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]

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

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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]

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*}

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Mathematica [A]  time = 0.10, 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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fricas [A]  time = 0.56, size = 39, normalized size = 0.91 \[ x \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right ) - \sqrt {x} \sqrt {-\frac {x - 1}{x}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arsech}\left (\sqrt {x}\right )\,{d x} \]

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)

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maple [A]  time = 0.06, size = 36, normalized size = 0.84 \[ x \,\mathrm {arcsech}\left (\sqrt {x}\right )-\sqrt {-\frac {-1+\sqrt {x}}{\sqrt {x}}}\, \sqrt {\frac {1+\sqrt {x}}{\sqrt {x}}}\, \sqrt {x} \]

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 = 0.39, size = 19, normalized size = 0.44 \[ x \operatorname {arsech}\left (\sqrt {x}\right ) - \sqrt {x} \sqrt {\frac {1}{x} - 1} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(1/x^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {asech}{\left (\sqrt {x} \right )}\, dx \]

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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