\(\int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx\) [501]

Optimal result

Integrand size = 24, antiderivative size = 109 \[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\frac {\arccos (a x)^{1+n}}{2 a^3 (1+n)}+\frac {i 2^{-3-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-2 i \arccos (a x))}{a^3}-\frac {i 2^{-3-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,2 i \arccos (a x))}{a^3} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2^{-3-n} \arccos (a x)^n \left (\arccos (a x)^2\right )^{-n} \left (2^{2+n} \arccos (a x) \left (\arccos (a x)^2\right )^n-i (1+n) (i \arccos (a x))^n \Gamma (1+n,-2 i \arccos (a x))+i (1+n) (-i \arccos (a x))^n \Gamma (1+n,2 i \arccos (a x))\right )}{a^3 (1+n)} \] Input:

Integrate[(x^2*ArcCos[a*x]^n)/Sqrt[1 - a^2*x^2],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5225, 3042, 3793, 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.

\(\displaystyle \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 5225

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \arccos (a x)^n \sin \left (\arccos (a x)+\frac {\pi }{2}\right )^2d\arccos (a x)}{a^3}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {\int \left (\frac {1}{2} \cos (2 \arccos (a x)) \arccos (a x)^n+\frac {1}{2} \arccos (a x)^n\right )d\arccos (a x)}{a^3}\)

\(\Big \downarrow \) 2009

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

Input:

Int[(x^2*ArcCos[a*x]^n)/Sqrt[1 - a^2*x^2],x]
 

Output:

-((ArcCos[a*x]^(1 + n)/(2*(1 + n)) - (I*2^(-3 - n)*ArcCos[a*x]^n*Gamma[1 + 
 n, (-2*I)*ArcCos[a*x]])/((-I)*ArcCos[a*x])^n + (I*2^(-3 - n)*ArcCos[a*x]^ 
n*Gamma[1 + n, (2*I)*ArcCos[a*x]])/(I*ArcCos[a*x])^n)/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 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5225

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c 
^2*x^2)^p]   Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], 
 x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e 
, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [F]

\[\int \frac {x^{2} \arccos \left (a x \right )^{n}}{\sqrt {-a^{2} x^{2}+1}}d x\]

Input:

int(x^2*arccos(a*x)^n/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(x^2*arccos(a*x)^n/(-a^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \arccos \left (a x\right )^{n}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^2*arccos(a*x)^n/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*x^2*arccos(a*x)^n/(a^2*x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{2} \operatorname {acos}^{n}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(x**2*acos(a*x)**n/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**2*acos(a*x)**n/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^2*arccos(a*x)^n/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \arccos \left (a x\right )^{n}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^2*arccos(a*x)^n/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^2*arccos(a*x)^n/sqrt(-a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {acos}\left (a\,x\right )}^n}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^2*acos(a*x)^n)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^2*acos(a*x)^n)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \arccos (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{n} x^{2}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^2*acos(a*x)^n/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acos(a*x)**n*x**2)/sqrt( - a**2*x**2 + 1),x)