Integrand size = 12, antiderivative size = 22 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}+\log (x)-\log (1+x) \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4946, 36, 29, 31} \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}+\log (x)-\log (x+1) \]
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Rule 29
Rule 31
Rule 36
Rule 4946
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}+\int \frac {1}{x (1+x)} \, dx \\ & = -\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}+\int \frac {1}{x} \, dx-\int \frac {1}{1+x} \, dx \\ & = -\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}+\log (x)-\log (1+x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}+\log (x)-\log (1+x) \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\ln \left (x \right )-\ln \left (x +1\right )-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}\) | \(19\) |
default | \(\ln \left (x \right )-\ln \left (x +1\right )-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}\) | \(19\) |
meijerg | \(\ln \left (x \right )-\ln \left (x +1\right )-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}\) | \(19\) |
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none
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {x \log \left (x + 1\right ) - x \log \left (x\right ) + 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right )}{x} \]
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Time = 0.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=\log {\left (x \right )} - \log {\left (x + 1 \right )} - \frac {2 \operatorname {atan}{\left (\sqrt {x} \right )}}{\sqrt {x}} \]
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none
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \, \arctan \left (\sqrt {x}\right )}{\sqrt {x}} - \log \left (x + 1\right ) + \log \left (x\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 \, \arctan \left (\sqrt {x}\right )}{\sqrt {x}} - \log \left (x + 1\right ) + \log \left (x\right ) \]
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Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{x^{3/2}} \, dx=2\,\ln \left (\sqrt {x}\right )-\ln \left (x+1\right )-\frac {2\,\mathrm {atan}\left (\sqrt {x}\right )}{\sqrt {x}} \]
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