Integrand size = 6, antiderivative size = 98 \[ \int \sqrt {\tan (x)} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )}{2 \sqrt {2}} \]
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Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \sqrt {\tan (x)} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}+\frac {\log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right ) \\ & = -\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )+\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {2}} \\ & = \frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}} \\ & = -\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (x)}+\tan (x)\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.41 \[ \int \sqrt {\tan (x)} \, dx=\frac {\left (\arctan \left (\sqrt [4]{-\tan ^2(x)}\right )-\text {arctanh}\left (\sqrt [4]{-\tan ^2(x)}\right )\right ) \sqrt [4]{-\tan (x)}}{\sqrt [4]{\tan (x)}} \]
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Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.50
method | result | size |
lookup | \(\frac {\left (\sqrt {\tan }\left (x \right )\right ) \cos \left (x \right ) \sqrt {2}\, \arccos \left (\cos \left (x \right )-\sin \left (x \right )\right )}{2 \sqrt {\cos \left (x \right ) \sin \left (x \right )}}-\frac {\sqrt {2}\, \ln \left (\cos \left (x \right )+\sqrt {2}\, \left (\sqrt {\tan }\left (x \right )\right ) \cos \left (x \right )+\sin \left (x \right )\right )}{2}\) | \(49\) |
default | \(\frac {\left (\sqrt {\tan }\left (x \right )\right ) \cos \left (x \right ) \sqrt {2}\, \arccos \left (\cos \left (x \right )-\sin \left (x \right )\right )}{2 \sqrt {\cos \left (x \right ) \sin \left (x \right )}}-\frac {\sqrt {2}\, \ln \left (\cos \left (x \right )+\sqrt {2}\, \left (\sqrt {\tan }\left (x \right )\right ) \cos \left (x \right )+\sin \left (x \right )\right )}{2}\) | \(49\) |
derivativedivides | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (x \right )\right )+\tan \left (x \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (x \right )\right )+\tan \left (x \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (x \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (x \right )\right )\right )\right )}{4}\) | \(62\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74 \[ \int \sqrt {\tan (x)} \, dx=\left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right ) - \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right ) + \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right ) - \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right ) \]
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\[ \int \sqrt {\tan (x)} \, dx=\int \sqrt {\tan {\left (x \right )}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \sqrt {\tan (x)} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \sqrt {\tan (x)} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) \]
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Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.66 \[ \int \sqrt {\tan (x)} \, dx=\frac {\sqrt {2}\,\left (\ln \left (\sqrt {2}\,\sqrt {\mathrm {tan}\left (x\right )}-\mathrm {tan}\left (x\right )-1\right )-\ln \left (\mathrm {tan}\left (x\right )+\sqrt {2}\,\sqrt {\mathrm {tan}\left (x\right )}+1\right )\right )}{4}+\frac {\sqrt {2}\,\left (\mathrm {atan}\left (\sqrt {2}\,\sqrt {\mathrm {tan}\left (x\right )}-1\right )+\mathrm {atan}\left (\sqrt {2}\,\sqrt {\mathrm {tan}\left (x\right )}+1\right )\right )}{2} \]
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