Integrand size = 10, antiderivative size = 136 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}+\frac {3 (1-x)}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{3/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3 \sqrt {1-x} \text {arctanh}\left (\sqrt {1-x}\right )}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \]
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Time = 0.02 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6480, 12, 44, 65, 212} \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {3 \sqrt {1-x} \text {arctanh}\left (\sqrt {1-x}\right )}{16 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} \sqrt {x}}+\frac {3 (1-x)}{16 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} x^{3/2}}+\frac {1-x}{8 \sqrt {\frac {1}{\sqrt {x}}-1} \sqrt {\frac {1}{\sqrt {x}}+1} x^{5/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2} \]
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Rule 12
Rule 44
Rule 65
Rule 212
Rule 6480
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\sqrt {1-x} \int \frac {1}{2 \sqrt {1-x} x^3} \, dx}{2 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = -\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\sqrt {1-x} \int \frac {1}{\sqrt {1-x} x^3} \, dx}{4 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\left (3 \sqrt {1-x}\right ) \int \frac {1}{\sqrt {1-x} x^2} \, dx}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}+\frac {3 (1-x)}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{3/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {\left (3 \sqrt {1-x}\right ) \int \frac {1}{\sqrt {1-x} x} \, dx}{32 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}+\frac {3 (1-x)}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{3/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {\left (3 \sqrt {1-x}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x}\right )}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ & = \frac {1-x}{8 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{5/2}}+\frac {3 (1-x)}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} x^{3/2}}-\frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {3 \sqrt {1-x} \text {arctanh}\left (\sqrt {1-x}\right )}{16 \sqrt {-1+\frac {1}{\sqrt {x}}} \sqrt {1+\frac {1}{\sqrt {x}}} \sqrt {x}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.92 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {1}{16} \left (\frac {\sqrt {\frac {1-\sqrt {x}}{1+\sqrt {x}}} \left (2+2 \sqrt {x}+3 x+3 x^{3/2}\right )}{x^2}-\frac {8 \text {sech}^{-1}\left (\sqrt {x}\right )}{x^2}+3 \log \left (1+\sqrt {\frac {1-\sqrt {x}}{1+\sqrt {x}}}+\sqrt {\frac {1-\sqrt {x}}{1+\sqrt {x}}} \sqrt {x}\right )-\frac {3 \log (x)}{2}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58
method | result | size |
derivativedivides | \(-\frac {\operatorname {arcsech}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-x}}\right ) x^{2}+3 \sqrt {1-x}\, x +2 \sqrt {1-x}\right )}{16 x^{\frac {3}{2}} \sqrt {1-x}}\) | \(79\) |
default | \(-\frac {\operatorname {arcsech}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-x}}\right ) x^{2}+3 \sqrt {1-x}\, x +2 \sqrt {1-x}\right )}{16 x^{\frac {3}{2}} \sqrt {1-x}}\) | \(79\) |
parts | \(-\frac {\operatorname {arcsech}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {-\frac {\sqrt {x}-1}{\sqrt {x}}}\, \sqrt {\frac {\sqrt {x}+1}{\sqrt {x}}}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-x}}\right ) x^{2}+3 \sqrt {1-x}\, x +2 \sqrt {1-x}\right )}{16 x^{\frac {3}{2}} \sqrt {1-x}}\) | \(79\) |
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.40 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\frac {{\left (3 \, x + 2\right )} \sqrt {x} \sqrt {-\frac {x - 1}{x}} + {\left (3 \, x^{2} - 8\right )} \log \left (\frac {x \sqrt {-\frac {x - 1}{x}} + \sqrt {x}}{x}\right )}{16 \, x^{2}} \]
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\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\operatorname {asech}{\left (\sqrt {x} \right )}}{x^{3}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=-\frac {3 \, x^{\frac {3}{2}} {\left (\frac {1}{x} - 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x} \sqrt {\frac {1}{x} - 1}}{16 \, {\left (x^{2} {\left (\frac {1}{x} - 1\right )}^{2} - 2 \, x {\left (\frac {1}{x} - 1\right )} + 1\right )}} - \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{2 \, x^{2}} + \frac {3}{32} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} - 1} + 1\right ) - \frac {3}{32} \, \log \left (\sqrt {x} \sqrt {\frac {1}{x} - 1} - 1\right ) \]
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\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\int { \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right )}{x^3} \,d x \]
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