### 3.17 $$\int \text{csch}^{-1}(\sqrt{x}) \, dx$$

Optimal. Leaf size=31 $\frac{\sqrt{-x-1} \sqrt{x}}{\sqrt{-x}}+x \text{csch}^{-1}\left (\sqrt{x}\right )$

[Out]

(Sqrt[-1 - x]*Sqrt[x])/Sqrt[-x] + x*ArcCsch[Sqrt[x]]

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Rubi [A]  time = 0.0076117, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {6344, 12, 32} $\frac{\sqrt{-x-1} \sqrt{x}}{\sqrt{-x}}+x \text{csch}^{-1}\left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[ArcCsch[Sqrt[x]],x]

[Out]

(Sqrt[-1 - x]*Sqrt[x])/Sqrt[-x] + x*ArcCsch[Sqrt[x]]

Rule 6344

Int[ArcCsch[u_], x_Symbol] :> Simp[x*ArcCsch[u], x] - Dist[u/Sqrt[-u^2], Int[SimplifyIntegrand[(x*D[u, x])/(u*
Sqrt[-1 - u^2]), x], x], x] /; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0094641, size = 24, normalized size = 0.77 $\sqrt{\frac{1}{x}+1} \sqrt{x}+x \text{csch}^{-1}\left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[ArcCsch[Sqrt[x]],x]

[Out]

Sqrt[1 + x^(-1)]*Sqrt[x] + x*ArcCsch[Sqrt[x]]

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Maple [A]  time = 0.113, size = 24, normalized size = 0.8 \begin{align*} x{\rm arccsch} \left (\sqrt{x}\right )+{(1+x){\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(arccsch(x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.00353, size = 24, normalized size = 0.77 \begin{align*} x \operatorname{arcsch}\left (\sqrt{x}\right ) + \sqrt{x} \sqrt{\frac{1}{x} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsch(x^(1/2)),x, algorithm="maxima")

[Out]

x*arccsch(sqrt(x)) + sqrt(x)*sqrt(1/x + 1)

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Fricas [A]  time = 2.57358, size = 89, normalized size = 2.87 \begin{align*} x \log \left (\frac{x \sqrt{\frac{x + 1}{x}} + \sqrt{x}}{x}\right ) + \sqrt{x} \sqrt{\frac{x + 1}{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsch(x^(1/2)),x, algorithm="fricas")

[Out]

x*log((x*sqrt((x + 1)/x) + sqrt(x))/x) + sqrt(x)*sqrt((x + 1)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acsch}{\left (\sqrt{x} \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsch(x**(1/2)),x)

[Out]

Integral(acsch(sqrt(x)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsch(x^(1/2)),x, algorithm="giac")

[Out]

integrate(arccsch(sqrt(x)), x)