### 3.22 $$\int \text{csch}^{-1}(\frac{1}{x}) \, dx$$

Optimal. Leaf size=16 $x \sinh ^{-1}(x)-\sqrt{x^2+1}$

[Out]

-Sqrt[1 + x^2] + x*ArcSinh[x]

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Rubi [A]  time = 0.0060857, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.75, Rules used = {6328, 5653, 261} $x \sinh ^{-1}(x)-\sqrt{x^2+1}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sqrt[1 + x^2] + x*ArcSinh[x]

Rule 6328

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

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \text{csch}^{-1}\left (\frac{1}{x}\right ) \, dx &=\int \sinh ^{-1}(x) \, dx\\ &=x \sinh ^{-1}(x)-\int \frac{x}{\sqrt{1+x^2}} \, dx\\ &=-\sqrt{1+x^2}+x \sinh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0027146, size = 18, normalized size = 1.12 $x \text{csch}^{-1}\left (\frac{1}{x}\right )-\sqrt{x^2+1}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sqrt[1 + x^2] + x*ArcCsch[x^(-1)]

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Maple [A]  time = 0.099, size = 29, normalized size = 1.8 \begin{align*} x{\rm arccsch} \left ({x}^{-1}\right )-{ \left ( 1+{x}^{-2} \right ){x}^{2}{\frac{1}{\sqrt{ \left ( 1+{x}^{-2} \right ){x}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(arccsch(1/x),x)

[Out]

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

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Maxima [A]  time = 1.00029, size = 22, normalized size = 1.38 \begin{align*} x \operatorname{arcsch}\left (\frac{1}{x}\right ) - \sqrt{x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccsch(1/x) - sqrt(x^2 + 1)

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Fricas [A]  time = 2.58506, size = 57, normalized size = 3.56 \begin{align*} x \log \left (x + \sqrt{x^{2} + 1}\right ) - \sqrt{x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.147368, size = 14, normalized size = 0.88 \begin{align*} x \operatorname{acsch}{\left (\frac{1}{x} \right )} - \sqrt{x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsch(1/x),x)

[Out]

x*acsch(1/x) - sqrt(x**2 + 1)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcsch}\left (\frac{1}{x}\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsch(1/x), x)