3.128 \(\int x^3 \cosh ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=117 \[ \frac{2^{-2 (n+3)} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-4 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \text{Gamma}\left (n+1,2 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \text{Gamma}\left (n+1,4 \cosh ^{-1}(a x)\right )}{a^4} \]

[Out]

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

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Rubi [A]  time = 0.19092, antiderivative size = 117, 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 = {5670, 5448, 3308, 2181} \[ \frac{2^{-2 (n+3)} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-4 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \text{Gamma}\left (n+1,2 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \text{Gamma}\left (n+1,4 \cosh ^{-1}(a x)\right )}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[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 3308

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 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^3 \cosh ^{-1}(a x)^n \, dx &=\frac{\operatorname{Subst}\left (\int x^n \cosh ^3(x) \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} x^n \sinh (2 x)+\frac{1}{8} x^n \sinh (4 x)\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int x^n \sinh (4 x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}+\frac{\operatorname{Subst}\left (\int x^n \sinh (2 x) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-4 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^4}+\frac{\operatorname{Subst}\left (\int e^{4 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^4}-\frac{\operatorname{Subst}\left (\int e^{-2 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}+\frac{\operatorname{Subst}\left (\int e^{2 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{4^{-3-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-4 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-4-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-2 \cosh ^{-1}(a x)\right )}{a^4}+\frac{2^{-4-n} \Gamma \left (1+n,2 \cosh ^{-1}(a x)\right )}{a^4}+\frac{4^{-3-n} \Gamma \left (1+n,4 \cosh ^{-1}(a x)\right )}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0950426, size = 97, normalized size = 0.83 \[ \frac{4^{-n-3} \left (-\cosh ^{-1}(a x)\right )^{-n} \left (\left (-\cosh ^{-1}(a x)\right )^n \left (2^{n+2} \text{Gamma}\left (n+1,2 \cosh ^{-1}(a x)\right )+\text{Gamma}\left (n+1,4 \cosh ^{-1}(a x)\right )\right )+\cosh ^{-1}(a x)^n \text{Gamma}\left (n+1,-4 \cosh ^{-1}(a x)\right )+2^{n+2} \cosh ^{-1}(a x)^n \text{Gamma}\left (n+1,-2 \cosh ^{-1}(a x)\right )\right )}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4^(-3 - n)*(ArcCosh[a*x]^n*Gamma[1 + n, -4*ArcCosh[a*x]] + 2^(2 + n)*ArcCosh[a*x]^n*Gamma[1 + n, -2*ArcCosh[a
*x]] + (-ArcCosh[a*x])^n*(2^(2 + n)*Gamma[1 + n, 2*ArcCosh[a*x]] + Gamma[1 + n, 4*ArcCosh[a*x]])))/(a^4*(-ArcC
osh[a*x])^n)

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\rm arccosh} \left (ax\right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arcosh}\left (a x\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arccosh(a*x)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{arcosh}\left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*arccosh(a*x)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{acosh}^{n}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*acosh(a*x)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccosh(a*x)^n,x, algorithm="giac")

[Out]

sage0*x