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

Optimal. Leaf size=30 \[ x \cosh ^{-1}(a x)-\frac{\sqrt{a x-1} \sqrt{a x+1}}{a} \]

[Out]

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

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Rubi [A]  time = 0.006411, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5654, 74} \[ x \cosh ^{-1}(a x)-\frac{\sqrt{a x-1} \sqrt{a x+1}}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x],x]

[Out]

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

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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

Mathematica [A]  time = 0.016369, size = 30, normalized size = 1. \[ x \cosh ^{-1}(a x)-\frac{\sqrt{a x-1} \sqrt{a x+1}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x],x]

[Out]

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

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Maple [A]  time = 0.001, size = 29, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax{\rm arccosh} \left (ax\right )-\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)

[Out]

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

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Maxima [A]  time = 1.15911, size = 34, normalized size = 1.13 \begin{align*} \frac{a x \operatorname{arcosh}\left (a x\right ) - \sqrt{a^{2} x^{2} - 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.36269, size = 78, normalized size = 2.6 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \sqrt{a^{2} x^{2} - 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.208176, size = 26, normalized size = 0.87 \begin{align*} \begin{cases} x \operatorname{acosh}{\left (a x \right )} - \frac{\sqrt{a^{2} x^{2} - 1}}{a} & \text{for}\: a \neq 0 \\\frac{i \pi x}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x),x)

[Out]

Piecewise((x*acosh(a*x) - sqrt(a**2*x**2 - 1)/a, Ne(a, 0)), (I*pi*x/2, True))

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Giac [A]  time = 1.37339, size = 47, normalized size = 1.57 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{\sqrt{a^{2} x^{2} - 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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