3.131 \(\int x^2 \cos ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=163 \[ \frac{\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,3 i \cos ^{-1}(a x)\right )}{8 a^3} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.14896, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4636, 4406, 3308, 2181} \[ \frac{\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,3 i \cos ^{-1}(a x)\right )}{8 a^3} \]

Antiderivative was successfully verified.

[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 4636

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

Rule 4406

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 3308

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 2181

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

Rubi steps

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

Mathematica [A]  time = 0.194486, size = 152, normalized size = 0.93 \[ \frac{\frac{1}{8} 3^{-n-1} \cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,3 i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-3 i \cos ^{-1}(a x)\right )\right )+\frac{1}{4} \left (\frac{1}{2} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )+\frac{1}{2} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )\right )}{a^3} \]

Antiderivative was successfully verified.

[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]  time = 0.152, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \arccos \left ( ax \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \arccos \left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*arccos(a*x)^n, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{acos}^{n}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*acos(a*x)**n, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \arccos \left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arccos(a*x)^n, x)