\(\int x^3 \text {arcsinh}(a x)^n \, dx\) [130]

Optimal result

Integrand size = 10, antiderivative size = 119 \[ \int x^3 \text {arcsinh}(a x)^n \, dx=\frac {2^{-2 (3+n)} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-4 \text {arcsinh}(a x))}{a^4}-\frac {2^{-4-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))}{a^4}-\frac {2^{-4-n} \Gamma (1+n,2 \text {arcsinh}(a x))}{a^4}+\frac {2^{-2 (3+n)} \Gamma (1+n,4 \text {arcsinh}(a x))}{a^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 119, 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 = {5780, 5556, 3389, 2212} \[ \int x^3 \text {arcsinh}(a x)^n \, dx=\frac {2^{-2 (n+3)} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-4 \text {arcsinh}(a x))}{a^4}-\frac {2^{-n-4} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-2 \text {arcsinh}(a x))}{a^4}-\frac {2^{-n-4} \Gamma (n+1,2 \text {arcsinh}(a x))}{a^4}+\frac {2^{-2 (n+3)} \Gamma (n+1,4 \text {arcsinh}(a x))}{a^4} \]

[In]

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

[Out]

(ArcSinh[a*x]^n*Gamma[1 + n, -4*ArcSinh[a*x]])/(2^(2*(3 + n))*a^4*(-ArcSinh[a*x])^n) - (2^(-4 - n)*ArcSinh[a*x
]^n*Gamma[1 + n, -2*ArcSinh[a*x]])/(a^4*(-ArcSinh[a*x])^n) - (2^(-4 - n)*Gamma[1 + n, 2*ArcSinh[a*x]])/a^4 + G
amma[1 + n, 4*ArcSinh[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 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sinh
[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[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 ^3(x) \, dx,x,\text {arcsinh}(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 {arcsinh}(a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int x^n \sinh (4 x) \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\text {arcsinh}(a x)\right )}{4 a^4} \\ & = -\frac {\text {Subst}\left (\int e^{-4 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int e^{4 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int e^{-2 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int e^{2 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{8 a^4} \\ & = \frac {4^{-3-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-4 \text {arcsinh}(a x))}{a^4}-\frac {2^{-4-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))}{a^4}-\frac {2^{-4-n} \Gamma (1+n,2 \text {arcsinh}(a x))}{a^4}+\frac {4^{-3-n} \Gamma (1+n,4 \text {arcsinh}(a x))}{a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83 \[ \int x^3 \text {arcsinh}(a x)^n \, dx=\frac {4^{-3-n} (-\text {arcsinh}(a x))^{-n} \left (\text {arcsinh}(a x)^n \Gamma (1+n,-4 \text {arcsinh}(a x))-2^{2+n} \text {arcsinh}(a x)^n \Gamma (1+n,-2 \text {arcsinh}(a x))+(-\text {arcsinh}(a x))^n \left (-2^{2+n} \Gamma (1+n,2 \text {arcsinh}(a x))+\Gamma (1+n,4 \text {arcsinh}(a x))\right )\right )}{a^4} \]

[In]

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

[Out]

(4^(-3 - n)*(ArcSinh[a*x]^n*Gamma[1 + n, -4*ArcSinh[a*x]] - 2^(2 + n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a
*x]] + (-ArcSinh[a*x])^n*(-(2^(2 + n)*Gamma[1 + n, 2*ArcSinh[a*x]]) + Gamma[1 + n, 4*ArcSinh[a*x]])))/(a^4*(-A
rcSinh[a*x])^n)

Maple [F]

\[\int x^{3} \operatorname {arcsinh}\left (a x \right )^{n}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^3*arcsinh(a*x)^n, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**3*asinh(a*x)**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^3*arcsinh(a*x)^n, x)

Giac [F(-2)]

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

[In]

integrate(x^3*arcsinh(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. \[ \int x^3 \text {arcsinh}(a x)^n \, dx=\int x^3\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]

[In]

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

[Out]

int(x^3*asinh(a*x)^n, x)