Integrand size = 8, antiderivative size = 88 \[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1-x}{2 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {(1-x)^2}{6 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6480, 12, 45} \[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {(1-x)^2}{6 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}-\frac {1-x}{2 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}} \]
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Rule 12
Rule 45
Rule 6480
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \frac {x}{2 \sqrt {1-x}} \, dx}{2 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \frac {x}{\sqrt {1-x}} \, dx}{4 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {1-x} \int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx}{4 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = -\frac {1-x}{2 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {(1-x)^2}{6 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}}+\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=-\frac {1}{6} \sqrt {\frac {1-\sqrt {x}}{1+\sqrt {x}}} \left (2+2 \sqrt {x}+x+x^{3/2}\right )+\frac {1}{2} x^2 \text {sech}^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.48
method | result | size |
derivativedivides | \(\frac {x^{2} \operatorname {arcsech}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (x +2\right )}{6}\) | \(42\) |
default | \(\frac {x^{2} \operatorname {arcsech}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (x +2\right )}{6}\) | \(42\) |
parts | \(\frac {x^{2} \operatorname {arcsech}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (x +2\right )}{6}\) | \(42\) |
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Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right ) - \frac {1}{6} \, {\left (x + 2\right )} \sqrt {x} \sqrt {-\frac {x - 1}{x}} \]
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\[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int x \operatorname {asech}{\left (\sqrt {x} \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.39 \[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{6} \, x^{\frac {3}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, x^{2} \operatorname {arsech}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x} \sqrt {\frac {1}{x} - 1} \]
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\[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int { x \operatorname {arsech}\left (\sqrt {x}\right ) \,d x } \]
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Timed out. \[ \int x \text {sech}^{-1}\left (\sqrt {x}\right ) \, dx=\int x\,\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
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