\(\int \text {sech}^{-1}(\frac {1}{x}) \, dx\) [28]

Optimal result

Integrand size = 4, antiderivative size = 21 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=-\sqrt {-1+x} \sqrt {1+x}+x \text {arccosh}(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6462, 5879, 75} \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x \text {arccosh}(x)-\sqrt {x-1} \sqrt {x+1} \]

[In]

Int[ArcSech[x^(-1)],x]

[Out]

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

Rule 6462

Int[ArcSech[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcCosh[a/c + b*(x^n/c)]^m, x] /
; FreeQ[{a, b, c, n, m}, x]

Rubi steps \begin{align*} \text {integral}& = \int \text {arccosh}(x) \, dx \\ & = x \text {arccosh}(x)-\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx \\ & = -\sqrt {-1+x} \sqrt {1+x}+x \text {arccosh}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=-\sqrt {\frac {-1+x}{1+x}} (1+x)+x \text {sech}^{-1}\left (\frac {1}{x}\right ) \]

[In]

Integrate[ArcSech[x^(-1)],x]

[Out]

-(Sqrt[(-1 + x)/(1 + x)]*(1 + x)) + x*ArcSech[x^(-1)]

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
parts	\(x \,\operatorname {arcsech}\left (\frac {1}{x}\right )-\sqrt {x -1}\, \sqrt {1+x}\)	\(20\)
derivativedivides	\(x \,\operatorname {arcsech}\left (\frac {1}{x}\right )-\sqrt {-\left (-1+\frac {1}{x}\right ) x}\, \sqrt {\left (1+\frac {1}{x}\right ) x}\)	\(29\)
default	\(x \,\operatorname {arcsech}\left (\frac {1}{x}\right )-\sqrt {-\left (-1+\frac {1}{x}\right ) x}\, \sqrt {\left (1+\frac {1}{x}\right ) x}\)	\(29\)

[In]

int(arcsech(1/x),x,method=_RETURNVERBOSE)

[Out]

x*arcsech(1/x)-(x-1)^(1/2)*(1+x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x \log \left (x + \sqrt {x^{2} - 1}\right ) - \sqrt {x^{2} - 1} \]

[In]

integrate(arcsech(1/x),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=\int \operatorname {asech}{\left (\frac {1}{x} \right )}\, dx \]

[In]

integrate(asech(1/x),x)

[Out]

Integral(asech(1/x), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x \operatorname {arsech}\left (\frac {1}{x}\right ) - \sqrt {x^{2} - 1} \]

[In]

integrate(arcsech(1/x),x, algorithm="maxima")

[Out]

x*arcsech(1/x) - sqrt(x^2 - 1)

Giac [F]

\[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=\int { \operatorname {arsech}\left (\frac {1}{x}\right ) \,d x } \]

[In]

integrate(arcsech(1/x),x, algorithm="giac")

[Out]

integrate(arcsech(1/x), x)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x\,\mathrm {acosh}\left (x\right )-\sqrt {x-1}\,\sqrt {x+1} \]

[In]

int(acosh(x),x)

[Out]

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