\(\int \text {arccosh}(a x)^n \, dx\) [131]

Optimal result

Integrand size = 6, antiderivative size = 49 \[ \int \text {arccosh}(a x)^n \, dx=\frac {(-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-\text {arccosh}(a x))}{2 a}+\frac {\Gamma (1+n,\text {arccosh}(a x))}{2 a} \]

[Out]

1/2*arccosh(a*x)^n*GAMMA(1+n,-arccosh(a*x))/a/((-arccosh(a*x))^n)+1/2*GAMMA(1+n,arccosh(a*x))/a

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5881, 3389, 2212} \[ \int \text {arccosh}(a x)^n \, dx=\frac {\text {arccosh}(a x)^n (-\text {arccosh}(a x))^{-n} \Gamma (n+1,-\text {arccosh}(a x))}{2 a}+\frac {\Gamma (n+1,\text {arccosh}(a x))}{2 a} \]

[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 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 5881

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \sinh (x) \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {\text {Subst}\left (\int e^{-x} x^n \, dx,x,\text {arccosh}(a x)\right )}{2 a}+\frac {\text {Subst}\left (\int e^x x^n \, dx,x,\text {arccosh}(a x)\right )}{2 a} \\ & = \frac {(-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-\text {arccosh}(a x))}{2 a}+\frac {\Gamma (1+n,\text {arccosh}(a x))}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \text {arccosh}(a x)^n \, dx=\frac {(-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-\text {arccosh}(a x))+\Gamma (1+n,\text {arccosh}(a x))}{2 a} \]

[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)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\operatorname {arccosh}\left (a x \right )^{2+n} \operatorname {hypergeom}\left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \frac {\operatorname {arccosh}\left (a x \right )^{2}}{4}\right )}{a \left (2+n \right )}\)	\(40\)

[In]

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

[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)

Fricas [F]

\[ \int \text {arccosh}(a x)^n \, dx=\int { \operatorname {arcosh}\left (a x\right )^{n} \,d x } \]

[In]

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

[Out]

integral(arccosh(a*x)^n, x)

Sympy [F]

\[ \int \text {arccosh}(a x)^n \, dx=\int \operatorname {acosh}^{n}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(acosh(a*x)**n, x)

Maxima [F]

\[ \int \text {arccosh}(a x)^n \, dx=\int { \operatorname {arcosh}\left (a x\right )^{n} \,d x } \]

[In]

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

[Out]

integrate(arccosh(a*x)^n, x)

Giac [F]

\[ \int \text {arccosh}(a x)^n \, dx=\int { \operatorname {arcosh}\left (a x\right )^{n} \,d x } \]

[In]

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

[Out]

integrate(arccosh(a*x)^n, x)

Mupad [F(-1)]

Timed out. \[ \int \text {arccosh}(a x)^n \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^n \,d x \]

[In]

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

[Out]

int(acosh(a*x)^n, x)