\(\int x^3 \text {arccosh}(a x)^n \, dx\) [128]

Optimal result

Integrand size = 10, antiderivative size = 117 \[ \int x^3 \text {arccosh}(a x)^n \, dx=\frac {2^{-2 (3+n)} (-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-4 \text {arccosh}(a x))}{a^4}+\frac {2^{-4-n} (-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-2 \text {arccosh}(a x))}{a^4}+\frac {2^{-4-n} \Gamma (1+n,2 \text {arccosh}(a x))}{a^4}+\frac {2^{-2 (3+n)} \Gamma (1+n,4 \text {arccosh}(a x))}{a^4} \]

[Out]

arccosh(a*x)^n*GAMMA(1+n,-4*arccosh(a*x))/(2^(6+2*n))/a^4/((-arccosh(a*x))^n)+2^(-4-n)*arccosh(a*x)^n*GAMMA(1+
n,-2*arccosh(a*x))/a^4/((-arccosh(a*x))^n)+2^(-4-n)*GAMMA(1+n,2*arccosh(a*x))/a^4+GAMMA(1+n,4*arccosh(a*x))/(2
^(6+2*n))/a^4

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5887, 5556, 3389, 2212} \[ \int x^3 \text {arccosh}(a x)^n \, dx=\frac {2^{-2 (n+3)} \text {arccosh}(a x)^n (-\text {arccosh}(a x))^{-n} \Gamma (n+1,-4 \text {arccosh}(a x))}{a^4}+\frac {2^{-n-4} \text {arccosh}(a x)^n (-\text {arccosh}(a x))^{-n} \Gamma (n+1,-2 \text {arccosh}(a x))}{a^4}+\frac {2^{-n-4} \Gamma (n+1,2 \text {arccosh}(a x))}{a^4}+\frac {2^{-2 (n+3)} \Gamma (n+1,4 \text {arccosh}(a x))}{a^4} \]

[In]

Int[x^3*ArcCosh[a*x]^n,x]

[Out]

(ArcCosh[a*x]^n*Gamma[1 + n, -4*ArcCosh[a*x]])/(2^(2*(3 + n))*a^4*(-ArcCosh[a*x])^n) + (2^(-4 - n)*ArcCosh[a*x
]^n*Gamma[1 + n, -2*ArcCosh[a*x]])/(a^4*(-ArcCosh[a*x])^n) + (2^(-4 - n)*Gamma[1 + n, 2*ArcCosh[a*x]])/a^4 + G
amma[1 + n, 4*ArcCosh[a*x]]/(2^(2*(3 + n))*a^4)

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 ^3(x) \sinh (x) \, dx,x,\text {arccosh}(a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{4} x^n \sinh (2 x)+\frac {1}{8} x^n \sinh (4 x)\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int x^n \sinh (4 x) \, dx,x,\text {arccosh}(a x)\right )}{8 a^4}+\frac {\text {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\text {arccosh}(a x)\right )}{4 a^4} \\ & = -\frac {\text {Subst}\left (\int e^{-4 x} x^n \, dx,x,\text {arccosh}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int e^{4 x} x^n \, dx,x,\text {arccosh}(a x)\right )}{16 a^4}-\frac {\text {Subst}\left (\int e^{-2 x} x^n \, dx,x,\text {arccosh}(a x)\right )}{8 a^4}+\frac {\text {Subst}\left (\int e^{2 x} x^n \, dx,x,\text {arccosh}(a x)\right )}{8 a^4} \\ & = \frac {4^{-3-n} (-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-4 \text {arccosh}(a x))}{a^4}+\frac {2^{-4-n} (-\text {arccosh}(a x))^{-n} \text {arccosh}(a x)^n \Gamma (1+n,-2 \text {arccosh}(a x))}{a^4}+\frac {2^{-4-n} \Gamma (1+n,2 \text {arccosh}(a x))}{a^4}+\frac {4^{-3-n} \Gamma (1+n,4 \text {arccosh}(a x))}{a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83 \[ \int x^3 \text {arccosh}(a x)^n \, dx=\frac {4^{-3-n} (-\text {arccosh}(a x))^{-n} \left (\text {arccosh}(a x)^n \Gamma (1+n,-4 \text {arccosh}(a x))+2^{2+n} \text {arccosh}(a x)^n \Gamma (1+n,-2 \text {arccosh}(a x))+(-\text {arccosh}(a x))^n \left (2^{2+n} \Gamma (1+n,2 \text {arccosh}(a x))+\Gamma (1+n,4 \text {arccosh}(a x))\right )\right )}{a^4} \]

[In]

Integrate[x^3*ArcCosh[a*x]^n,x]

[Out]

(4^(-3 - n)*(ArcCosh[a*x]^n*Gamma[1 + n, -4*ArcCosh[a*x]] + 2^(2 + n)*ArcCosh[a*x]^n*Gamma[1 + n, -2*ArcCosh[a
*x]] + (-ArcCosh[a*x])^n*(2^(2 + n)*Gamma[1 + n, 2*ArcCosh[a*x]] + Gamma[1 + n, 4*ArcCosh[a*x]])))/(a^4*(-ArcC
osh[a*x])^n)

Maple [C] (verified)

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

Time = 0.69 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {\operatorname {arccosh}\left (a x \right )^{2+n} \operatorname {hypergeom}\left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], \operatorname {arccosh}\left (a x \right )^{2}\right )}{2 a^{4} \left (2+n \right )}+\frac {\operatorname {arccosh}\left (a x \right )^{2+n} \operatorname {hypergeom}\left (\left [1+\frac {n}{2}\right ], \left [\frac {3}{2}, 2+\frac {n}{2}\right ], 4 \operatorname {arccosh}\left (a x \right )^{2}\right )}{2 a^{4} \left (2+n \right )}\)	\(80\)

[In]

int(x^3*arccosh(a*x)^n,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int x^3 \text {arccosh}(a x)^n \, dx=\int { x^{3} \operatorname {arcosh}\left (a x\right )^{n} \,d x } \]

[In]

integrate(x^3*arccosh(a*x)^n,x, algorithm="fricas")

[Out]

integral(x^3*arccosh(a*x)^n, x)

Sympy [F]

\[ \int x^3 \text {arccosh}(a x)^n \, dx=\int x^{3} \operatorname {acosh}^{n}{\left (a x \right )}\, dx \]

[In]

integrate(x**3*acosh(a*x)**n,x)

[Out]

Integral(x**3*acosh(a*x)**n, x)

Maxima [F]

\[ \int x^3 \text {arccosh}(a x)^n \, dx=\int { x^{3} \operatorname {arcosh}\left (a x\right )^{n} \,d x } \]

[In]

integrate(x^3*arccosh(a*x)^n,x, algorithm="maxima")

[Out]

integrate(x^3*arccosh(a*x)^n, x)

Giac [F(-2)]

Exception generated. \[ \int x^3 \text {arccosh}(a x)^n \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*arccosh(a*x)^n,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^3 \text {arccosh}(a x)^n \, dx=\int x^3\,{\mathrm {acosh}\left (a\,x\right )}^n \,d x \]

[In]

int(x^3*acosh(a*x)^n,x)

[Out]

int(x^3*acosh(a*x)^n, x)