\(\int \frac {x^3}{\arcsin (a x)^4} \, dx\) [61]

Optimal result

Integrand size = 10, antiderivative size = 144 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {2 x^4}{3 \arcsin (a x)^2}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)}-\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{3 a^4}+\frac {4 \operatorname {CosIntegral}(4 \arcsin (a x))}{3 a^4} \] Output:

-1/3*x^3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^3-1/2*x^2/a^2/arcsin(a*x)^2+2/3* 
x^4/arcsin(a*x)^2-x*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)+8/3*x^3*(-a^2*x^2+1 
)^(1/2)/a/arcsin(a*x)-1/3*Ci(2*arcsin(a*x))/a^4+4/3*Ci(4*arcsin(a*x))/a^4
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.74 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\frac {\frac {a x \left (-2 a^2 x^2 \sqrt {1-a^2 x^2}+a x \left (-3+4 a^2 x^2\right ) \arcsin (a x)+2 \sqrt {1-a^2 x^2} \left (-3+8 a^2 x^2\right ) \arcsin (a x)^2\right )}{\arcsin (a x)^3}-2 \operatorname {CosIntegral}(2 \arcsin (a x))+8 \operatorname {CosIntegral}(4 \arcsin (a x))}{6 a^4} \] Input:

Integrate[x^3/ArcSin[a*x]^4,x]
 

Output:

((a*x*(-2*a^2*x^2*Sqrt[1 - a^2*x^2] + a*x*(-3 + 4*a^2*x^2)*ArcSin[a*x] + 2 
*Sqrt[1 - a^2*x^2]*(-3 + 8*a^2*x^2)*ArcSin[a*x]^2))/ArcSin[a*x]^3 - 2*CosI 
ntegral[2*ArcSin[a*x]] + 8*CosIntegral[4*ArcSin[a*x]])/(6*a^4)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5144, 5222, 5142, 2009, 3042, 3783}

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.

\(\displaystyle \int \frac {x^3}{\arcsin (a x)^4} \, dx\)

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5142

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

Input:

Int[x^3/ArcSin[a*x]^4,x]
 

Output:

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

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 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 = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}+\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{3}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{24 \arcsin \left (a x \right )^{3}}+\frac {\cos \left (4 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}-\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{3 \arcsin \left (a x \right )}+\frac {4 \,\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{3}}{a^{4}}\)	\(114\)
default	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}+\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{3}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{24 \arcsin \left (a x \right )^{3}}+\frac {\cos \left (4 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}-\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{3 \arcsin \left (a x \right )}+\frac {4 \,\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{3}}{a^{4}}\)	\(114\)

Input:

int(x^3/arcsin(a*x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a^4*(-1/12*sin(2*arcsin(a*x))/arcsin(a*x)^3-1/12/arcsin(a*x)^2*cos(2*arc 
sin(a*x))+1/6*sin(2*arcsin(a*x))/arcsin(a*x)-1/3*Ci(2*arcsin(a*x))+1/24*si 
n(4*arcsin(a*x))/arcsin(a*x)^3+1/12/arcsin(a*x)^2*cos(4*arcsin(a*x))-1/3*s 
in(4*arcsin(a*x))/arcsin(a*x)+4/3*Ci(4*arcsin(a*x)))
 

Fricas [F]

\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{4}} \,d x } \] Input:

integrate(x^3/arcsin(a*x)^4,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(x^3/arcsin(a*x)^4, x)
 

Sympy [F]

\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int \frac {x^{3}}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \] Input:

integrate(x**3/asin(a*x)**4,x)
 

Output:

Integral(x**3/asin(a*x)**4, x)
 

Maxima [F]

\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{4}} \,d x } \] Input:

integrate(x^3/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(1/3*(32 
*a^4*x^4 - 30*a^2*x^2 + 3)*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^5*x^2 - a^3)*a 
rctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + 2*(a^2*x^3 - (8*a^2*x^3 - 
 3*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a* 
x + 1) - (4*a^3*x^4 - 3*a*x^2)*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 = 174, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=-\frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{3 \, a^{3} \arcsin \left (a x\right )} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x}{3 \, a^{3} \arcsin \left (a x\right )} + \frac {4 \, \operatorname {Ci}\left (4 \, \arcsin \left (a x\right )\right )}{3 \, a^{4}} - \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{3 \, a^{4}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{3 \, a^{3} \arcsin \left (a x\right )^{3}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{3 \, a^{4} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{3 \, a^{3} \arcsin \left (a x\right )^{3}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )}}{6 \, a^{4} \arcsin \left (a x\right )^{2}} + \frac {1}{6 \, a^{4} \arcsin \left (a x\right )^{2}} \] Input:

integrate(x^3/arcsin(a*x)^4,x, algorithm="giac")
 

Output:

-8/3*(-a^2*x^2 + 1)^(3/2)*x/(a^3*arcsin(a*x)) + 5/3*sqrt(-a^2*x^2 + 1)*x/( 
a^3*arcsin(a*x)) + 4/3*cos_integral(4*arcsin(a*x))/a^4 - 1/3*cos_integral( 
2*arcsin(a*x))/a^4 + 1/3*(-a^2*x^2 + 1)^(3/2)*x/(a^3*arcsin(a*x)^3) + 2/3* 
(a^2*x^2 - 1)^2/(a^4*arcsin(a*x)^2) - 1/3*sqrt(-a^2*x^2 + 1)*x/(a^3*arcsin 
(a*x)^3) + 5/6*(a^2*x^2 - 1)/(a^4*arcsin(a*x)^2) + 1/6/(a^4*arcsin(a*x)^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \] Input:

int(x^3/asin(a*x)^4,x)
 

Output:

int(x^3/asin(a*x)^4, x)
 

Reduce [F]

\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int \frac {x^{3}}{\mathit {asin} \left (a x \right )^{4}}d x \] Input:

int(x^3/asin(a*x)^4,x)
 

Output:

int(x**3/asin(a*x)**4,x)