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

Optimal. Leaf size=75 \[ \frac{\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )}{2 a}+\frac{\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0493503, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4624, 3308, 2181} \[ \frac{\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )}{2 a}+\frac{\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4624

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

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

Mathematica [A]  time = 0.0303296, size = 70, normalized size = 0.93 \[ \frac{\cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.095, size = 148, normalized size = 2. \begin{align*} -{\frac{{2}^{n}\sqrt{\pi }}{a} \left ({\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{1+n}{2}^{-n}}{\sqrt{\pi } \left ( 2+n \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{2}^{-n}}{\sqrt{\pi } \left ( 2+n \right ) }\sqrt{\arccos \left ( ax \right ) }{\it LommelS1} \left ( n+{\frac{3}{2}},{\frac{3}{2}},\arccos \left ( ax \right ) \right ) \sqrt{-{a}^{2}{x}^{2}+1}}-3\,{\frac{{2}^{-1-n} \left ( 4/3+2/3\,n \right ) \left ( ax\arccos \left ( ax \right ) -\sqrt{-{a}^{2}{x}^{2}+1} \right ){\it LommelS1} \left ( n+1/2,1/2,\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(arccos(a*x)^n,x)

[Out]

-2^n*Pi^(1/2)/a*(1/Pi^(1/2)/(2+n)*arccos(a*x)^(1+n)*2^(-n)*(-a^2*x^2+1)^(1/2)-2^(-n)/Pi^(1/2)/(2+n)*arccos(a*x
)^(1/2)*LommelS1(n+3/2,3/2,arccos(a*x))*(-a^2*x^2+1)^(1/2)-3*2^(-1-n)/Pi^(1/2)/(2+n)/arccos(a*x)^(1/2)*(4/3+2/
3*n)*(a*x*arccos(a*x)-(-a^2*x^2+1)^(1/2))*LommelS1(n+1/2,1/2,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(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 (\arccos \left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccos(a*x)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x)^n, x)