\(\int \text {arccosh}(a x) \, dx\) [5]

Optimal result

Integrand size = 4, antiderivative size = 30 \[ \int \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a}+x \text {arccosh}(a x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5879, 75} \[ \int \text {arccosh}(a x) \, dx=x \text {arccosh}(a x)-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a} \]

[In]

Int[ArcCosh[a*x],x]

[Out]

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

Rule 75

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 5879

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*} \text {integral}& = x \text {arccosh}(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 \text {arccosh}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a}+x \text {arccosh}(a x) \]

[In]

Integrate[ArcCosh[a*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90

method	result	size
parts	\(x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{a}\)	\(27\)
derivativedivides	\(\frac {a x \,\operatorname {arccosh}\left (a x \right )-\sqrt {a x -1}\, \sqrt {a x +1}}{a}\)	\(29\)
default	\(\frac {a x \,\operatorname {arccosh}\left (a x \right )-\sqrt {a x -1}\, \sqrt {a x +1}}{a}\)	\(29\)

[In]

int(arccosh(a*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \text {arccosh}(a x) \, dx=\frac {a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \sqrt {a^{2} x^{2} - 1}}{a} \]

[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

Sympy [F]

\[ \int \text {arccosh}(a x) \, dx=\int \operatorname {acosh}{\left (a x \right )}\, dx \]

[In]

integrate(acosh(a*x),x)

[Out]

Integral(acosh(a*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \text {arccosh}(a x) \, dx=\frac {a x \operatorname {arcosh}\left (a x\right ) - \sqrt {a^{2} x^{2} - 1}}{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \text {arccosh}(a x) \, dx=x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{a} \]

[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

Mupad [B] (verification not implemented)

Time = 2.68 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \text {arccosh}(a x) \, dx=x\,\mathrm {acosh}\left (a\,x\right )-\frac {\sqrt {a\,x-1}\,\sqrt {a\,x+1}}{a} \]

[In]

int(acosh(a*x),x)

[Out]

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