Integrand size = 14, antiderivative size = 16 \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \sqrt {x}+2 \tan \left (\sqrt {x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3832, 3554, 8} \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \tan \left (\sqrt {x}\right )-2 \sqrt {x} \]
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Rule 8
Rule 3554
Rule 3832
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \tan ^2(x) \, dx,x,\sqrt {x}\right ) \\ & = 2 \tan \left (\sqrt {x}\right )-2 \text {Subst}\left (\int 1 \, dx,x,\sqrt {x}\right ) \\ & = -2 \sqrt {x}+2 \tan \left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \arctan \left (\tan \left (\sqrt {x}\right )\right )+2 \tan \left (\sqrt {x}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(2 \tan \left (\sqrt {x}\right )-2 \arctan \left (\tan \left (\sqrt {x}\right )\right )\) | \(15\) |
default | \(2 \tan \left (\sqrt {x}\right )-2 \arctan \left (\tan \left (\sqrt {x}\right )\right )\) | \(15\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \, \sqrt {x} + 2 \, \tan \left (\sqrt {x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=- 2 \sqrt {x} + 2 \tan {\left (\sqrt {x} \right )} \]
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Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \, \sqrt {x} + 2 \, \tan \left (\sqrt {x}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \, \sqrt {x} + 2 \, \tan \left (\sqrt {x}\right ) \]
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Time = 26.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\tan ^2\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-2\,\sqrt {x}+\frac {4{}\mathrm {i}}{{\mathrm {e}}^{\sqrt {x}\,2{}\mathrm {i}}+1} \]
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