3.7 \(\int \frac{\cosh ^{-1}(a x)}{x^2} \, dx\)

Optimal. Leaf size=32 \[ a \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )-\frac{\cosh ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0168724, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5662, 92, 205} \[ a \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )-\frac{\cosh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0245135, size = 57, normalized size = 1.78 \[ \frac{a \sqrt{a^2 x^2-1} \tan ^{-1}\left (\sqrt{a^2 x^2-1}\right )}{\sqrt{a x-1} \sqrt{a x+1}}-\frac{\cosh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 51, normalized size = 1.6 \begin{align*} -{\frac{{\rm arccosh} \left (ax\right )}{x}}-{a\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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arccosh(a*x)/x-a*(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))

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Maxima [A]  time = 1.78092, size = 32, normalized size = 1. \begin{align*} -a \arcsin \left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{\operatorname{arcosh}\left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.40256, size = 158, normalized size = 4.94 \begin{align*} \frac{2 \, a x \arctan \left (-a x + \sqrt{a^{2} x^{2} - 1}\right ) +{\left (x - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + x \log \left (-a x + \sqrt{a^{2} x^{2} - 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)/x**2, x)

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Giac [A]  time = 1.39625, size = 49, normalized size = 1.53 \begin{align*} a \arctan \left (\sqrt{a^{2} x^{2} - 1}\right ) - \frac{\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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