3.11 \(\int \frac{\cosh ^{-1}(a x)}{x^6} \, dx\)

Optimal. Leaf size=93 \[ \frac{3 a^3 \sqrt{a x-1} \sqrt{a x+1}}{40 x^2}+\frac{3}{40} a^5 \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5} \]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*x^4) + (3*a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(40*x^2) - ArcCosh[a*x]/(5*x^
5) + (3*a^5*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/40

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Rubi [A]  time = 0.0407888, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5662, 103, 12, 92, 205} \[ \frac{3 a^3 \sqrt{a x-1} \sqrt{a x+1}}{40 x^2}+\frac{3}{40} a^5 \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/x^6,x]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*x^4) + (3*a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(40*x^2) - ArcCosh[a*x]/(5*x^
5) + (3*a^5*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/40

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 103

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)}{x^6} \, dx &=-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{5} a \int \frac{1}{x^5 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{20} a \int \frac{3 a^2}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{20} \left (3 a^3\right ) \int \frac{1}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^3\right ) \int \frac{a^2}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^5\right ) \int \frac{1}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{a+a x^2} \, dx,x,\sqrt{-1+a x} \sqrt{1+a x}\right )\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{3}{40} a^5 \tan ^{-1}\left (\sqrt{-1+a x} \sqrt{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0436838, size = 104, normalized size = 1.12 \[ -\frac{-3 a^5 x^5+a^3 x^3-3 a^5 x^5 \sqrt{a^2 x^2-1} \tan ^{-1}\left (\sqrt{a^2 x^2-1}\right )+2 a x+8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{40 x^5 \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]/x^6,x]

[Out]

-(2*a*x + a^3*x^3 - 3*a^5*x^5 + 8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] - 3*a^5*x^5*Sqrt[-1 + a^2*x^2]*Arc
Tan[Sqrt[-1 + a^2*x^2]])/(40*x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

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Maple [A]  time = 0.016, size = 95, normalized size = 1. \begin{align*} -{\frac{{\rm arccosh} \left (ax\right )}{5\,{x}^{5}}}-{\frac{3\,{a}^{5}}{40}\sqrt{ax-1}\sqrt{ax+1}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}}}+{\frac{3\,{a}^{3}}{40\,{x}^{2}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{a}{20\,{x}^{4}}\sqrt{ax-1}\sqrt{ax+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)/x^6,x)

[Out]

-1/5*arccosh(a*x)/x^5-3/40*a^5*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))+3/40*
a^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^2+1/20*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^4

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Maxima [A]  time = 1.76618, size = 88, normalized size = 0.95 \begin{align*} -\frac{1}{40} \,{\left (3 \, a^{4} \arcsin \left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{3 \, \sqrt{a^{2} x^{2} - 1} a^{2}}{x^{2}} - \frac{2 \, \sqrt{a^{2} x^{2} - 1}}{x^{4}}\right )} a - \frac{\operatorname{arcosh}\left (a x\right )}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/x^6,x, algorithm="maxima")

[Out]

-1/40*(3*a^4*arcsin(1/(sqrt(a^2)*abs(x))) - 3*sqrt(a^2*x^2 - 1)*a^2/x^2 - 2*sqrt(a^2*x^2 - 1)/x^4)*a - 1/5*arc
cosh(a*x)/x^5

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Fricas [A]  time = 2.51877, size = 238, normalized size = 2.56 \begin{align*} \frac{6 \, a^{5} x^{5} \arctan \left (-a x + \sqrt{a^{2} x^{2} - 1}\right ) + 8 \, x^{5} \log \left (-a x + \sqrt{a^{2} x^{2} - 1}\right ) + 8 \,{\left (x^{5} - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) +{\left (3 \, a^{3} x^{3} + 2 \, a x\right )} \sqrt{a^{2} x^{2} - 1}}{40 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/x^6,x, algorithm="fricas")

[Out]

1/40*(6*a^5*x^5*arctan(-a*x + sqrt(a^2*x^2 - 1)) + 8*x^5*log(-a*x + sqrt(a^2*x^2 - 1)) + 8*(x^5 - 1)*log(a*x +
sqrt(a^2*x^2 - 1)) + (3*a^3*x^3 + 2*a*x)*sqrt(a^2*x^2 - 1))/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}{\left (a x \right )}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)/x**6,x)

[Out]

Integral(acosh(a*x)/x**6, x)

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Giac [A]  time = 1.3486, size = 103, normalized size = 1.11 \begin{align*} \frac{1}{40} \, a^{5}{\left (\frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{a^{2} x^{2} - 1}}{a^{4} x^{4}} + 3 \, \arctan \left (\sqrt{a^{2} x^{2} - 1}\right )\right )} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)/x^6,x, algorithm="giac")

[Out]

1/40*a^5*((3*(a^2*x^2 - 1)^(3/2) + 5*sqrt(a^2*x^2 - 1))/(a^4*x^4) + 3*arctan(sqrt(a^2*x^2 - 1))) - 1/5*log(a*x
+ sqrt(a^2*x^2 - 1))/x^5