\(\int x^3 \arccos (a x)^n \, dx\) [130]

Optimal result

Integrand size = 10, antiderivative size = 165 \[ \int x^3 \arccos (a x)^n \, dx=\frac {2^{-4-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-2 i \arccos (a x))}{a^4}+\frac {2^{-4-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,2 i \arccos (a x))}{a^4}+\frac {4^{-3-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-4 i \arccos (a x))}{a^4}+\frac {4^{-3-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,4 i \arccos (a x))}{a^4} \] Output:

2^(-4-n)*arccos(a*x)^n*GAMMA(1+n,-2*I*arccos(a*x))/a^4/((-I*arccos(a*x))^n 
)+2^(-4-n)*arccos(a*x)^n*GAMMA(1+n,2*I*arccos(a*x))/a^4/((I*arccos(a*x))^n 
)+4^(-3-n)*arccos(a*x)^n*GAMMA(1+n,-4*I*arccos(a*x))/a^4/((-I*arccos(a*x)) 
^n)+4^(-3-n)*arccos(a*x)^n*GAMMA(1+n,4*I*arccos(a*x))/a^4/((I*arccos(a*x)) 
^n)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.79 \[ \int x^3 \arccos (a x)^n \, dx=\frac {2^{-2 (3+n)} \arccos (a x)^n \left (\arccos (a x)^2\right )^{-n} \left (2^{2+n} (i \arccos (a x))^n \Gamma (1+n,-2 i \arccos (a x))+2^{2+n} (-i \arccos (a x))^n \Gamma (1+n,2 i \arccos (a x))+(i \arccos (a x))^n \Gamma (1+n,-4 i \arccos (a x))+(-i \arccos (a x))^n \Gamma (1+n,4 i \arccos (a x))\right )}{a^4} \] Input:

Integrate[x^3*ArcCos[a*x]^n,x]
 

Output:

(ArcCos[a*x]^n*(2^(2 + n)*(I*ArcCos[a*x])^n*Gamma[1 + n, (-2*I)*ArcCos[a*x 
]] + 2^(2 + n)*((-I)*ArcCos[a*x])^n*Gamma[1 + n, (2*I)*ArcCos[a*x]] + (I*A 
rcCos[a*x])^n*Gamma[1 + n, (-4*I)*ArcCos[a*x]] + ((-I)*ArcCos[a*x])^n*Gamm 
a[1 + n, (4*I)*ArcCos[a*x]]))/(2^(2*(3 + n))*a^4*(ArcCos[a*x]^2)^n)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5147, 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.

\(\displaystyle \int x^3 \arccos (a x)^n \, dx\)

\(\Big \downarrow \) 5147

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

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {\int \left (\frac {1}{4} \sin (2 \arccos (a x)) \arccos (a x)^n+\frac {1}{8} \sin (4 \arccos (a x)) \arccos (a x)^n\right )d\arccos (a x)}{a^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-2^{-n-4} \arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-2 i \arccos (a x))-2^{-2 (n+3)} \arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-4 i \arccos (a x))-2^{-n-4} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,2 i \arccos (a x))-2^{-2 (n+3)} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,4 i \arccos (a x))}{a^4}\)

Input:

Int[x^3*ArcCos[a*x]^n,x]
 

Output:

-((-((2^(-4 - n)*ArcCos[a*x]^n*Gamma[1 + n, (-2*I)*ArcCos[a*x]])/((-I)*Arc 
Cos[a*x])^n) - (2^(-4 - n)*ArcCos[a*x]^n*Gamma[1 + n, (2*I)*ArcCos[a*x]])/ 
(I*ArcCos[a*x])^n - (ArcCos[a*x]^n*Gamma[1 + n, (-4*I)*ArcCos[a*x]])/(2^(2 
*(3 + n))*((-I)*ArcCos[a*x])^n) - (ArcCos[a*x]^n*Gamma[1 + n, (4*I)*ArcCos 
[a*x]])/(2^(2*(3 + n))*(I*ArcCos[a*x])^n))/a^4)
 

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 5147

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- 
(b*c^(m + 1))^(-1)   Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x 
, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.59 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.74

method	result	size
default	\(-\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \cos \left (2 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{8 a^{4}}-\frac {2^{-5-n} \sqrt {\pi }\, \left (\frac {2^{2+n} \arccos \left (a x \right )^{1+n} \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{-n +1} \sqrt {\arccos \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 4 \arccos \left (a x \right )\right ) \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-n -2} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (4 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-\sin \left (4 \arccos \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{a^{4}}\)	\(287\)

Input:

int(x^3*arccos(a*x)^n,x,method=_RETURNVERBOSE)
 

Output:

-1/8*Pi^(1/2)/a^4*(2/Pi^(1/2)/(2+n)*arccos(a*x)^(1+n)*sin(2*arccos(a*x))-2 
^(1/2-n)/Pi^(1/2)/(2+n)*arccos(a*x)^(1/2)*LommelS1(n+3/2,3/2,2*arccos(a*x) 
)*sin(2*arccos(a*x))-3*2^(-3/2-n)/Pi^(1/2)/(2+n)/arccos(a*x)^(1/2)*(4/3+2/ 
3*n)*(2*cos(2*arccos(a*x))*arccos(a*x)-sin(2*arccos(a*x)))*LommelS1(n+1/2, 
1/2,2*arccos(a*x)))-2^(-5-n)*Pi^(1/2)/a^4*(2^(2+n)/Pi^(1/2)/(2+n)*arccos(a 
*x)^(1+n)*sin(4*arccos(a*x))-2^(-n+1)/Pi^(1/2)/(2+n)*arccos(a*x)^(1/2)*Lom 
melS1(n+3/2,3/2,4*arccos(a*x))*sin(4*arccos(a*x))-3*2^(-n-2)/Pi^(1/2)/(2+n 
)/arccos(a*x)^(1/2)*(4/3+2/3*n)*(4*arccos(a*x)*cos(4*arccos(a*x))-sin(4*ar 
ccos(a*x)))*LommelS1(n+1/2,1/2,4*arccos(a*x)))
 

Fricas [F]

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

integrate(x^3*arccos(a*x)^n,x, algorithm="fricas")
 

Output:

integral(x^3*arccos(a*x)^n, x)
 

Sympy [F]

\[ \int x^3 \arccos (a x)^n \, dx=\int x^{3} \operatorname {acos}^{n}{\left (a x \right )}\, dx \] Input:

integrate(x**3*acos(a*x)**n,x)
 

Output:

Integral(x**3*acos(a*x)**n, x)
 

Maxima [F(-2)]

Exception generated. \[ \int x^3 \arccos (a x)^n \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3*arccos(a*x)^n,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

integrate(x^3*arccos(a*x)^n,x, algorithm="giac")
 

Output:

integrate(x^3*arccos(a*x)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \arccos (a x)^n \, dx=\int x^3\,{\mathrm {acos}\left (a\,x\right )}^n \,d x \] Input:

int(x^3*acos(a*x)^n,x)
 

Output:

int(x^3*acos(a*x)^n, x)
 

Reduce [F]

\[ \int x^3 \arccos (a x)^n \, dx=\int \mathit {acos} \left (a x \right )^{n} x^{3}d x \] Input:

int(x^3*acos(a*x)^n,x)
 

Output:

int(acos(a*x)**n*x**3,x)