Integrand size = 10, antiderivative size = 126 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1-x}{3 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {2 (1-x)^2}{9 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}-\frac {(1-x)^3}{15 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 126, 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 = {6480, 12, 45} \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )-\frac {(1-x)^3}{15 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}+\frac {2 (1-x)^2}{9 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}-\frac {1-x}{3 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}} \]
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Rule 12
Rule 45
Rule 6480
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \frac {x^2}{2 \sqrt {1-x}} \, dx}{3 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \frac {x^2}{\sqrt {1-x}} \, dx}{6 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \left (\frac {1}{\sqrt {1-x}}-2 \sqrt {1-x}+(1-x)^{3/2}\right ) \, dx}{6 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = -\frac {1-x}{3 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {2 (1-x)^2}{9 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}-\frac {(1-x)^3}{15 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.57 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{45} \sqrt {\frac {1-\sqrt {x}}{1+\sqrt {x}}} \left (8+8 \sqrt {x}+4 x+4 x^{3/2}+3 x^2+3 x^{5/2}\right )+\frac {1}{3} x^3 \text {sech}^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.39
method | result | size |
derivativedivides | \(\frac {x^{3} \operatorname {arcsech}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 x^{2}+4 x +8\right )}{45}\) | \(49\) |
default | \(\frac {x^{3} \operatorname {arcsech}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 x^{2}+4 x +8\right )}{45}\) | \(49\) |
parts | \(\frac {x^{3} \operatorname {arcsech}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 x^{2}+4 x +8\right )}{45}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.41 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} \, x^{3} \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right ) - \frac {1}{45} \, {\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt {x} \sqrt {-\frac {x - 1}{x}} \]
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\[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^{2} \operatorname {asech}{\left (\sqrt {x} \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.37 \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{15} \, x^{\frac {5}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {5}{2}} + \frac {1}{3} \, x^{3} \operatorname {arsech}\left (\sqrt {x}\right ) + \frac {2}{9} \, x^{\frac {3}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {3}{2}} - \frac {1}{3} \, \sqrt {x} \sqrt {\frac {1}{x} - 1} \]
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\[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int { x^{2} \operatorname {arsech}\left (\sqrt {x}\right ) \,d x } \]
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Timed out. \[ \int x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int x^2\,\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
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