Integrand size = 6, antiderivative size = 35 \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=-\frac {1}{2} \sqrt {x} \sqrt {1+x}+\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2}+x \text {arcsinh}\left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5874, 12, 1978, 52, 56, 221} \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=x \text {arcsinh}\left (\sqrt {x}\right )+\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2}-\frac {1}{2} \sqrt {x} \sqrt {x+1} \]
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Rule 12
Rule 52
Rule 56
Rule 221
Rule 1978
Rule 5874
Rubi steps \begin{align*} \text {integral}& = x \text {arcsinh}\left (\sqrt {x}\right )-\int \frac {1}{2} \sqrt {\frac {x}{1+x}} \, dx \\ & = x \text {arcsinh}\left (\sqrt {x}\right )-\frac {1}{2} \int \sqrt {\frac {x}{1+x}} \, dx \\ & = x \text {arcsinh}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {1+x}} \, dx \\ & = -\frac {1}{2} \sqrt {x} \sqrt {1+x}+x \text {arcsinh}\left (\sqrt {x}\right )+\frac {1}{4} \int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx \\ & = -\frac {1}{2} \sqrt {x} \sqrt {1+x}+x \text {arcsinh}\left (\sqrt {x}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {1}{2} \sqrt {x} \sqrt {1+x}+\frac {\text {arcsinh}\left (\sqrt {x}\right )}{2}+x \text {arcsinh}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \left (-\frac {x}{\sqrt {\frac {x}{1+x}}}+2 x \text {arcsinh}\left (\sqrt {x}\right )-\log \left (-\sqrt {x}+\sqrt {1+x}\right )\right ) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{2}+x \,\operatorname {arcsinh}\left (\sqrt {x}\right )-\frac {\sqrt {x}\, \sqrt {1+x}}{2}\) | \(24\) |
default | \(\frac {\operatorname {arcsinh}\left (\sqrt {x}\right )}{2}+x \,\operatorname {arcsinh}\left (\sqrt {x}\right )-\frac {\sqrt {x}\, \sqrt {1+x}}{2}\) | \(24\) |
parts | \(x \,\operatorname {arcsinh}\left (\sqrt {x}\right )-\frac {\sqrt {x}\, \sqrt {1+x}}{2}+\frac {\sqrt {x \left (1+x \right )}\, \ln \left (\frac {1}{2}+x +\sqrt {x^{2}+x}\right )}{4 \sqrt {x}\, \sqrt {1+x}}\) | \(46\) |
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, {\left (2 \, x + 1\right )} \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x + 1} \sqrt {x} \]
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Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=- \frac {\sqrt {x} \sqrt {x + 1}}{2} + x \operatorname {asinh}{\left (\sqrt {x} \right )} + \frac {\operatorname {asinh}{\left (\sqrt {x} \right )}}{2} \]
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Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=x \operatorname {arsinh}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x + 1} \sqrt {x} + \frac {1}{2} \, \operatorname {arsinh}\left (\sqrt {x}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=x \log \left (\sqrt {x + 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x^{2} + x} - \frac {1}{4} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
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Time = 3.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \text {arcsinh}\left (\sqrt {x}\right ) \, dx=\mathrm {atanh}\left (\frac {\sqrt {x}}{\sqrt {x+1}-1}\right )+x\,\mathrm {asinh}\left (\sqrt {x}\right )-\frac {\sqrt {x}\,\sqrt {x+1}}{2} \]
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