\(\int \arccos (a x)^n \, dx\) [133]

Optimal result

Integrand size = 6, antiderivative size = 75 \[ \int \arccos (a x)^n \, dx=\frac {(-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-i \arccos (a x))}{2 a}+\frac {(i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,i \arccos (a x))}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4720, 3389, 2212} \[ \int \arccos (a x)^n \, dx=\frac {\arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-i \arccos (a x))}{2 a}+\frac {(i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,i \arccos (a x))}{2 a} \]

[In]

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

[Out]

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

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 4720

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^n \sin (x) \, dx,x,\arccos (a x)\right )}{a} \\ & = -\frac {i \text {Subst}\left (\int e^{-i x} x^n \, dx,x,\arccos (a x)\right )}{2 a}+\frac {i \text {Subst}\left (\int e^{i x} x^n \, dx,x,\arccos (a x)\right )}{2 a} \\ & = \frac {(-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-i \arccos (a x))}{2 a}+\frac {(i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,i \arccos (a x))}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int \arccos (a x)^n \, dx=\frac {\arccos (a x)^n \left (\arccos (a x)^2\right )^{-n} \left ((i \arccos (a x))^n \Gamma (1+n,-i \arccos (a x))+(-i \arccos (a x))^n \Gamma (1+n,i \arccos (a x))\right )}{2 a} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.87 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.97

method	result	size
default	\(-\frac {2^{n} \sqrt {\pi }\, \left (\frac {\arccos \left (a x \right )^{1+n} 2^{-n} \sqrt {-a^{2} x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{-n} \sqrt {\arccos \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-1-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (\arccos \left (a x \right ) a x -\sqrt {-a^{2} x^{2}+1}\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{a}\)	\(148\)

[In]

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

[Out]

-2^n*Pi^(1/2)/a*(1/Pi^(1/2)/(2+n)*arccos(a*x)^(1+n)*2^(-n)*(-a^2*x^2+1)^(1/2)-2^(-n)/Pi^(1/2)/(2+n)*arccos(a*x
)^(1/2)*LommelS1(n+3/2,3/2,arccos(a*x))*(-a^2*x^2+1)^(1/2)-3*2^(-1-n)/Pi^(1/2)/(2+n)/arccos(a*x)^(1/2)*(4/3+2/
3*n)*(arccos(a*x)*a*x-(-a^2*x^2+1)^(1/2))*LommelS1(n+1/2,1/2,arccos(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(arccos(a*x)^n, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acos(a*x)**n, x)

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(arccos(a*x)^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(acos(a*x)^n, x)