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

Optimal. Leaf size=93 \[ -\frac{4 x^2 \sqrt{a x-1} \sqrt{a x+1}}{75 a^3}-\frac{8 \sqrt{a x-1} \sqrt{a x+1}}{75 a^5}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x) \]

[Out]

(-8*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^5) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^3) - (x^4*Sqrt[-1 + a*
x]*Sqrt[1 + a*x])/(25*a) + (x^5*ArcCosh[a*x])/5

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Rubi [A]  time = 0.0374411, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5662, 100, 12, 74} \[ -\frac{4 x^2 \sqrt{a x-1} \sqrt{a x+1}}{75 a^3}-\frac{8 \sqrt{a x-1} \sqrt{a x+1}}{75 a^5}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^5) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^3) - (x^4*Sqrt[-1 + a*
x]*Sqrt[1 + a*x])/(25*a) + (x^5*ArcCosh[a*x])/5

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin{align*} \int x^4 \cosh ^{-1}(a x) \, dx &=\frac{1}{5} x^5 \cosh ^{-1}(a x)-\frac{1}{5} a \int \frac{x^5}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)-\frac{\int \frac{4 x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)-\frac{4 \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{25 a}\\ &=-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{75 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)-\frac{4 \int \frac{2 x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{75 a^3}\\ &=-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{75 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)-\frac{8 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{75 a^3}\\ &=-\frac{8 \sqrt{-1+a x} \sqrt{1+a x}}{75 a^5}-\frac{4 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{75 a^3}-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{25 a}+\frac{1}{5} x^5 \cosh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0328575, size = 55, normalized size = 0.59 \[ \frac{1}{5} x^5 \cosh ^{-1}(a x)-\frac{\sqrt{a x-1} \sqrt{a x+1} \left (3 a^4 x^4+4 a^2 x^2+8\right )}{75 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(8 + 4*a^2*x^2 + 3*a^4*x^4))/(75*a^5) + (x^5*ArcCosh[a*x])/5

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Maple [A]  time = 0.014, size = 52, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{a}^{5}{x}^{5}{\rm arccosh} \left (ax\right )}{5}}-{\frac{3\,{x}^{4}{a}^{4}+4\,{a}^{2}{x}^{2}+8}{75}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/5*a^5*x^5*arccosh(a*x)-1/75*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(3*a^4*x^4+4*a^2*x^2+8))

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Maxima [A]  time = 1.09191, size = 92, normalized size = 0.99 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arcosh}\left (a x\right ) - \frac{1}{75} \,{\left (\frac{3 \, \sqrt{a^{2} x^{2} - 1} x^{4}}{a^{2}} + \frac{4 \, \sqrt{a^{2} x^{2} - 1} x^{2}}{a^{4}} + \frac{8 \, \sqrt{a^{2} x^{2} - 1}}{a^{6}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arccosh(a*x) - 1/75*(3*sqrt(a^2*x^2 - 1)*x^4/a^2 + 4*sqrt(a^2*x^2 - 1)*x^2/a^4 + 8*sqrt(a^2*x^2 - 1)/a
^6)*a

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Fricas [A]  time = 2.5049, size = 135, normalized size = 1.45 \begin{align*} \frac{15 \, a^{5} x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt{a^{2} x^{2} - 1}}{75 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/75*(15*a^5*x^5*log(a*x + sqrt(a^2*x^2 - 1)) - (3*a^4*x^4 + 4*a^2*x^2 + 8)*sqrt(a^2*x^2 - 1))/a^5

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Sympy [A]  time = 2.48224, size = 76, normalized size = 0.82 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acosh}{\left (a x \right )}}{5} - \frac{x^{4} \sqrt{a^{2} x^{2} - 1}}{25 a} - \frac{4 x^{2} \sqrt{a^{2} x^{2} - 1}}{75 a^{3}} - \frac{8 \sqrt{a^{2} x^{2} - 1}}{75 a^{5}} & \text{for}\: a \neq 0 \\\frac{i \pi x^{5}}{10} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5*acosh(a*x)/5 - x**4*sqrt(a**2*x**2 - 1)/(25*a) - 4*x**2*sqrt(a**2*x**2 - 1)/(75*a**3) - 8*sqrt
(a**2*x**2 - 1)/(75*a**5), Ne(a, 0)), (I*pi*x**5/10, True))

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Giac [A]  time = 1.22981, size = 90, normalized size = 0.97 \begin{align*} \frac{1}{5} \, x^{5} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{a^{2} x^{2} - 1}}{75 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*log(a*x + sqrt(a^2*x^2 - 1)) - 1/75*(3*(a^2*x^2 - 1)^(5/2) + 10*(a^2*x^2 - 1)^(3/2) + 15*sqrt(a^2*x^2
- 1))/a^5