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

Optimal. Leaf size=49 \[ \frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{2 a}+\frac{\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0492336, antiderivative size = 49, 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 = {5658, 3308, 2181} \[ \frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{2 a}+\frac{\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5658

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Dist[(b*c)^(-1), Subst[Int[x^n*Sinh[a/b - x/b], x]
, x, a + b*ArcCosh[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 \cosh ^{-1}(a x)^n \, dx &=\frac{\operatorname{Subst}\left (\int x^n \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{2 a}+\frac{\operatorname{Subst}\left (\int e^x x^n \, dx,x,\cosh ^{-1}(a x)\right )}{2 a}\\ &=\frac{\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )}{2 a}+\frac{\Gamma \left (1+n,\cosh ^{-1}(a x)\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0241498, size = 43, normalized size = 0.88 \[ \frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )+\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.041, size = 40, normalized size = 0.8 \begin{align*}{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2+n}}{a \left ( 2+n \right ) }{\mbox{$_1$F$_2$}(1+{\frac{n}{2}};\,{\frac{3}{2}},2+{\frac{n}{2}};\,{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a/(2+n)*arccosh(a*x)^(2+n)*hypergeom([1+1/2*n],[3/2,2+1/2*n],1/4*arccosh(a*x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(a*x)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccosh(a*x)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x