Integrand size = 10, antiderivative size = 58 \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{4} \sqrt {-1+x}-\frac {1}{4} (-1+x)^{3/2}-\frac {3}{20} (-1+x)^{5/2}-\frac {1}{28} (-1+x)^{7/2}+\frac {1}{4} x^4 \sec ^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5378, 12, 45} \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{4} x^4 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{28} (x-1)^{7/2}-\frac {3}{20} (x-1)^{5/2}-\frac {1}{4} (x-1)^{3/2}-\frac {\sqrt {x-1}}{4} \]
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Rule 12
Rule 45
Rule 5378
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x^3}{2 \sqrt {-1+x}} \, dx \\ & = \frac {1}{4} x^4 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{8} \int \frac {x^3}{\sqrt {-1+x}} \, dx \\ & = \frac {1}{4} x^4 \sec ^{-1}\left (\sqrt {x}\right )-\frac {1}{8} \int \left (\frac {1}{\sqrt {-1+x}}+3 \sqrt {-1+x}+3 (-1+x)^{3/2}+(-1+x)^{5/2}\right ) \, dx \\ & = -\frac {1}{4} \sqrt {-1+x}-\frac {1}{4} (-1+x)^{3/2}-\frac {3}{20} (-1+x)^{5/2}-\frac {1}{28} (-1+x)^{7/2}+\frac {1}{4} x^4 \sec ^{-1}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{140} \sqrt {-1+x} \left (16+8 x+6 x^2+5 x^3\right )+\frac {1}{4} x^4 \sec ^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69
method | result | size |
parts | \(\frac {x^{4} \operatorname {arcsec}\left (\sqrt {x}\right )}{4}-\frac {\sqrt {\frac {x -1}{x}}\, \sqrt {x}\, \left (5 x^{3}+6 x^{2}+8 x +16\right )}{140}\) | \(40\) |
derivativedivides | \(\frac {x^{4} \operatorname {arcsec}\left (\sqrt {x}\right )}{4}-\frac {\left (x -1\right ) \left (5 x^{3}+6 x^{2}+8 x +16\right )}{140 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(43\) |
default | \(\frac {x^{4} \operatorname {arcsec}\left (\sqrt {x}\right )}{4}-\frac {\left (x -1\right ) \left (5 x^{3}+6 x^{2}+8 x +16\right )}{140 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(43\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.55 \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{4} \, x^{4} \operatorname {arcsec}\left (\sqrt {x}\right ) - \frac {1}{140} \, {\left (5 \, x^{3} + 6 \, x^{2} + 8 \, x + 16\right )} \sqrt {x - 1} \]
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Result contains complex when optimal does not.
Time = 61.84 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.05 \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x^{4} \operatorname {asec}{\left (\sqrt {x} \right )}}{4} - \frac {\begin {cases} \frac {2 x^{3} \sqrt {x - 1}}{7} + \frac {12 x^{2} \sqrt {x - 1}}{35} + \frac {16 x \sqrt {x - 1}}{35} + \frac {32 \sqrt {x - 1}}{35} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x^{3} \sqrt {1 - x}}{7} + \frac {12 i x^{2} \sqrt {1 - x}}{35} + \frac {16 i x \sqrt {1 - x}}{35} + \frac {32 i \sqrt {1 - x}}{35} & \text {otherwise} \end {cases}}{8} \]
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Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.14 \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{28} \, x^{\frac {7}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {7}{2}} - \frac {3}{20} \, x^{\frac {5}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, x^{4} \operatorname {arcsec}\left (\sqrt {x}\right ) - \frac {1}{4} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (38) = 76\).
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 2.62 \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{3584} \, x^{\frac {7}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{7} - \frac {7}{2560} \, x^{\frac {5}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{5} + \frac {1}{4} \, x^{4} \arccos \left (\frac {1}{\sqrt {x}}\right ) - \frac {7}{512} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} - \frac {35}{512} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} + \frac {1225 \, x^{3} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{6} + 245 \, x^{2} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{4} + 49 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 5}{17920 \, x^{\frac {7}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{7}} \]
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Timed out. \[ \int x^3 \sec ^{-1}\left (\sqrt {x}\right ) \, dx=\int x^3\,\mathrm {acos}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
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