\(\int \frac {x^4}{\arcsin (a x)^4} \, dx\) [60]

Optimal result

Integrand size = 10, antiderivative size = 158 \[ \int \frac {x^4}{\arcsin (a x)^4} \, dx=-\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {2 x^3}{3 a^2 \arcsin (a x)^2}+\frac {5 x^5}{6 \arcsin (a x)^2}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{48 a^5}-\frac {27 \text {Si}(3 \arcsin (a x))}{32 a^5}+\frac {125 \text {Si}(5 \arcsin (a x))}{96 a^5} \] Output:

-1/3*x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^3-2/3*x^3/a^2/arcsin(a*x)^2+5/6* 
x^5/arcsin(a*x)^2-2*x^2*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)+25/6*x^4*(-a^2* 
x^2+1)^(1/2)/a/arcsin(a*x)+1/48*Si(arcsin(a*x))/a^5-27/32*Si(3*arcsin(a*x) 
)/a^5+125/96*Si(5*arcsin(a*x))/a^5
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{\arcsin (a x)^4} \, dx=\frac {-32 a^4 x^4 \sqrt {1-a^2 x^2}-64 a^3 x^3 \arcsin (a x)+80 a^5 x^5 \arcsin (a x)-192 a^2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2+400 a^4 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2+2 \arcsin (a x)^3 \text {Si}(\arcsin (a x))-81 \arcsin (a x)^3 \text {Si}(3 \arcsin (a x))+125 \arcsin (a x)^3 \text {Si}(5 \arcsin (a x))}{96 a^5 \arcsin (a x)^3} \] Input:

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

Output:

(-32*a^4*x^4*Sqrt[1 - a^2*x^2] - 64*a^3*x^3*ArcSin[a*x] + 80*a^5*x^5*ArcSi 
n[a*x] - 192*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + 400*a^4*x^4*Sqrt[1 
- a^2*x^2]*ArcSin[a*x]^2 + 2*ArcSin[a*x]^3*SinIntegral[ArcSin[a*x]] - 81*A 
rcSin[a*x]^3*SinIntegral[3*ArcSin[a*x]] + 125*ArcSin[a*x]^3*SinIntegral[5* 
ArcSin[a*x]])/(96*a^5*ArcSin[a*x]^3)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5144, 5222, 5142, 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.

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

\(\Big \downarrow \) 5144

\(\displaystyle -\frac {5}{3} a \int \frac {x^5}{\sqrt {1-a^2 x^2} \arcsin (a x)^3}dx+\frac {4 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^3}dx}{3 a}-\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\)

\(\Big \downarrow \) 5222

\(\displaystyle -\frac {5}{3} a \left (\frac {5 \int \frac {x^4}{\arcsin (a x)^2}dx}{2 a}-\frac {x^5}{2 a \arcsin (a x)^2}\right )+\frac {4 \left (\frac {3 \int \frac {x^2}{\arcsin (a x)^2}dx}{2 a}-\frac {x^3}{2 a \arcsin (a x)^2}\right )}{3 a}-\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\)

\(\Big \downarrow \) 5142

\(\displaystyle -\frac {5}{3} a \left (\frac {5 \left (\frac {\int \left (-\frac {a x}{8 \arcsin (a x)}+\frac {9 \sin (3 \arcsin (a x))}{16 \arcsin (a x)}-\frac {5 \sin (5 \arcsin (a x))}{16 \arcsin (a x)}\right )d\arcsin (a x)}{a^5}-\frac {x^4 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{2 a}-\frac {x^5}{2 a \arcsin (a x)^2}\right )+\frac {4 \left (\frac {3 \left (\frac {\int \left (\frac {3 \sin (3 \arcsin (a x))}{4 \arcsin (a x)}-\frac {a x}{4 \arcsin (a x)}\right )d\arcsin (a x)}{a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{2 a}-\frac {x^3}{2 a \arcsin (a x)^2}\right )}{3 a}-\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {5}{3} a \left (\frac {5 \left (\frac {-\frac {1}{8} \text {Si}(\arcsin (a x))+\frac {9}{16} \text {Si}(3 \arcsin (a x))-\frac {5}{16} \text {Si}(5 \arcsin (a x))}{a^5}-\frac {x^4 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{2 a}-\frac {x^5}{2 a \arcsin (a x)^2}\right )+\frac {4 \left (\frac {3 \left (\frac {\frac {3}{4} \text {Si}(3 \arcsin (a x))-\frac {1}{4} \text {Si}(\arcsin (a x))}{a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{2 a}-\frac {x^3}{2 a \arcsin (a x)^2}\right )}{3 a}\)

