\(\int \frac {x^4}{(a+b \arcsin (c x))^3} \, dx\) [53]

Optimal result

Integrand size = 14, antiderivative size = 261 \[ \int \frac {x^4}{(a+b \arcsin (c x))^3} \, dx=-\frac {x^4 \sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}-\frac {2 x^3}{b^2 c^2 (a+b \arcsin (c x))}+\frac {5 x^5}{2 b^2 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{16 b^3 c^5}+\frac {27 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b^3 c^5}-\frac {25 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{32 b^3 c^5}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{16 b^3 c^5}+\frac {27 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b^3 c^5}-\frac {25 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{32 b^3 c^5} \] Output:

-1/2*x^4*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))^2-2*x^3/b^2/c^2/(a+b*arc 
sin(c*x))+5/2*x^5/b^2/(a+b*arcsin(c*x))-1/16*cos(a/b)*Ci((a+b*arcsin(c*x)) 
/b)/b^3/c^5+27/32*cos(3*a/b)*Ci(3*(a+b*arcsin(c*x))/b)/b^3/c^5-25/32*cos(5 
*a/b)*Ci(5*(a+b*arcsin(c*x))/b)/b^3/c^5-1/16*sin(a/b)*Si((a+b*arcsin(c*x)) 
/b)/b^3/c^5+27/32*sin(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^3/c^5-25/32*sin(5 
*a/b)*Si(5*(a+b*arcsin(c*x))/b)/b^3/c^5
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{(a+b \arcsin (c x))^3} \, dx=-\frac {\frac {16 b^2 x^4 \sqrt {1-c^2 x^2}}{c (a+b \arcsin (c x))^2}+\frac {64 b x^3}{c^2 (a+b \arcsin (c x))}-\frac {80 b x^5}{a+b \arcsin (c x)}+\frac {2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{c^5}-\frac {27 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^5}+\frac {25 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^5}+\frac {2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{c^5}-\frac {27 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^5}+\frac {25 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{c^5}}{32 b^3} \] Input:

Integrate[x^4/(a + b*ArcSin[c*x])^3,x]
 

Output:

-1/32*((16*b^2*x^4*Sqrt[1 - c^2*x^2])/(c*(a + b*ArcSin[c*x])^2) + (64*b*x^ 
3)/(c^2*(a + b*ArcSin[c*x])) - (80*b*x^5)/(a + b*ArcSin[c*x]) + (2*Cos[a/b 
]*CosIntegral[a/b + ArcSin[c*x]])/c^5 - (27*Cos[(3*a)/b]*CosIntegral[3*(a/ 
b + ArcSin[c*x])])/c^5 + (25*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x] 
)])/c^5 + (2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/c^5 - (27*Sin[(3*a)/ 
b]*SinIntegral[3*(a/b + ArcSin[c*x])])/c^5 + (25*Sin[(5*a)/b]*SinIntegral[ 
5*(a/b + ArcSin[c*x])])/c^5)/b^3
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.37, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, 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.

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

\(\Big \downarrow \) 5144

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^4/(a + b*ArcSin[c*x])^3,x]
 

Output:

-1/2*(x^4*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])^2) + (2*(-(x^3/(b*c* 
(a + b*ArcSin[c*x]))) + (3*((Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/ 
4 - (Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/4 + (Sin[a/b]*Si 
nIntegral[(a + b*ArcSin[c*x])/b])/4 - (Sin[(3*a)/b]*SinIntegral[(3*(a + b* 
ArcSin[c*x]))/b])/4))/(b^2*c^4)))/(b*c) - (5*c*(-(x^5/(b*c*(a + b*ArcSin[c 
*x]))) + (5*((Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/8 - (3*Cos[(3*a 
)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/16 + (Cos[(5*a)/b]*CosIntegra 
l[(5*(a + b*ArcSin[c*x]))/b])/16 + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x 
])/b])/8 - (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/16 + (S 
in[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/16))/(b^2*c^6)))/(2*b)
 

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]
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.69

method	result	size
derivativedivides	\(\frac {\frac {3 \cos \left (3 \arcsin \left (c x \right )\right )}{32 \left (a +b \arcsin \left (c x \right )\right )^{2} b}+\frac {\frac {27 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{32}+\frac {27 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{32}+\frac {27 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{32}+\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{32}-\frac {9 \sin \left (3 \arcsin \left (c x \right )\right ) b}{32}}{\left (a +b \arcsin \left (c x \right )\right ) b^{3}}-\frac {\cos \left (5 \arcsin \left (c x \right )\right )}{32 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {5 \left (5 \arcsin \left (c x \right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +5 \arcsin \left (c x \right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +5 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +5 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -\sin \left (5 \arcsin \left (c x \right )\right ) b \right )}{32 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{16 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{16 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{5}}\)	\(442\)
default	\(\frac {\frac {3 \cos \left (3 \arcsin \left (c x \right )\right )}{32 \left (a +b \arcsin \left (c x \right )\right )^{2} b}+\frac {\frac {27 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{32}+\frac {27 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{32}+\frac {27 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{32}+\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{32}-\frac {9 \sin \left (3 \arcsin \left (c x \right )\right ) b}{32}}{\left (a +b \arcsin \left (c x \right )\right ) b^{3}}-\frac {\cos \left (5 \arcsin \left (c x \right )\right )}{32 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {5 \left (5 \arcsin \left (c x \right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +5 \arcsin \left (c x \right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +5 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +5 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -\sin \left (5 \arcsin \left (c x \right )\right ) b \right )}{32 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{16 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{16 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c^{5}}\)	\(442\)

