3.1.63 \(\int \frac {x}{\arcsin (a x)^3} \, dx\) [63]

3.1.63.1 Optimal result

Integrand size = 8, antiderivative size = 64 \[ \int \frac {x}{\arcsin (a x)^3} \, dx=-\frac {x \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {1}{2 a^2 \arcsin (a x)}+\frac {x^2}{\arcsin (a x)}-\frac {\text {Si}(2 \arcsin (a x))}{a^2} \]

output

-1/2/a^2/arcsin(a*x)+x^2/arcsin(a*x)-Si(2*arcsin(a*x))/a^2-1/2*x*(-a^2*x^2 
+1)^(1/2)/a/arcsin(a*x)^2
 

3.1.63.2 Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\arcsin (a x)^3} \, dx=-\frac {a x \sqrt {1-a^2 x^2}+\left (1-2 a^2 x^2\right ) \arcsin (a x)+2 \arcsin (a x)^2 \text {Si}(2 \arcsin (a x))}{2 a^2 \arcsin (a x)^2} \]

input

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

output

-1/2*(a*x*Sqrt[1 - a^2*x^2] + (1 - 2*a^2*x^2)*ArcSin[a*x] + 2*ArcSin[a*x]^ 
2*SinIntegral[2*ArcSin[a*x]])/(a^2*ArcSin[a*x]^2)
 

3.1.63.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5144, 5152, 5222, 5146, 4906, 27, 3042, 3780}

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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

\(\displaystyle -a \left (\frac {\text {Si}(2 \arcsin (a x))}{a^3}-\frac {x^2}{a \arcsin (a x)}\right )-\frac {x \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {1}{2 a^2 \arcsin (a x)}\)

input

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

output

-1/2*(x*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^2) - 1/(2*a^2*ArcSin[a*x]) - a*( 
-(x^2/(a*ArcSin[a*x])) + SinIntegral[2*ArcSin[a*x]]/a^3)
 

3.1.63.3.1 Defintions of rubi rules used

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 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]
 

3.1.63.4 Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{4 \arcsin \left (a x \right )^{2}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{2 \arcsin \left (a x \right )}-\operatorname {Si}\left (2 \arcsin \left (a x \right )\right )}{a^{2}}\)	\(45\)
default	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{4 \arcsin \left (a x \right )^{2}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{2 \arcsin \left (a x \right )}-\operatorname {Si}\left (2 \arcsin \left (a x \right )\right )}{a^{2}}\)	\(45\)

input

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

output

1/a^2*(-1/4/arcsin(a*x)^2*sin(2*arcsin(a*x))-1/2/arcsin(a*x)*cos(2*arcsin( 
a*x))-Si(2*arcsin(a*x)))
 

3.1.63.5 Fricas [F]

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

input

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

output

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

3.1.63.6 Sympy [F]

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

input

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

output

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

3.1.63.7 Maxima [F]

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

input

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

output

-1/2*(4*a^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2*integrate(x/arcta 
n2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)), x) + sqrt(a*x + 1)*sqrt(-a*x + 1)*a 
*x - (2*a^2*x^2 - 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)))/(a^2*arct 
an2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)
 

3.1.63.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\arcsin (a x)^3} \, dx=-\frac {\operatorname {Si}\left (2 \, \arcsin \left (a x\right )\right )}{a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a \arcsin \left (a x\right )^{2}} + \frac {a^{2} x^{2} - 1}{a^{2} \arcsin \left (a x\right )} + \frac {1}{2 \, a^{2} \arcsin \left (a x\right )} \]

input

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

output

-sin_integral(2*arcsin(a*x))/a^2 - 1/2*sqrt(-a^2*x^2 + 1)*x/(a*arcsin(a*x) 
^2) + (a^2*x^2 - 1)/(a^2*arcsin(a*x)) + 1/2/(a^2*arcsin(a*x))
 

3.1.63.9 Mupad [F(-1)]

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

input

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

output

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