\(\int x^2 \text {arcsinh}(a x)^n \, dx\) [131]

Optimal result

Integrand size = 10, antiderivative size = 113 \[ \int x^2 \text {arcsinh}(a x)^n \, dx=\frac {3^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))}{8 a^3}-\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{8 a^3}+\frac {\Gamma (1+n,\text {arcsinh}(a x))}{8 a^3}-\frac {3^{-1-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{8 a^3} \]

[Out]

1/8*3^(-1-n)*arcsinh(a*x)^n*GAMMA(1+n,-3*arcsinh(a*x))/a^3/((-arcsinh(a*x))^n)-1/8*arcsinh(a*x)^n*GAMMA(1+n,-a
rcsinh(a*x))/a^3/((-arcsinh(a*x))^n)+1/8*GAMMA(1+n,arcsinh(a*x))/a^3-1/8*3^(-1-n)*GAMMA(1+n,3*arcsinh(a*x))/a^
3

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 113, 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, 3388, 2212} \[ \int x^2 \text {arcsinh}(a x)^n \, dx=\frac {3^{-n-1} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-3 \text {arcsinh}(a x))}{8 a^3}-\frac {\text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-\text {arcsinh}(a x))}{8 a^3}+\frac {\Gamma (n+1,\text {arcsinh}(a x))}{8 a^3}-\frac {3^{-n-1} \Gamma (n+1,3 \text {arcsinh}(a x))}{8 a^3} \]

[In]

Int[x^2*ArcSinh[a*x]^n,x]

[Out]

(3^(-1 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/(8*a^3*(-ArcSinh[a*x])^n) - (ArcSinh[a*x]^n*Gamma[1
+ n, -ArcSinh[a*x]])/(8*a^3*(-ArcSinh[a*x])^n) + Gamma[1 + n, ArcSinh[a*x]]/(8*a^3) - (3^(-1 - n)*Gamma[1 + n,
3*ArcSinh[a*x]])/(8*a^3)

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 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int x^2 \text {arcsinh}(a x)^n \, dx=\frac {3^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))-(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))+\Gamma (1+n,\text {arcsinh}(a x))-3^{-1-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{8 a^3} \]

[In]

Integrate[x^2*ArcSinh[a*x]^n,x]

[Out]

((3^(-1 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/(-ArcSinh[a*x])^n - (ArcSinh[a*x]^n*Gamma[1 + n, -A
rcSinh[a*x]])/(-ArcSinh[a*x])^n + Gamma[1 + n, ArcSinh[a*x]] - 3^(-1 - n)*Gamma[1 + n, 3*ArcSinh[a*x]])/(8*a^3
)

Maple [F]

\[\int x^{2} \operatorname {arcsinh}\left (a x \right )^{n}d x\]

[In]

int(x^2*arcsinh(a*x)^n,x)

[Out]

int(x^2*arcsinh(a*x)^n,x)

Fricas [F]

\[ \int x^2 \text {arcsinh}(a x)^n \, dx=\int { x^{2} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]

[In]

integrate(x^2*arcsinh(a*x)^n,x, algorithm="fricas")

[Out]

integral(x^2*arcsinh(a*x)^n, x)

Sympy [F]

\[ \int x^2 \text {arcsinh}(a x)^n \, dx=\int x^{2} \operatorname {asinh}^{n}{\left (a x \right )}\, dx \]

[In]

integrate(x**2*asinh(a*x)**n,x)

[Out]

Integral(x**2*asinh(a*x)**n, x)

Maxima [F]

\[ \int x^2 \text {arcsinh}(a x)^n \, dx=\int { x^{2} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]

[In]

integrate(x^2*arcsinh(a*x)^n,x, algorithm="maxima")

[Out]

integrate(x^2*arcsinh(a*x)^n, x)

Giac [F]

\[ \int x^2 \text {arcsinh}(a x)^n \, dx=\int { x^{2} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]

[In]

integrate(x^2*arcsinh(a*x)^n,x, algorithm="giac")

[Out]

integrate(x^2*arcsinh(a*x)^n, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \text {arcsinh}(a x)^n \, dx=\int x^2\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]

[In]

int(x^2*asinh(a*x)^n,x)

[Out]

int(x^2*asinh(a*x)^n, x)