∫ sec −1 (√_x ) ________________

Optimal. Leaf size=38 \[ \frac {\sqrt {-1+x}}{2 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \text {ArcTan}\left (\sqrt {-1+x}\right ) \]

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5378, 12, 44, 65, 209} \begin {gather*} \frac {1}{2} \text {ArcTan}\left (\sqrt {x-1}\right )+\frac {\sqrt {x-1}}{2 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{x} \end {gather*}

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

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1

)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,

x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] && !IntegerQ[n] && LtQ[n, 0]

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;

Int[((a_.) + ArcSec[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcSec[

u])/(d*(m + 1))), x] - Dist[b*(u/(d*(m + 1)*Sqrt[u^2])), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(u*S

qrt[u^2 - 1])), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !Funct

\begin {align*} \int \frac {\sec ^{-1}\left (\sqrt {x}\right )}{x^2} \, dx &=-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{x}+\int \frac {1}{2 \sqrt {-1+x} x^2} \, dx\\ &=-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{\sqrt {-1+x} x^2} \, dx\\ &=\frac {\sqrt {-1+x}}{2 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{4} \int \frac {1}{\sqrt {-1+x} x} \, dx\\ &=\frac {\sqrt {-1+x}}{2 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {\sqrt {-1+x}}{2 x}-\frac {\sec ^{-1}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 32, normalized size = 0.84 \begin {gather*} \frac {\sqrt {-1+x}-2 \sec ^{-1}\left (\sqrt {x}\right )-x \text {ArcSin}\left (\frac {1}{\sqrt {x}}\right )}{2 x} \end {gather*}

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method	result	size

derivativedivides	\(-\frac {\mathrm {arcsec}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {x -1}\, \left (\arctan \left (\frac {1}{\sqrt {x -1}}\right ) x -\sqrt {x -1}\right )}{2 \sqrt {\frac {x -1}{x}}\, x^{\frac {3}{2}}}\)	\(46\)
default	\(-\frac {\mathrm {arcsec}\left (\sqrt {x}\right )}{x}-\frac {\sqrt {x -1}\, \left (\arctan \left (\frac {1}{\sqrt {x -1}}\right ) x -\sqrt {x -1}\right )}{2 \sqrt {\frac {x -1}{x}}\, x^{\frac {3}{2}}}\)	\(46\)

-arcsec(x^(1/2))/x-1/2*(x-1)^(1/2)*(arctan(1/(x-1)^(1/2))*x-(x-1)^(1/2))/((x-1)/x)^(1/2)/x^(3/2)

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Maxima [A]
time = 0.47, size = 51, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {x} \sqrt {-\frac {1}{x} + 1}}{2 \, {\left (x {\left (\frac {1}{x} - 1\right )} - 1\right )}} - \frac {\operatorname {arcsec}\left (\sqrt {x}\right )}{x} + \frac {1}{2} \, \arctan \left (\sqrt {x} \sqrt {-\frac {1}{x} + 1}\right ) \end {gather*}

-1/2*sqrt(x)*sqrt(-1/x + 1)/(x*(1/x - 1) - 1) - arcsec(sqrt(x))/x + 1/2*arctan(sqrt(x)*sqrt(-1/x + 1))

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Fricas [A]
time = 3.88, size = 19, normalized size = 0.50 \begin {gather*} \frac {{\left (x - 2\right )} \operatorname {arcsec}\left (\sqrt {x}\right ) + \sqrt {x - 1}}{2 \, x} \end {gather*}

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Sympy [C] Result contains complex when optimal does not.
time = 15.32, size = 75, normalized size = 1.97 \begin {gather*} \frac {\begin {cases} i \operatorname {acosh}{\left (\frac {1}{\sqrt {x}} \right )} - \frac {i}{\sqrt {x} \sqrt {-1 + \frac {1}{x}}} + \frac {i}{x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x}}} & \text {for}\: \frac {1}{\left |{x}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )} + \frac {\sqrt {1 - \frac {1}{x}}}{\sqrt {x}} & \text {otherwise} \end {cases}}{2} - \frac {\operatorname {asec}{\left (\sqrt {x} \right )}}{x} \end {gather*}

Piecewise((I*acosh(1/sqrt(x)) - I/(sqrt(x)*sqrt(-1 + 1/x)) + I/(x**(3/2)*sqrt(-1 + 1/x)), 1/Abs(x) > 1), (-asi

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Giac [A]
time = 0.44, size = 30, normalized size = 0.79 \begin {gather*} \frac {\sqrt {-\frac {1}{x} + 1}}{2 \, \sqrt {x}} - \frac {\arccos \left (\frac {1}{\sqrt {x}}\right )}{x} + \frac {1}{2} \, \arccos \left (\frac {1}{\sqrt {x}}\right ) \end {gather*}

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Mupad [B]
time = 0.68, size = 28, normalized size = 0.74 \begin {gather*} \frac {\sqrt {1-\frac {1}{x}}}{2\,\sqrt {x}}-\frac {\mathrm {acos}\left (\frac {1}{\sqrt {x}}\right )\,\left (\frac {2}{x}-1\right )}{2} \end {gather*}

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3.1.7 \(\int \frac {\sec ^{-1}(\sqrt {x})}{x^2} \, dx\) [7]