3.2.30 \(\int x^3 \text {ArcCos}(a x)^n \, dx\) [130]

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

[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^(6+2*n))/a^4/((-I*arccos(
a*x))^n)+arccos(a*x)^n*GAMMA(1+n,4*I*arccos(a*x))/(2^(6+2*n))/a^4/((I*arccos(a*x))^n)

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Rubi [A]
time = 0.13, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4732, 4491, 3389, 2212} \begin {gather*} \frac {2^{-n-4} \text {ArcCos}(a x)^n (-i \text {ArcCos}(a x))^{-n} \text {Gamma}(n+1,-2 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-2 (n+3)} \text {ArcCos}(a x)^n (-i \text {ArcCos}(a x))^{-n} \text {Gamma}(n+1,-4 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-n-4} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(n+1,2 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-2 (n+3)} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(n+1,4 i \text {ArcCos}(a x))}{a^4} \end {gather*}

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 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*} \int x^3 \cos ^{-1}(a x)^n \, dx &=-\frac {\text {Subst}\left (\int x^n \cos ^3(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\text {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 {\text {Subst}\left (\int x^n \sin (4 x) \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int x^n \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {i \text {Subst}\left (\int e^{-4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}+\frac {i \text {Subst}\left (\int e^{4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}-\frac {i \text {Subst}\left (\int e^{-2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}+\frac {i \text {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*}

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Mathematica [A]
time = 0.07, size = 130, normalized size = 0.79 \begin {gather*} \frac {2^{-2 (3+n)} \text {ArcCos}(a x)^n \left (\text {ArcCos}(a x)^2\right )^{-n} \left (2^{2+n} (i \text {ArcCos}(a x))^n \text {Gamma}(1+n,-2 i \text {ArcCos}(a x))+2^{2+n} (-i \text {ArcCos}(a x))^n \text {Gamma}(1+n,2 i \text {ArcCos}(a x))+(i \text {ArcCos}(a x))^n \text {Gamma}(1+n,-4 i \text {ArcCos}(a x))+(-i \text {ArcCos}(a x))^n \text {Gamma}(1+n,4 i \text {ArcCos}(a x))\right )}{a^4} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.73, size = 287, normalized size = 1.74

method	result	size
default	\(-\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{8 a^{4}}-\frac {2^{-5-n} \sqrt {\pi }\, \left (\frac {2^{2+n} \arccos \left (a x \right )^{1+n} \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{1-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, 4 \arccos \left (a x \right )\right ) \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-2-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (4 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-\sin \left (4 \arccos \left (a x \right )\right )\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, 4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{a^{4}}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccos(a*x)^n,x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \operatorname {acos}^{n}{\left (a x \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\mathrm {acos}\left (a\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*acos(a*x)^n, x)

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