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

Optimal. Leaf size=83 \[ \frac {2^{-n-3} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (n+1,2 i \cos ^{-1}(a x)\right )}{a^2} \]

[Out]

2^(-3-n)*arccos(a*x)^n*GAMMA(1+n,-2*I*arccos(a*x))/a^2/((-I*arccos(a*x))^n)+2^(-3-n)*arccos(a*x)^n*GAMMA(1+n,2
*I*arccos(a*x))/a^2/((I*arccos(a*x))^n)

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Rubi [A]  time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4636, 4406, 12, 3308, 2181} \[ \frac {2^{-n-3} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text {Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 74, normalized size = 0.89 \[ \frac {2^{-n-3} \cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(a x)\right )^n \Gamma \left (n+1,2 i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \Gamma \left (n+1,-2 i \cos ^{-1}(a x)\right )\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \arccos \left (a x\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*arccos(a*x)^n, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \arccos \left (a x\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*arccos(a*x)^n, x)

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maple [C]  time = 0.15, size = 138, normalized size = 1.66 \[ -\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{4 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*Pi^(1/2)/a^2*(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)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*acos(a*x)^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {acos}^{n}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*acos(a*x)**n, x)

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