Input:

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

Output:

-1/3*(x^4*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^3) + (4*(-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*ArcSin[a*x]])/4)/a^3))/(2*a)))/(3*a) - (5 
*a*(-1/2*x^5/(a*ArcSin[a*x]^2) + (5*(-((x^4*Sqrt[1 - a^2*x^2])/(a*ArcSin[a 
*x])) + (-1/8*SinIntegral[ArcSin[a*x]] + (9*SinIntegral[3*ArcSin[a*x]])/16 
 - (5*SinIntegral[5*ArcSin[a*x]])/16)/a^5))/(2*a)))/3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )^{3}}+\frac {a x}{48 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{48}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )^{3}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{48 \arcsin \left (a x \right )^{3}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )^{2}}+\frac {25 \cos \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )}+\frac {125 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{96}}{a^{5}}\)	\(171\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )^{3}}+\frac {a x}{48 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{48}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )^{3}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{48 \arcsin \left (a x \right )^{3}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )^{2}}+\frac {25 \cos \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )}+\frac {125 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{96}}{a^{5}}\)	\(171\)

Input:

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

Output:

1/a^5*(-1/24*(-a^2*x^2+1)^(1/2)/arcsin(a*x)^3+1/48*a*x/arcsin(a*x)^2+1/48/ 
arcsin(a*x)*(-a^2*x^2+1)^(1/2)+1/48*Si(arcsin(a*x))+1/16*cos(3*arcsin(a*x) 
)/arcsin(a*x)^3-3/32*sin(3*arcsin(a*x))/arcsin(a*x)^2-9/32/arcsin(a*x)*cos 
(3*arcsin(a*x))-27/32*Si(3*arcsin(a*x))-1/48*cos(5*arcsin(a*x))/arcsin(a*x 
)^3+5/96*sin(5*arcsin(a*x))/arcsin(a*x)^2+25/96/arcsin(a*x)*cos(5*arcsin(a 
*x))+125/96*Si(5*arcsin(a*x)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^4/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*(12 
5*a^4*x^5 - 136*a^2*x^3 + 24*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^5*x^2 - a 
^3)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + (2*a^2*x^4 - (25*a^2 
*x^4 - 12*x^2)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1) 
*sqrt(-a*x + 1) - (5*a^3*x^5 - 4*a*x^3)*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 = 250, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arcsin (a x)^4} \, dx=\frac {5 \, {\left (a^{2} x^{2} - 1\right )}^{2} x}{6 \, a^{4} \arcsin \left (a x\right )^{2}} + \frac {25 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{5} \arcsin \left (a x\right )} + \frac {{\left (a^{2} x^{2} - 1\right )} x}{a^{4} \arcsin \left (a x\right )^{2}} + \frac {125 \, \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{96 \, a^{5}} - \frac {27 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{32 \, a^{5}} + \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{48 \, a^{5}} - \frac {19 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{5} \arcsin \left (a x\right )} + \frac {x}{6 \, a^{4} \arcsin \left (a x\right )^{2}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} + \frac {13 \, \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{5} \arcsin \left (a x\right )} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} \] Input:

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

Output:

5/6*(a^2*x^2 - 1)^2*x/(a^4*arcsin(a*x)^2) + 25/6*(a^2*x^2 - 1)^2*sqrt(-a^2 
*x^2 + 1)/(a^5*arcsin(a*x)) + (a^2*x^2 - 1)*x/(a^4*arcsin(a*x)^2) + 125/96 
*sin_integral(5*arcsin(a*x))/a^5 - 27/32*sin_integral(3*arcsin(a*x))/a^5 + 
 1/48*sin_integral(arcsin(a*x))/a^5 - 19/3*(-a^2*x^2 + 1)^(3/2)/(a^5*arcsi 
n(a*x)) + 1/6*x/(a^4*arcsin(a*x)^2) - 1/3*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 
1)/(a^5*arcsin(a*x)^3) + 13/6*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)) + 2/3*( 
-a^2*x^2 + 1)^(3/2)/(a^5*arcsin(a*x)^3) - 1/3*sqrt(-a^2*x^2 + 1)/(a^5*arcs 
in(a*x)^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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