3.130 \(\int x^3 \cos ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=165 \[ \frac{2^{-n-4} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-4 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,4 i \cos ^{-1}(a x)\right )}{a^4} \]

[Out]

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

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Rubi [A]  time = 0.175624, antiderivative size = 165, 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} \[ \frac{2^{-n-4} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-4 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,4 i \cos ^{-1}(a x)\right )}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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) + (ArcCos[a*x]^n*Gamma[1 + n, (-4*I)*ArcCos[a*x]
])/(2^(2*(3 + n))*a^4*((-I)*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, (4*I)*ArcCos[a*x]])/(2^(2*(3 + n))*a
^4*(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^3 \cos ^{-1}(a x)^n \, dx &=-\frac{\operatorname{Subst}\left (\int x^n \cos ^3(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} x^n \sin (2 x)+\frac{1}{8} x^n \sin (4 x)\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int x^n \sin (4 x) \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int x^n \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{i \operatorname{Subst}\left (\int e^{-4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}+\frac{i \operatorname{Subst}\left (\int e^{4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}-\frac{i \operatorname{Subst}\left (\int e^{-2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}+\frac{i \operatorname{Subst}\left (\int e^{2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{2^{-4-n} \left (-i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,-2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-4-n} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{4^{-3-n} \left (-i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,-4 i \cos ^{-1}(a x)\right )}{a^4}+\frac{4^{-3-n} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,4 i \cos ^{-1}(a x)\right )}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.101475, size = 130, normalized size = 0.79 \[ \frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (2^{n+2} \left (-i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )+\left (-i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,4 i \cos ^{-1}(a x)\right )+2^{n+2} \left (i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-4 i \cos ^{-1}(a x)\right )\right )}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.215, size = 287, normalized size = 1.7 \begin{align*} -{\frac{\sqrt{\pi }}{8\,{a}^{4}} \left ( 2\,{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{1+n}\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }}-{\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }{2}^{{\frac{1}{2}}-n}\sqrt{\arccos \left ( ax \right ) }{\it LommelS1} \left ( n+{\frac{3}{2}},{\frac{3}{2}},2\,\arccos \left ( ax \right ) \right ) }-3\,{\frac{{2}^{-3/2-n} \left ( 4/3+2/3\,n \right ) \left ( 2\,\arccos \left ( ax \right ) \cos \left ( 2\,\arccos \left ( ax \right ) \right ) -\sin \left ( 2\,\arccos \left ( ax \right ) \right ) \right ){\it LommelS1} \left ( n+1/2,1/2,2\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) \sqrt{\arccos \left ( ax \right ) }}} \right ) }-{\frac{{2}^{-5-n}\sqrt{\pi }}{{a}^{4}} \left ({\frac{{2}^{2+n} \left ( \arccos \left ( ax \right ) \right ) ^{1+n}\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }}-{\frac{{2}^{1-n}\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }\sqrt{\arccos \left ( ax \right ) }{\it LommelS1} \left ( n+{\frac{3}{2}},{\frac{3}{2}},4\,\arccos \left ( ax \right ) \right ) }-3\,{\frac{{2}^{-2-n} \left ( 4/3+2/3\,n \right ) \left ( 4\,\arccos \left ( ax \right ) \cos \left ( 4\,\arccos \left ( ax \right ) \right ) -\sin \left ( 4\,\arccos \left ( ax \right ) \right ) \right ){\it LommelS1} \left ( n+1/2,1/2,4\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) \sqrt{\arccos \left ( ax \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \arccos \left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*arccos(a*x)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{acos}^{n}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*acos(a*x)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \arccos \left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arccos(a*x)^n, x)