\(\int x^2 \arccos (a x)^n \, dx\) [131]

Optimal result

Integrand size = 10, antiderivative size = 163 \[ \int x^2 \arccos (a x)^n \, dx=\frac {(-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-i \arccos (a x))}{8 a^3}+\frac {(i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,i \arccos (a x))}{8 a^3}+\frac {3^{-1-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-3 i \arccos (a x))}{8 a^3}+\frac {3^{-1-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,3 i \arccos (a x))}{8 a^3} \]

[Out]

1/8*arccos(a*x)^n*GAMMA(1+n,-I*arccos(a*x))/a^3/((-I*arccos(a*x))^n)+1/8*arccos(a*x)^n*GAMMA(1+n,I*arccos(a*x)
)/a^3/((I*arccos(a*x))^n)+1/8*3^(-1-n)*arccos(a*x)^n*GAMMA(1+n,-3*I*arccos(a*x))/a^3/((-I*arccos(a*x))^n)+1/8*
3^(-1-n)*arccos(a*x)^n*GAMMA(1+n,3*I*arccos(a*x))/a^3/((I*arccos(a*x))^n)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4732, 4491, 3389, 2212} \[ \int x^2 \arccos (a x)^n \, dx=\frac {\arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-i \arccos (a x))}{8 a^3}+\frac {3^{-n-1} \arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-3 i \arccos (a x))}{8 a^3}+\frac {(i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,i \arccos (a x))}{8 a^3}+\frac {3^{-n-1} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,3 i \arccos (a x))}{8 a^3} \]

[In]

Int[x^2*ArcCos[a*x]^n,x]

[Out]

(ArcCos[a*x]^n*Gamma[1 + n, (-I)*ArcCos[a*x]])/(8*a^3*((-I)*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, I*Ar
cCos[a*x]])/(8*a^3*(I*ArcCos[a*x])^n) + (3^(-1 - n)*ArcCos[a*x]^n*Gamma[1 + n, (-3*I)*ArcCos[a*x]])/(8*a^3*((-
I)*ArcCos[a*x])^n) + (3^(-1 - n)*ArcCos[a*x]^n*Gamma[1 + n, (3*I)*ArcCos[a*x]])/(8*a^3*(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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 4732

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(b*c^(m + 1))^(-1), Subst[Int[x^n*C
os[-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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^n \cos ^2(x) \sin (x) \, dx,x,\arccos (a x)\right )}{a^3} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{4} x^n \sin (x)+\frac {1}{4} x^n \sin (3 x)\right ) \, dx,x,\arccos (a x)\right )}{a^3} \\ & = -\frac {\text {Subst}\left (\int x^n \sin (x) \, dx,x,\arccos (a x)\right )}{4 a^3}-\frac {\text {Subst}\left (\int x^n \sin (3 x) \, dx,x,\arccos (a x)\right )}{4 a^3} \\ & = -\frac {i \text {Subst}\left (\int e^{-i x} x^n \, dx,x,\arccos (a x)\right )}{8 a^3}+\frac {i \text {Subst}\left (\int e^{i x} x^n \, dx,x,\arccos (a x)\right )}{8 a^3}-\frac {i \text {Subst}\left (\int e^{-3 i x} x^n \, dx,x,\arccos (a x)\right )}{8 a^3}+\frac {i \text {Subst}\left (\int e^{3 i x} x^n \, dx,x,\arccos (a x)\right )}{8 a^3} \\ & = \frac {(-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-i \arccos (a x))}{8 a^3}+\frac {(i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,i \arccos (a x))}{8 a^3}+\frac {3^{-1-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-3 i \arccos (a x))}{8 a^3}+\frac {3^{-1-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,3 i \arccos (a x))}{8 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93 \[ \int x^2 \arccos (a x)^n \, dx=\frac {\frac {1}{4} \left (\frac {1}{2} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-i \arccos (a x))+\frac {1}{2} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,i \arccos (a x))\right )+\frac {1}{8} 3^{-1-n} \arccos (a x)^n \left (\arccos (a x)^2\right )^{-n} \left ((i \arccos (a x))^n \Gamma (1+n,-3 i \arccos (a x))+(-i \arccos (a x))^n \Gamma (1+n,3 i \arccos (a x))\right )}{a^3} \]

[In]

Integrate[x^2*ArcCos[a*x]^n,x]

[Out]

(((ArcCos[a*x]^n*Gamma[1 + n, (-I)*ArcCos[a*x]])/(2*((-I)*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, I*ArcC
os[a*x]])/(2*(I*ArcCos[a*x])^n))/4 + (3^(-1 - n)*ArcCos[a*x]^n*((I*ArcCos[a*x])^n*Gamma[1 + n, (-3*I)*ArcCos[a
*x]] + ((-I)*ArcCos[a*x])^n*Gamma[1 + n, (3*I)*ArcCos[a*x]]))/(8*(ArcCos[a*x]^2)^n))/a^3

Maple [F]

\[\int x^{2} \arccos \left (a x \right )^{n}d x\]

[In]

int(x^2*arccos(a*x)^n,x)

[Out]

int(x^2*arccos(a*x)^n,x)

Fricas [F]

\[ \int x^2 \arccos (a x)^n \, dx=\int { x^{2} \arccos \left (a x\right )^{n} \,d x } \]

[In]

integrate(x^2*arccos(a*x)^n,x, algorithm="fricas")

[Out]

integral(x^2*arccos(a*x)^n, x)

Sympy [F]

\[ \int x^2 \arccos (a x)^n \, dx=\int x^{2} \operatorname {acos}^{n}{\left (a x \right )}\, dx \]

[In]

integrate(x**2*acos(a*x)**n,x)

[Out]

Integral(x**2*acos(a*x)**n, x)

Maxima [F(-2)]

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

[In]

integrate(x^2*arccos(a*x)^n,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x^2 \arccos (a x)^n \, dx=\int { x^{2} \arccos \left (a x\right )^{n} \,d x } \]

[In]

integrate(x^2*arccos(a*x)^n,x, algorithm="giac")

[Out]

integrate(x^2*arccos(a*x)^n, x)

Mupad [F(-1)]

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

[In]

int(x^2*acos(a*x)^n,x)

[Out]

int(x^2*acos(a*x)^n, x)