\(\int \text {arcsinh}(a x)^n \, dx\) [133]

Optimal result

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

[Out]

1/2*arcsinh(a*x)^n*GAMMA(1+n,-arcsinh(a*x))/a/((-arcsinh(a*x))^n)-1/2*GAMMA(1+n,arcsinh(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 = {5774, 3388, 2212} \[ \int \text {arcsinh}(a x)^n \, dx=\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (n+1,-\text {arcsinh}(a x))}{2 a}-\frac {\Gamma (n+1,\text {arcsinh}(a x))}{2 a} \]

[In]

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

[Out]

(ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(2*a*(-ArcSinh[a*x])^n) - Gamma[1 + n, ArcSinh[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 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 5774

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \text {arcsinh}(a x)^n \, dx=\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))-\Gamma (1+n,\text {arcsinh}(a x))}{2 a} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(arcsinh(a*x)^n, x)

Sympy [F]

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

[In]

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

[Out]

Integral(asinh(a*x)**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate(arcsinh(a*x)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate(arcsinh(a*x)^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(asinh(a*x)^n, x)