Input:

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

Output:

1/c^5*(3/32*cos(3*arcsin(c*x))/(a+b*arcsin(c*x))^2/b+9/32*(3*arcsin(c*x)*C 
i(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b+3*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)* 
sin(3*a/b)*b+3*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+3*Si(3*arcsin(c*x)+3*a 
/b)*sin(3*a/b)*a-sin(3*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^3-1/32*cos(5*ar 
csin(c*x))/(a+b*arcsin(c*x))^2/b-5/32*(5*arcsin(c*x)*Si(5*arcsin(c*x)+5*a/ 
b)*sin(5*a/b)*b+5*arcsin(c*x)*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*b+5*Si(5* 
arcsin(c*x)+5*a/b)*sin(5*a/b)*a+5*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*a-sin 
(5*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^3-1/16*(-c^2*x^2+1)^(1/2)/(a+b*arcs 
in(c*x))^2/b-1/16*(arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+arcsin(c*x)* 
Ci(arcsin(c*x)+a/b)*cos(a/b)*b+Si(arcsin(c*x)+a/b)*sin(a/b)*a+Ci(arcsin(c* 
x)+a/b)*cos(a/b)*a-x*b*c)/(a+b*arcsin(c*x))/b^3)
 

Fricas [F]

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

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

Output:

integral(x^4/(b^3*arcsin(c*x)^3 + 3*a*b^2*arcsin(c*x)^2 + 3*a^2*b*arcsin(c 
*x) + a^3), x)
 

Sympy [F]

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

integrate(x**4/(a+b*asin(c*x))**3,x)
 

Output:

Integral(x**4/(a + b*asin(c*x))**3, x)
 

Maxima [F]

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

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

Output:

1/2*(5*a*c^2*x^5 - sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x^4 - 4*a*x^3 + (5*b*c 
^2*x^5 - 4*b*x^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - 2*(b^4*c^2* 
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*arctan2(c*x, sq 
rt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2)*integrate(1/2*(25*c^2*x^4 - 12* 
x^2)/(b^3*c^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b^2*c^2), x)) 
/(b^4*c^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*arcta 
n2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2988 vs. \(2 (243) = 486\).

Time = 0.23 (sec) , antiderivative size = 2988, normalized size of antiderivative = 11.45 \[ \int \frac {x^4}{(a+b \arcsin (c x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-25/2*b^2*arcsin(c*x)^2*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b^ 
5*c^5*arcsin(c*x)^2 + 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) - 25/2*b^2*ar 
csin(c*x)^2*cos(a/b)^4*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^5*c 
^5*arcsin(c*x)^2 + 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) - 25*a*b*arcsin( 
c*x)*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b^5*c^5*arcsin(c*x)^2 
 + 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) - 25*a*b*arcsin(c*x)*cos(a/b)^4* 
sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^5*c^5*arcsin(c*x)^2 + 2*a* 
b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) + 125/8*b^2*arcsin(c*x)^2*cos(a/b)^3*co 
s_integral(5*a/b + 5*arcsin(c*x))/(b^5*c^5*arcsin(c*x)^2 + 2*a*b^4*c^5*arc 
sin(c*x) + a^2*b^3*c^5) - 25/2*a^2*cos(a/b)^5*cos_integral(5*a/b + 5*arcsi 
n(c*x))/(b^5*c^5*arcsin(c*x)^2 + 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) + 
27/8*b^2*arcsin(c*x)^2*cos(a/b)^3*cos_integral(3*a/b + 3*arcsin(c*x))/(b^5 
*c^5*arcsin(c*x)^2 + 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) + 75/8*b^2*arc 
sin(c*x)^2*cos(a/b)^2*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^5*c^ 
5*arcsin(c*x)^2 + 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) - 25/2*a^2*cos(a/ 
b)^4*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^5*c^5*arcsin(c*x)^2 + 
 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) + 27/8*b^2*arcsin(c*x)^2*cos(a/b)^ 
2*sin(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^5*c^5*arcsin(c*x)^2 + 2* 
a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) + 5/2*(c^2*x^2 - 1)^2*b^2*c*x*arcsin( 
c*x)/(b^5*c^5*arcsin(c*x)^2 + 2*a*b^4*c^5*arcsin(c*x) + a^2*b^3*c^5) + ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4}{(a+b \arcsin (c x))^3} \, dx=\int \frac {x^{4}}{\mathit {asin} \left (c x \right )^{3} b^{3}+3 \mathit {asin} \left (c x \right )^{2} a \,b^{2}+3 \mathit {asin} \left (c x \right ) a^{2} b +a^{3}}d x \] Input:

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

Output:

int(x**4/(asin(c*x)**3*b**3 + 3*asin(c*x)**2*a*b**2 + 3*asin(c*x)*a**2*b + 
 a**3),x)