3.131 \(\int x^2 \sinh ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=113 \[ \frac{3^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{8 a^3}-\frac{\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{8 a^3}+\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac{3^{-n-1} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^3} \]

[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)

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Rubi [A]  time = 0.145062, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5669, 5448, 3307, 2181} \[ \frac{3^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{8 a^3}-\frac{\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{8 a^3}+\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{8 a^3}-\frac{3^{-n-1} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^3} \]

Antiderivative was successfully verified.

[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 5669

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Sinh[x]^m*Cosh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5448

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

Mathematica [A]  time = 0.0702013, size = 97, normalized size = 0.86 \[ \frac{3^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )-\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )+\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )-3^{-n-1} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^3} \]

Antiderivative was successfully verified.

[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
)

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arsinh}\left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arcsinh(a*x)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{arsinh}\left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*arcsinh(a*x)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{asinh}^{n}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*asinh(a*x)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arsinh}\left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arcsinh(a*x)^n, x)