Optimal. Leaf size=36 \[ -\frac {1}{2} \sqrt {-1+x}-\frac {1}{6} (-1+x)^{3/2}+\frac {1}{2} x^2 \sec ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5378, 12, 45}
\begin {gather*} \frac {1}{2} x^2 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} (x-1)^{3/2}-\frac {\sqrt {x-1}}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 5378
Rubi steps
\begin {align*} \int x \sec ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{2} x^2 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {x}{2 \sqrt {-1+x}} \, dx\\ &=\frac {1}{2} x^2 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x}{\sqrt {-1+x}} \, dx\\ &=\frac {1}{2} x^2 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \left (\frac {1}{\sqrt {-1+x}}+\sqrt {-1+x}\right ) \, dx\\ &=-\frac {1}{2} \sqrt {-1+x}-\frac {1}{6} (-1+x)^{3/2}+\frac {1}{2} x^2 \sec ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 28, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \sqrt {-1+x} (2+x)+\frac {1}{2} x^2 \sec ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 31, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {x^{2} \mathrm {arcsec}\left (\sqrt {x}\right )}{2}-\frac {\left (x -1\right ) \left (x +2\right )}{6 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(31\) |
default | \(\frac {x^{2} \mathrm {arcsec}\left (\sqrt {x}\right )}{2}-\frac {\left (x -1\right ) \left (x +2\right )}{6 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 38, normalized size = 1.06 \begin {gather*} -\frac {1}{6} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, x^{2} \operatorname {arcsec}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.52, size = 20, normalized size = 0.56 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arcsec}\left (\sqrt {x}\right ) - \frac {1}{6} \, {\left (x + 2\right )} \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 6.74, size = 58, normalized size = 1.61 \begin {gather*} \frac {x^{2} \operatorname {asec}{\left (\sqrt {x} \right )}}{2} - \frac {\begin {cases} \frac {x \sqrt {x - 1}}{3} + \frac {2 \sqrt {x - 1}}{3} & \text {for}\: \left |{x}\right | > 1 \\\frac {i x \sqrt {1 - x}}{3} + \frac {2 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (24) = 48\).
time = 0.47, size = 80, normalized size = 2.22 \begin {gather*} -\frac {1}{48} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} + \frac {1}{2} \, x^{2} \arccos \left (\frac {1}{\sqrt {x}}\right ) - \frac {3}{16} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} + \frac {9 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 1}{48 \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x\,\mathrm {acos}\left (\frac {1}{\sqrt {x}}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________