Optimal. Leaf size=98 \[ -\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}+\frac {\log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.07, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (x)}+1\right )}{\sqrt {2}}+\frac {\log \left (\tan (x)-\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\sqrt {2} \sqrt {\tan (x)}+1\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3476
Rubi steps
\begin {align*} \int \sqrt {\tan (x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )\\ &=-\operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )+\operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (x)}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (x)}\right )+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sqrt {2}}+\frac {\operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (x)}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\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*}
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Mathematica [C] time = 0.01, size = 24, normalized size = 0.24 \[ \frac {2}{3} \tan ^{\frac {3}{2}}(x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.92, size = 180, normalized size = 1.84 \[ -\sqrt {2} \arctan \left (\sqrt {2} \sqrt {\frac {\sqrt {2} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x)}} \cos \relax (x) + \cos \relax (x) + \sin \relax (x)}{\cos \relax (x)}} - \sqrt {2} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x)}} - 1\right ) - \sqrt {2} \arctan \left (\sqrt {2} \sqrt {-\frac {\sqrt {2} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x)}} \cos \relax (x) - \cos \relax (x) - \sin \relax (x)}{\cos \relax (x)}} - \sqrt {2} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x)}} + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x)}} \cos \relax (x) + \cos \relax (x) + \sin \relax (x)\right )}}{\cos \relax (x)}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x)}} \cos \relax (x) - \cos \relax (x) - \sin \relax (x)\right )}}{\cos \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 80, normalized size = 0.82 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \relax (x)}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \relax (x)}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 49, normalized size = 0.50
method | result | size |
lookup | \(\frac {\left (\sqrt {\tan }\relax (x )\right ) \cos \relax (x ) \sqrt {2}\, \arccos \left (\cos \relax (x )-\sin \relax (x )\right )}{2 \sqrt {\cos \relax (x ) \sin \relax (x )}}-\frac {\sqrt {2}\, \ln \left (\cos \relax (x )+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right ) \cos \relax (x )+\sin \relax (x )\right )}{2}\) | \(49\) |
default | \(\frac {\left (\sqrt {\tan }\relax (x )\right ) \cos \relax (x ) \sqrt {2}\, \arccos \left (\cos \relax (x )-\sin \relax (x )\right )}{2 \sqrt {\cos \relax (x ) \sin \relax (x )}}-\frac {\sqrt {2}\, \ln \left (\cos \relax (x )+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right ) \cos \relax (x )+\sin \relax (x )\right )}{2}\) | \(49\) |
derivativedivides | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )+\tan \relax (x )}{1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )+\tan \relax (x )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )\right )\right )}{4}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 80, normalized size = 0.82 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \relax (x)}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \relax (x)}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 65, normalized size = 0.66 \[ \frac {\sqrt {2}\,\left (\ln \left (\sqrt {2}\,\sqrt {\mathrm {tan}\relax (x)}-\mathrm {tan}\relax (x)-1\right )-\ln \left (\mathrm {tan}\relax (x)+\sqrt {2}\,\sqrt {\mathrm {tan}\relax (x)}+1\right )\right )}{4}+\frac {\sqrt {2}\,\left (\mathrm {atan}\left (\sqrt {2}\,\sqrt {\mathrm {tan}\relax (x)}-1\right )+\mathrm {atan}\left (\sqrt {2}\,\sqrt {\mathrm {tan}\relax (x)}+1\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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