3.23 \(\int \text{sech}^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=43 \[ x \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1-x}{\sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0072149, antiderivative size = 43, 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 = {6343, 12, 32} \[ x \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1-x}{\sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSech[Sqrt[x]],x]

[Out]

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

Rule 6343

Int[ArcSech[u_], x_Symbol] :> Simp[x*ArcSech[u], x] + Dist[Sqrt[1 - u^2]/(u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u]), Int
[SimplifyIntegrand[(x*D[u, x])/(u*Sqrt[1 - u^2]), x], x], x] /; InverseFunctionFreeQ[u, x] &&  !FunctionOfExpo
nentialQ[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{sech}^{-1}\left (\sqrt{x}\right ) \, dx &=x \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{1}{2 \sqrt{1-x}} \, dx}{\sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=x \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x}} \, dx}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{1-x}{\sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+x \text{sech}^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0958415, size = 67, normalized size = 1.56 \[ x \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \sqrt{\sqrt{x}+1} \sqrt{1-x}}{\sqrt{1-\sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSech[Sqrt[x]],x]

[Out]

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

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Maple [A]  time = 0.123, size = 36, normalized size = 0.8 \begin{align*} x{\rm arcsech} \left (\sqrt{x}\right )-\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.971505, size = 26, normalized size = 0.6 \begin{align*} x \operatorname{arsech}\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(arcsech(x^(1/2)),x, algorithm="maxima")

[Out]

x*arcsech(sqrt(x)) - sqrt(x)*sqrt(1/x - 1)

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Fricas [A]  time = 1.80622, size = 92, normalized size = 2.14 \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(arcsech(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{asech}{\left (\sqrt{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asech(sqrt(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsech(sqrt(x)), x)