3.5 \(\int \sec ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=18 \[ x \sec ^{-1}\left (\sqrt{x}\right )-\sqrt{x-1} \]

[Out]

-Sqrt[-1 + x] + x*ArcSec[Sqrt[x]]

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Rubi [A]  time = 0.0043606, antiderivative size = 18, 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 = {5268, 12, 32} \[ x \sec ^{-1}\left (\sqrt{x}\right )-\sqrt{x-1} \]

Antiderivative was successfully verified.

[In]

Int[ArcSec[Sqrt[x]],x]

[Out]

-Sqrt[-1 + x] + x*ArcSec[Sqrt[x]]

Rule 5268

Int[ArcSec[u_], x_Symbol] :> Simp[x*ArcSec[u], x] - Dist[u/Sqrt[u^2], Int[SimplifyIntegrand[(x*D[u, x])/(u*Sqr
t[u^2 - 1]), 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 \sec ^{-1}\left (\sqrt{x}\right ) \, dx &=x \sec ^{-1}\left (\sqrt{x}\right )-\int \frac{1}{2 \sqrt{-1+x}} \, dx\\ &=x \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{1}{\sqrt{-1+x}} \, dx\\ &=-\sqrt{-1+x}+x \sec ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0052079, size = 18, normalized size = 1. \[ x \sec ^{-1}\left (\sqrt{x}\right )-\sqrt{x-1} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSec[Sqrt[x]],x]

[Out]

-Sqrt[-1 + x] + x*ArcSec[Sqrt[x]]

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Maple [A]  time = 0.107, size = 25, normalized size = 1.4 \begin{align*} x{\rm arcsec} \left (\sqrt{x}\right )-{(x-1){\frac{1}{\sqrt{{\frac{x-1}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arcsec(x^(1/2))-1/((x-1)/x)^(1/2)/x^(1/2)*(x-1)

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Maxima [A]  time = 0.970726, size = 28, normalized size = 1.56 \begin{align*} x \operatorname{arcsec}\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(arcsec(x^(1/2)),x, algorithm="maxima")

[Out]

x*arcsec(sqrt(x)) - sqrt(x)*sqrt(-1/x + 1)

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Fricas [A]  time = 2.17848, size = 45, normalized size = 2.5 \begin{align*} x \operatorname{arcsec}\left (\sqrt{x}\right ) - \sqrt{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arcsec(sqrt(x)) - sqrt(x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asec(sqrt(x)), x)

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Giac [A]  time = 1.11703, size = 20, normalized size = 1.11 \begin{align*} x \arccos \left (\frac{1}{\sqrt{x}}\right ) + i - \sqrt{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccos(1/sqrt(x)) + i - sqrt(x - 1)