3.2.29 \(\int x^2 \cosh ^{-1}(a x)^n \, dx\) [129]

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

[Out]

1/8*3^(-1-n)*arccosh(a*x)^n*GAMMA(1+n,-3*arccosh(a*x))/a^3/((-arccosh(a*x))^n)+1/8*arccosh(a*x)^n*GAMMA(1+n,-a
rccosh(a*x))/a^3/((-arccosh(a*x))^n)+1/8*GAMMA(1+n,arccosh(a*x))/a^3+1/8*3^(-1-n)*GAMMA(1+n,3*arccosh(a*x))/a^
3

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Rubi [A]
time = 0.11, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5887, 5556, 3389, 2212} \begin {gather*} \frac {3^{-n-1} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac {\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {\text {Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {3^{-n-1} \text {Gamma}\left (n+1,3 \cosh ^{-1}(a x)\right )}{8 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3^(-1 - n)*ArcCosh[a*x]^n*Gamma[1 + n, -3*ArcCosh[a*x]])/(8*a^3*(-ArcCosh[a*x])^n) + (ArcCosh[a*x]^n*Gamma[1
+ n, -ArcCosh[a*x]])/(8*a^3*(-ArcCosh[a*x])^n) + Gamma[1 + n, ArcCosh[a*x]]/(8*a^3) + (3^(-1 - n)*Gamma[1 + n,
3*ArcCosh[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 3389

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 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 5887

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

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 95, normalized size = 0.84 \begin {gather*} \frac {3^{-1-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-3 \cosh ^{-1}(a x)\right )+\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )+\Gamma \left (1+n,\cosh ^{-1}(a x)\right )+3^{-1-n} \Gamma \left (1+n,3 \cosh ^{-1}(a x)\right )}{8 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 1.49, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {arccosh}\left (a x \right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arccosh(a*x)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*arccosh(a*x)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {acosh}^{n}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*acosh(a*x)**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arccosh(a*x)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\mathrm {acosh}\left (a\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*acosh(a*x)^n, x)

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