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

Optimal result

Integrand size = 8, antiderivative size = 59 \[ \int x \text {arccosh}(a x)^n \, dx=\frac {2^{-3-n} (-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-2 \text {arccosh}(a x))}{a^2}+\frac {2^{-3-n} \Gamma (1+n,2 \text {arccosh}(a x))}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5887, 5556, 12, 3389, 2212} \[ \int x \text {arccosh}(a x)^n \, dx=\frac {2^{-n-3} \text {arccosh}(a x)^n (-\text {arccosh}(a x))^{-n} \Gamma (n+1,-2 \text {arccosh}(a x))}{a^2}+\frac {2^{-n-3} \Gamma (n+1,2 \text {arccosh}(a x))}{a^2} \]

[In]

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

[Out]

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

Rule 12

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

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 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \cosh (x) \sinh (x) \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {1}{2} x^n \sinh (2 x) \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\text {arccosh}(a x)\right )}{2 a^2} \\ & = -\frac {\text {Subst}\left (\int e^{-2 x} x^n \, dx,x,\text {arccosh}(a x)\right )}{4 a^2}+\frac {\text {Subst}\left (\int e^{2 x} x^n \, dx,x,\text {arccosh}(a x)\right )}{4 a^2} \\ & = \frac {2^{-3-n} (-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-2 \text {arccosh}(a x))}{a^2}+\frac {2^{-3-n} \Gamma (1+n,2 \text {arccosh}(a x))}{a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int x \text {arccosh}(a x)^n \, dx=\frac {2^{-3-n} (-\text {arccosh}(a x))^{-n} \left (\text {arccosh}(a x)^n \Gamma (1+n,-2 \text {arccosh}(a x))+(-\text {arccosh}(a x))^n \Gamma (1+n,2 \text {arccosh}(a x))\right )}{a^2} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64

method	result	size
default	\(\frac {\operatorname {arccosh}\left (a x \right )^{2+n} \operatorname {hypergeom}\left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \operatorname {arccosh}\left (a x \right )^{2}\right )}{a^{2} \left (2+n \right )}\)	\(38\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x*arccosh(a*x)^n, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*acosh(a*x)**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate(x*arccosh(a*x)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate(x*arccosh(a*x)^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x*acosh(a*x)^n, x)