 Mupad [F(-1)]

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

input

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

output

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