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]

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

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.00 \[ x \cosh ^{-1}(a x)-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a} \]

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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fricas [A]  time = 0.53, size = 37, normalized size = 1.23 \[ \frac {a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \sqrt {a^{2} x^{2} - 1}}{a} \]

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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giac [A]  time = 0.20, size = 35, normalized size = 1.17 \[ x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{a} \]

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

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maple [A]  time = 0.00, size = 29, normalized size = 0.97 \[ \frac {a x \,\mathrm {arccosh}\left (a x \right )-\sqrt {a x -1}\, \sqrt {a x +1}}{a} \]

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 = 0.62, size = 25, normalized size = 0.83 \[ \frac {a x \operatorname {arcosh}\left (a x\right ) - \sqrt {a^{2} x^{2} - 1}}{a} \]

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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mupad [B]  time = 0.32, size = 26, normalized size = 0.87 \[ x\,\mathrm {acosh}\left (a\,x\right )-\frac {\sqrt {a\,x-1}\,\sqrt {a\,x+1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x),x)

[Out]

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

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sympy [A]  time = 0.15, size = 26, normalized size = 0.87 \[ \begin {cases} x \operatorname {acosh}{\left (a x \right )} - \frac {\sqrt {a^{2} x^{2} - 1}}{a} & \text {for}\: a \neq 0 \\\frac {i \pi x}{2} & \text {otherwise} \end {cases} \]

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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