3.8 \(\int \frac{\cosh ^{-1}(a x)}{x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac{a \sqrt{a x-1} \sqrt{a x+1}}{2 x}-\frac{\cosh ^{-1}(a x)}{2 x^2} \]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*x) - ArcCosh[a*x]/(2*x^2)

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Rubi [A]  time = 0.0134166, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5662, 95} \[ \frac{a \sqrt{a x-1} \sqrt{a x+1}}{2 x}-\frac{\cosh ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*x) - ArcCosh[a*x]/(2*x^2)

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 95

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] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.007299, size = 35, normalized size = 0.92 \[ \frac{a x \sqrt{a x-1} \sqrt{a x+1}-\cosh ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] - ArcCosh[a*x])/(2*x^2)

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Maple [A]  time = 0.012, size = 40, normalized size = 1.1 \begin{align*}{a}^{2} \left ( -{\frac{{\rm arccosh} \left (ax\right )}{2\,{a}^{2}{x}^{2}}}+{\frac{1}{2\,ax}\sqrt{ax-1}\sqrt{ax+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(-1/2*arccosh(a*x)/a^2/x^2+1/2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/x)

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Maxima [A]  time = 1.72844, size = 36, normalized size = 0.95 \begin{align*} \frac{\sqrt{a^{2} x^{2} - 1} a}{2 \, x} - \frac{\operatorname{arcosh}\left (a x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(a^2*x^2 - 1)*a/x - 1/2*arccosh(a*x)/x^2

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Fricas [A]  time = 2.4195, size = 86, normalized size = 2.26 \begin{align*} \frac{\sqrt{a^{2} x^{2} - 1} a x - \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(a^2*x^2 - 1)*a*x - log(a*x + sqrt(a^2*x^2 - 1)))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)/x**3, x)

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Giac [A]  time = 1.20903, size = 68, normalized size = 1.79 \begin{align*} \frac{a{\left | a \right |}}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*abs(a)/((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1) - 1/2*log(a*x + sqrt(a^2*x^2 - 1))/x^2