Optimal. Leaf size=47 \[ \frac {1}{3} x^3 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{15} (x-1)^{5/2}-\frac {2}{9} (x-1)^{3/2}-\frac {\sqrt {x-1}}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5270, 12, 43} \[ \frac {1}{3} x^3 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{15} (x-1)^{5/2}-\frac {2}{9} (x-1)^{3/2}-\frac {\sqrt {x-1}}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 5270
Rubi steps
\begin {align*} \int x^2 \sec ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x^2}{2 \sqrt {-1+x}} \, dx\\ &=\frac {1}{3} x^3 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^2}{\sqrt {-1+x}} \, dx\\ &=\frac {1}{3} x^3 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \left (\frac {1}{\sqrt {-1+x}}+2 \sqrt {-1+x}+(-1+x)^{3/2}\right ) \, dx\\ &=-\frac {1}{3} \sqrt {-1+x}-\frac {2}{9} (-1+x)^{3/2}-\frac {1}{15} (-1+x)^{5/2}+\frac {1}{3} x^3 \sec ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 35, normalized size = 0.74 \[ \frac {1}{3} x^3 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{45} \sqrt {x-1} \left (3 x^2+4 x+8\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 27, normalized size = 0.57 \[ \frac {1}{3} \, x^{3} \operatorname {arcsec}\left (\sqrt {x}\right ) - \frac {1}{45} \, {\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt {x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.14, size = 116, normalized size = 2.47 \[ -\frac {1}{480} \, x^{\frac {5}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{5} - \frac {5}{288} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} + \frac {1}{3} \, x^{3} \arccos \left (\frac {1}{\sqrt {x}}\right ) - \frac {5}{48} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} + \frac {150 \, x^{2} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{4} + 25 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 3}{1440 \, x^{\frac {5}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 38, normalized size = 0.81 \[ \frac {x^{3} \mathrm {arcsec}\left (\sqrt {x}\right )}{3}-\frac {\left (x -1\right ) \left (3 x^{2}+4 x +8\right )}{45 \sqrt {\frac {x -1}{x}}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 52, normalized size = 1.11 \[ -\frac {1}{15} \, x^{\frac {5}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{3} \, x^{3} \operatorname {arcsec}\left (\sqrt {x}\right ) - \frac {2}{9} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} - \frac {1}{3} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,\mathrm {acos}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 25.69, size = 90, normalized size = 1.91 \[ \frac {x^{3} \operatorname {asec}{\left (\sqrt {x} \right )}}{3} - \frac {\begin {cases} \frac {2 x^{2} \sqrt {x - 1}}{5} + \frac {8 x \sqrt {x - 1}}{15} + \frac {16 \sqrt {x - 1}}{15} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x^{2} \sqrt {1 - x}}{5} + \frac {8 i x \sqrt {1 - x}}{15} + \frac {16 i \sqrt {1 - x}}{15} & \text {otherwise} \end {cases}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________