\(\int \text {sech}^{-1}(\sqrt {x}) \, dx\) [23]

Optimal result

Integrand size = 6, antiderivative size = 43 \[ \int \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1-x}{\sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+x \text {sech}^{-1}\left (\sqrt {x}\right ) \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6478, 12, 32} \[ \int \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=x \text {sech}^{-1}\left (\sqrt {x}\right )-\frac {1-x}{\sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}} \]

[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 6478

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*} \text {integral}& = 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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(118\) vs. \(2(43)=86\).

Time = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.74 \[ \int \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {2 \left (-1+\sqrt {1-\sqrt {x}}\right )^2 \left (-1+\sqrt {1+\sqrt {x}}\right )^2 \sqrt {\frac {1-\sqrt {x}}{1+\sqrt {x}}} \sqrt {1+\sqrt {x}}}{\left (-2+\sqrt {1-\sqrt {x}}+\sqrt {1+\sqrt {x}}\right )^2 \sqrt {1-\sqrt {x}}}+x \text {sech}^{-1}\left (\sqrt {x}\right ) \]

[In]

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

[Out]

(-2*(-1 + Sqrt[1 - Sqrt[x]])^2*(-1 + Sqrt[1 + Sqrt[x]])^2*Sqrt[(1 - Sqrt[x])/(1 + Sqrt[x])]*Sqrt[1 + Sqrt[x]])
/((-2 + Sqrt[1 - Sqrt[x]] + Sqrt[1 + Sqrt[x]])^2*Sqrt[1 - Sqrt[x]]) + x*ArcSech[Sqrt[x]]

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(x \,\operatorname {arcsech}\left (\sqrt {x}\right )-\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\)	\(36\)
default	\(x \,\operatorname {arcsech}\left (\sqrt {x}\right )-\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\)	\(36\)
parts	\(x \,\operatorname {arcsech}\left (\sqrt {x}\right )-\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\)	\(36\)

[In]

int(arcsech(x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=x \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right ) - \sqrt {x} \sqrt {-\frac {x - 1}{x}} \]

[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)

Sympy [F]

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

[In]

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

[Out]

Integral(asech(sqrt(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.44 \[ \int \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=x \operatorname {arsech}\left (\sqrt {x}\right ) - \sqrt {x} \sqrt {\frac {1}{x} - 1} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(arcsech(sqrt(x)), x)

Mupad [F(-1)]

Timed out. \[ \int \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int \mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]

[In]

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

[Out]

int(acosh(1/x^(1/2)), x)