3.1.60 \(\int \frac {x^4}{\arcsin (a x)^3} \, dx\) [60]

3.1.60.1 Optimal result

Integrand size = 10, antiderivative size = 98 \[ \int \frac {x^4}{\arcsin (a x)^3} \, dx=-\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {2 x^3}{a^2 \arcsin (a x)}+\frac {5 x^5}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{16 a^5}+\frac {27 \operatorname {CosIntegral}(3 \arcsin (a x))}{32 a^5}-\frac {25 \operatorname {CosIntegral}(5 \arcsin (a x))}{32 a^5} \]

output

-2*x^3/a^2/arcsin(a*x)+5/2*x^5/arcsin(a*x)-1/16*Ci(arcsin(a*x))/a^5+27/32* 
Ci(3*arcsin(a*x))/a^5-25/32*Ci(5*arcsin(a*x))/a^5-1/2*x^4*(-a^2*x^2+1)^(1/ 
2)/a/arcsin(a*x)^2
 

3.1.60.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int \frac {x^4}{\arcsin (a x)^3} \, dx=-\frac {16 a^4 x^4 \sqrt {1-a^2 x^2}+64 a^3 x^3 \arcsin (a x)-80 a^5 x^5 \arcsin (a x)+2 \arcsin (a x)^2 \operatorname {CosIntegral}(\arcsin (a x))-27 \arcsin (a x)^2 \operatorname {CosIntegral}(3 \arcsin (a x))+25 \arcsin (a x)^2 \operatorname {CosIntegral}(5 \arcsin (a x))}{32 a^5 \arcsin (a x)^2} \]

input

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

output

-1/32*(16*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] + 2*ArcSin[a*x]^2*CosIntegral[ArcSin[a*x]] - 27*ArcSin[a*x]^2* 
CosIntegral[3*ArcSin[a*x]] + 25*ArcSin[a*x]^2*CosIntegral[5*ArcSin[a*x]])/ 
(a^5*ArcSin[a*x]^2)
 

3.1.60.3 Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5144, 5222, 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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5146

\(\displaystyle \frac {2 \left (\frac {3 \int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\arcsin (a x)}d\arcsin (a x)}{a^4}-\frac {x^3}{a \arcsin (a x)}\right )}{a}-\frac {5}{2} a \left (\frac {5 \int \frac {a^4 x^4 \sqrt {1-a^2 x^2}}{\arcsin (a x)}d\arcsin (a x)}{a^6}-\frac {x^5}{a \arcsin (a x)}\right )-\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {5}{2} a \left (\frac {5 \int \left (-\frac {3 \cos (3 \arcsin (a x))}{16 \arcsin (a x)}+\frac {\cos (5 \arcsin (a x))}{16 \arcsin (a x)}+\frac {\sqrt {1-a^2 x^2}}{8 \arcsin (a x)}\right )d\arcsin (a x)}{a^6}-\frac {x^5}{a \arcsin (a x)}\right )+\frac {2 \left (\frac {3 \int \left (\frac {\sqrt {1-a^2 x^2}}{4 \arcsin (a x)}-\frac {\cos (3 \arcsin (a x))}{4 \arcsin (a x)}\right )d\arcsin (a x)}{a^4}-\frac {x^3}{a \arcsin (a x)}\right )}{a}-\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5}{2} a \left (\frac {5 \left (\frac {1}{8} \operatorname {CosIntegral}(\arcsin (a x))-\frac {3}{16} \operatorname {CosIntegral}(3 \arcsin (a x))+\frac {1}{16} \operatorname {CosIntegral}(5 \arcsin (a x))\right )}{a^6}-\frac {x^5}{a \arcsin (a x)}\right )+\frac {2 \left (\frac {3 \left (\frac {1}{4} \operatorname {CosIntegral}(\arcsin (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arcsin (a x))\right )}{a^4}-\frac {x^3}{a \arcsin (a x)}\right )}{a}-\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}\)

input

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

output

-1/2*(x^4*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^2) + (2*(-(x^3/(a*ArcSin[a*x]) 
) + (3*(CosIntegral[ArcSin[a*x]]/4 - CosIntegral[3*ArcSin[a*x]]/4))/a^4))/ 
a - (5*a*(-(x^5/(a*ArcSin[a*x])) + (5*(CosIntegral[ArcSin[a*x]]/8 - (3*Cos 
Integral[3*ArcSin[a*x]])/16 + CosIntegral[5*ArcSin[a*x]]/16))/a^6))/2
 

3.1.60.3.1 Defintions of rubi rules used

rule 2009

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

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 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]
 

3.1.60.4 Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{16 \arcsin \left (a x \right )^{2}}+\frac {a x}{16 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{16}+\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )}{32}}{a^{5}}\)	\(121\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{16 \arcsin \left (a x \right )^{2}}+\frac {a x}{16 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{16}+\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )}{32}}{a^{5}}\)	\(121\)

input

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

output

1/a^5*(-1/16/arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+1/16*a*x/arcsin(a*x)-1/16*Ci 
(arcsin(a*x))+3/32/arcsin(a*x)^2*cos(3*arcsin(a*x))-9/32/arcsin(a*x)*sin(3 
*arcsin(a*x))+27/32*Ci(3*arcsin(a*x))-1/32/arcsin(a*x)^2*cos(5*arcsin(a*x) 
)+5/32/arcsin(a*x)*sin(5*arcsin(a*x))-25/32*Ci(5*arcsin(a*x)))
 

3.1.60.5 Fricas [F]

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

input

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

output

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

3.1.60.6 Sympy [F]

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

input

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

output

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

3.1.60.7 Maxima [F]

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

input

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

output

-1/2*(sqrt(a*x + 1)*sqrt(-a*x + 1)*a*x^4 + arctan2(a*x, sqrt(a*x + 1)*sqrt 
(-a*x + 1))^2*integrate((25*a^2*x^4 - 12*x^2)/arctan2(a*x, sqrt(a*x + 1)*s 
qrt(-a*x + 1)), x) - (5*a^2*x^5 - 4*x^3)*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)
 

3.1.60.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.73 \[ \int \frac {x^4}{\arcsin (a x)^3} \, dx=\frac {5 \, {\left (a^{2} x^{2} - 1\right )}^{2} x}{2 \, a^{4} \arcsin \left (a x\right )} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )} x}{a^{4} \arcsin \left (a x\right )} + \frac {x}{2 \, a^{4} \arcsin \left (a x\right )} - \frac {25 \, \operatorname {Ci}\left (5 \, \arcsin \left (a x\right )\right )}{32 \, a^{5}} + \frac {27 \, \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{32 \, a^{5}} - \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{5} \arcsin \left (a x\right )^{2}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a^{5} \arcsin \left (a x\right )^{2}} \]

input

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

output

5/2*(a^2*x^2 - 1)^2*x/(a^4*arcsin(a*x)) + 3*(a^2*x^2 - 1)*x/(a^4*arcsin(a* 
x)) + 1/2*x/(a^4*arcsin(a*x)) - 25/32*cos_integral(5*arcsin(a*x))/a^5 + 27 
/32*cos_integral(3*arcsin(a*x))/a^5 - 1/16*cos_integral(arcsin(a*x))/a^5 - 
 1/2*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^2) + (-a^2*x^2 + 
1)^(3/2)/(a^5*arcsin(a*x)^2) - 1/2*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^2)
 

3.1.60.9 Mupad [F(-1)]

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

input

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

output

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