3.133 \(\int \sinh ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=49 \[ \frac{\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{2 a}-\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a} \]

[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)

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Rubi [A]  time = 0.0467243, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5657, 3307, 2181} \[ \frac{\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{2 a}-\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a} \]

Antiderivative was successfully verified.

[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 5657

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]

Rule 3307

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 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

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

Mathematica [A]  time = 0.0251728, size = 45, normalized size = 0.92 \[ \frac{\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )-\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a} \]

Antiderivative was successfully verified.

[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)

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Maple [C]  time = 0.053, size = 40, normalized size = 0.8 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{1+n}}{a \left ( 1+n \right ) }{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{n}{2}};\,{\frac{1}{2}},{\frac{3}{2}}+{\frac{n}{2}};\,{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arsinh}\left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{arsinh}\left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsinh(a*x)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{asinh}^{n}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asinh(a*x)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arsinh}\left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x)^n, x)