Optimal. Leaf size=17 \[ -\tanh ^{-1}\left (\frac {\tan (x)}{\sqrt {\tan (x) \tan (2 x)}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4397, 3774, 207} \[ -\tanh ^{-1}\left (\frac {\tan (2 x)}{\sqrt {\sec (2 x)-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 3774
Rule 4397
Rubi steps
\begin {align*} \int \sqrt {\tan (x) \tan (2 x)} \, dx &=\int \sqrt {-1+\sec (2 x)} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,-\frac {\tan (2 x)}{\sqrt {-1+\sec (2 x)}}\right )\\ &=-\tanh ^{-1}\left (\frac {\tan (2 x)}{\sqrt {-1+\sec (2 x)}}\right )\\ \end {align*}
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Mathematica [B] time = 0.07, size = 45, normalized size = 2.65 \[ -\frac {\sqrt {\cos (2 x)} \sqrt {\tan (x) \tan (2 x)} \csc (x) \tanh ^{-1}\left (\frac {\sqrt {2} \cos (x)}{\sqrt {\cos (2 x)}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan (x) \tan (2 x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.41, size = 50, normalized size = 2.94 \[ \frac {1}{2} \, \log \left (-\frac {\tan \relax (x)^{3} - 2 \, \sqrt {2} {\left (\tan \relax (x)^{2} - 1\right )} \sqrt {-\frac {\tan \relax (x)^{2}}{\tan \relax (x)^{2} - 1}} - 3 \, \tan \relax (x)}{\tan \relax (x)^{3} + \tan \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.77, size = 85, normalized size = 5.00 \[ \frac {1}{4} \, \sqrt {2} {\left ({\left (\sqrt {2} \log \left (\sqrt {2} + \sqrt {-\tan \relax (x)^{2} + 1}\right ) - \sqrt {2} \log \left (\sqrt {2} - \sqrt {-\tan \relax (x)^{2} + 1}\right )\right )} \mathrm {sgn}\left (\tan \relax (x)^{2} - 1\right ) \mathrm {sgn}\left (\tan \relax (x)\right ) + {\left (\sqrt {2} \log \left (\sqrt {2} + 1\right ) - \sqrt {2} \log \left (\sqrt {2} - 1\right )\right )} \mathrm {sgn}\left (\tan \relax (x)\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 88, normalized size = 5.18
method | result | size |
default | \(-\frac {\sqrt {4}\, \sqrt {\frac {1-\left (\cos ^{2}\relax (x )\right )}{2 \left (\cos ^{2}\relax (x )\right )-1}}\, \sin \relax (x ) \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}\, \arctanh \left (\frac {\sqrt {2}\, \cos \relax (x ) \sqrt {4}\, \left (-1+\cos \relax (x )\right )}{2 \sqrt {\frac {2 \left (\cos ^{2}\relax (x )\right )-1}{\left (1+\cos \relax (x )\right )^{2}}}\, \sin \relax (x )^{2}}\right )}{2 \left (-1+\cos \relax (x )\right )}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 259, normalized size = 15.24 \[ \frac {1}{4} \, \log \left (4 \, \sqrt {\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + 4 \, \sqrt {\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + 8 \, {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + 4\right ) - \frac {1}{4} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + \sqrt {\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1} {\left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2}\right )} + 2 \, {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ) \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \sqrt {\mathrm {tan}\left (2\,x\right )\,\mathrm {tan}\relax (x)} \,d x \]
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 \[ \int \sqrt {\tan {\relax (x )} \tan {\left (2 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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