3.127 \(\int x^4 \cosh ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=173 \[ \frac {5^{-n-1} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {\Gamma \left (n+1,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \Gamma \left (n+1,3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {5^{-n-1} \Gamma \left (n+1,5 \cosh ^{-1}(a x)\right )}{32 a^5} \]

[Out]

1/32*5^(-1-n)*arccosh(a*x)^n*GAMMA(1+n,-5*arccosh(a*x))/a^5/((-arccosh(a*x))^n)+1/32*arccosh(a*x)^n*GAMMA(1+n,
-3*arccosh(a*x))/(3^n)/a^5/((-arccosh(a*x))^n)+1/16*arccosh(a*x)^n*GAMMA(1+n,-arccosh(a*x))/a^5/((-arccosh(a*x
))^n)+1/16*GAMMA(1+n,arccosh(a*x))/a^5+1/32*GAMMA(1+n,3*arccosh(a*x))/(3^n)/a^5+1/32*5^(-1-n)*GAMMA(1+n,5*arcc
osh(a*x))/a^5

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Rubi [A]  time = 0.25, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5670, 5448, 3308, 2181} \[ \frac {5^{-n-1} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {\text {Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \text {Gamma}\left (n+1,3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {5^{-n-1} \text {Gamma}\left (n+1,5 \cosh ^{-1}(a x)\right )}{32 a^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCosh[a*x]^n,x]

[Out]

(5^(-1 - n)*ArcCosh[a*x]^n*Gamma[1 + n, -5*ArcCosh[a*x]])/(32*a^5*(-ArcCosh[a*x])^n) + (ArcCosh[a*x]^n*Gamma[1
+ n, -3*ArcCosh[a*x]])/(32*3^n*a^5*(-ArcCosh[a*x])^n) + (ArcCosh[a*x]^n*Gamma[1 + n, -ArcCosh[a*x]])/(16*a^5*
(-ArcCosh[a*x])^n) + Gamma[1 + n, ArcCosh[a*x]]/(16*a^5) + Gamma[1 + n, 3*ArcCosh[a*x]]/(32*3^n*a^5) + (5^(-1
- n)*Gamma[1 + n, 5*ArcCosh[a*x]])/(32*a^5)

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]

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 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 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]

Rubi steps

\begin {align*} \int x^4 \cosh ^{-1}(a x)^n \, dx &=\frac {\operatorname {Subst}\left (\int x^n \cosh ^4(x) \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{8} x^n \sinh (x)+\frac {3}{16} x^n \sinh (3 x)+\frac {1}{16} x^n \sinh (5 x)\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\operatorname {Subst}\left (\int x^n \sinh (5 x) \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {\operatorname {Subst}\left (\int x^n \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac {3 \operatorname {Subst}\left (\int x^n \sinh (3 x) \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac {\operatorname {Subst}\left (\int e^{-5 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}+\frac {\operatorname {Subst}\left (\int e^{5 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}-\frac {\operatorname {Subst}\left (\int e^{-x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {\operatorname {Subst}\left (\int e^x x^n \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}-\frac {3 \operatorname {Subst}\left (\int e^{-3 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}+\frac {3 \operatorname {Subst}\left (\int e^{3 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac {5^{-1-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {3^{-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {\Gamma \left (1+n,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \Gamma \left (1+n,3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {5^{-1-n} \Gamma \left (1+n,5 \cosh ^{-1}(a x)\right )}{32 a^5}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 144, normalized size = 0.83 \[ \frac {5^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-5 \cosh ^{-1}(a x)\right )+5\ 3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-3 \cosh ^{-1}(a x)\right )+10 \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\cosh ^{-1}(a x)\right )+10 \Gamma \left (n+1,\cosh ^{-1}(a x)\right )+5\ 3^{-n} \Gamma \left (n+1,3 \cosh ^{-1}(a x)\right )+5^{-n} \Gamma \left (n+1,5 \cosh ^{-1}(a x)\right )}{160 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCosh[a*x]^n,x]

[Out]

((ArcCosh[a*x]^n*Gamma[1 + n, -5*ArcCosh[a*x]])/(5^n*(-ArcCosh[a*x])^n) + (5*ArcCosh[a*x]^n*Gamma[1 + n, -3*Ar
cCosh[a*x]])/(3^n*(-ArcCosh[a*x])^n) + (10*ArcCosh[a*x]^n*Gamma[1 + n, -ArcCosh[a*x]])/(-ArcCosh[a*x])^n + 10*
Gamma[1 + n, ArcCosh[a*x]] + (5*Gamma[1 + n, 3*ArcCosh[a*x]])/3^n + Gamma[1 + n, 5*ArcCosh[a*x]]/5^n)/(160*a^5
)

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fricas [F]  time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {arcosh}\left (a x\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^n,x, algorithm="fricas")

[Out]

integral(x^4*arccosh(a*x)^n, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arcosh}\left (a x\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^n,x, algorithm="giac")

[Out]

integrate(x^4*arccosh(a*x)^n, x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{4} \mathrm {arccosh}\left (a x \right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccosh(a*x)^n,x)

[Out]

int(x^4*arccosh(a*x)^n,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arcosh}\left (a x\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccosh(a*x)^n,x, algorithm="maxima")

[Out]

integrate(x^4*arccosh(a*x)^n, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {acosh}\left (a\,x\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*acosh(a*x)^n,x)

[Out]

int(x^4*acosh(a*x)^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {acosh}^{n}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acosh(a*x)**n,x)

[Out]

Integral(x**4*acosh(a*x)**n, x)

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