\(\int \frac {x^2}{\arcsin (a x)^4} \, dx\) [62]

Optimal result

Integrand size = 10, antiderivative size = 141 \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x}{3 a^2 \arcsin (a x)^2}+\frac {x^3}{2 \arcsin (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arcsin (a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{24 a^3}-\frac {9 \text {Si}(3 \arcsin (a x))}{8 a^3} \] Output:

-1/3*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^3-1/3*x/a^2/arcsin(a*x)^2+1/2*x^ 
3/arcsin(a*x)^2-1/3*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)+3/2*x^2*(-a^2*x^2+1 
)^(1/2)/a/arcsin(a*x)+1/24*Si(arcsin(a*x))/a^3-9/8*Si(3*arcsin(a*x))/a^3
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\frac {-\frac {8 a^2 x^2 \sqrt {1-a^2 x^2}}{\arcsin (a x)^3}+\frac {4 a x \left (-2+3 a^2 x^2\right )}{\arcsin (a x)^2}+\frac {4 \sqrt {1-a^2 x^2} \left (-2+9 a^2 x^2\right )}{\arcsin (a x)}-80 \text {Si}(\arcsin (a x))-27 (-3 \text {Si}(\arcsin (a x))+\text {Si}(3 \arcsin (a x)))}{24 a^3} \] Input:

Integrate[x^2/ArcSin[a*x]^4,x]
 

Output:

((-8*a^2*x^2*Sqrt[1 - a^2*x^2])/ArcSin[a*x]^3 + (4*a*x*(-2 + 3*a^2*x^2))/A 
rcSin[a*x]^2 + (4*Sqrt[1 - a^2*x^2]*(-2 + 9*a^2*x^2))/ArcSin[a*x] - 80*Sin 
Integral[ArcSin[a*x]] - 27*(-3*SinIntegral[ArcSin[a*x]] + SinIntegral[3*Ar 
cSin[a*x]]))/(24*a^3)
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 177, 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, 5132, 5142, 2009, 5224, 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.

\(\displaystyle \int \frac {x^2}{\arcsin (a x)^4} \, dx\)

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5132

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

\(\Big \downarrow \) 5142

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

\(\displaystyle \frac {2 \left (\frac {-\frac {\sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{a}}{2 a}-\frac {x}{2 a \arcsin (a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-a \left (\frac {3 \left (\frac {\frac {3}{4} \text {Si}(3 \arcsin (a x))-\frac {1}{4} \text {Si}(\arcsin (a x))}{a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{2 a}-\frac {x^3}{2 a \arcsin (a x)^2}\right )\)

Input:

Int[x^2/ArcSin[a*x]^4,x]
 

Output:

-1/3*(x^2*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^3) + (2*(-1/2*x/(a*ArcSin[a*x] 
^2) + (-(Sqrt[1 - a^2*x^2]/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/a)/ 
(2*a)))/(3*a) - a*(-1/2*x^3/(a*ArcSin[a*x]^2) + (3*(-((x^2*Sqrt[1 - a^2*x^ 
2])/(a*ArcSin[a*x])) + (-1/4*SinIntegral[ArcSin[a*x]] + (3*SinIntegral[3*A 
rcSin[a*x]])/4)/a^3))/(2*a))
 

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 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 5132

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2 
*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + Simp[c/(b*(n + 1)) 
  Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a 
, b, c}, x] && LtQ[n, -1]
 

rule 5142

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] - Simp 
[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[-a/b 
 + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c* 
x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
 

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 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]
 

rule 5224

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arcsin \left (a x \right )^{3}}+\frac {a x}{24 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{24}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\)	\(117\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arcsin \left (a x \right )^{3}}+\frac {a x}{24 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{24}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\)	\(117\)

Input:

int(x^2/arcsin(a*x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a^3*(-1/12*(-a^2*x^2+1)^(1/2)/arcsin(a*x)^3+1/24*a*x/arcsin(a*x)^2+1/24/ 
arcsin(a*x)*(-a^2*x^2+1)^(1/2)+1/24*Si(arcsin(a*x))+1/12*cos(3*arcsin(a*x) 
)/arcsin(a*x)^3-1/8*sin(3*arcsin(a*x))/arcsin(a*x)^2-3/8/arcsin(a*x)*cos(3 
*arcsin(a*x))-9/8*Si(3*arcsin(a*x)))
 

Fricas [F]

\[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )^{4}} \,d x } \] Input:

integrate(x^2/arcsin(a*x)^4,x, algorithm="fricas")
 

Output:

integral(x^2/arcsin(a*x)^4, x)
 

Sympy [F]

\[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\int \frac {x^{2}}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \] Input:

integrate(x**2/asin(a*x)**4,x)
 

Output:

Integral(x**2/asin(a*x)**4, x)
 

Maxima [F]

\[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )^{4}} \,d x } \] Input:

integrate(x^2/arcsin(a*x)^4,x, algorithm="maxima")
 

Output:

-1/6*(6*a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3*integrate(1/6*(27 
*a^2*x^3 - 20*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^3*x^2 - a)*arctan2(a*x, 
sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + (2*a^2*x^2 - (9*a^2*x^2 - 2)*arctan2( 
a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) - (3*a^ 
3*x^3 - 2*a*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)))/(a^3*arctan2(a* 
x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\frac {{\left (a^{2} x^{2} - 1\right )} x}{2 \, a^{2} \arcsin \left (a x\right )^{2}} - \frac {9 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{24 \, a^{3}} - \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, a^{3} \arcsin \left (a x\right )} + \frac {x}{6 \, a^{2} \arcsin \left (a x\right )^{2}} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{3} \arcsin \left (a x\right )} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} \] Input:

integrate(x^2/arcsin(a*x)^4,x, algorithm="giac")
 

Output:

1/2*(a^2*x^2 - 1)*x/(a^2*arcsin(a*x)^2) - 9/8*sin_integral(3*arcsin(a*x))/ 
a^3 + 1/24*sin_integral(arcsin(a*x))/a^3 - 3/2*(-a^2*x^2 + 1)^(3/2)/(a^3*a 
rcsin(a*x)) + 1/6*x/(a^2*arcsin(a*x)^2) + 7/6*sqrt(-a^2*x^2 + 1)/(a^3*arcs 
in(a*x)) + 1/3*(-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x)^3) - 1/3*sqrt(-a^2*x^ 
2 + 1)/(a^3*arcsin(a*x)^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \] Input:

int(x^2/asin(a*x)^4,x)
 

Output:

int(x^2/asin(a*x)^4, x)
 

Reduce [F]

\[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\int \frac {x^{2}}{\mathit {asin} \left (a x \right )^{4}}d x \] Input:

int(x^2/asin(a*x)^4,x)
 

Output:

int(x**2/asin(a*x)**4,x)