Optimal. Leaf size=17 \[ -\tanh ^{-1}\left (\frac{\tan (x)}{\sqrt{\tan (x) \tan (2 x)}}\right ) \]
[Out]
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Rubi [A] time = 0.0531911, 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 \[ -\tanh ^{-1}\left (\frac{\tan (2 x)}{\sqrt{\sec (2 x)-1}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[Tan[x]*Tan[2*x]],x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((tan(x)*tan(2*x))**(1/2),x)
[Out]
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Mathematica [B] time = 0.0789664, 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.
[In] Integrate[Sqrt[Tan[x]*Tan[2*x]],x]
[Out]
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Maple [B] time = 0.158, size = 88, normalized size = 5.2 \[ -{\frac{\sqrt{4}\sin \left ( x \right ) }{2\,\cos \left ( x \right ) -2}\sqrt{{\frac{- \left ( \cos \left ( x \right ) \right ) ^{2}+1}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}\sqrt{{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}{\it Artanh} \left ({\frac{\cos \left ( x \right ) \sqrt{2}\sqrt{4} \left ( \cos \left ( x \right ) -1 \right ) }{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((tan(x)*tan(2*x))^(1/2),x)
[Out]
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Maxima [A] time = 1.67975, size = 350, normalized size = 20.59 \[ \frac{1}{4} \, \log \left (4 \, \sqrt{\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + 4 \, \sqrt{\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + 8 \,{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac{1}{4}} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + 4\right ) - \frac{1}{4} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + \sqrt{\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1}{\left (\cos \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2} + \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )^{2}\right )} + 2 \,{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac{1}{4}}{\left (\cos \left (2 \, x\right ) \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ) \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(tan(2*x)*tan(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241358, size = 85, normalized size = 5. \[ \frac{1}{2} \, \log \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (\tan \left (x\right )^{2} - 3\right )} \sqrt{-\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} - 1}} + 4 \, \tan \left (x\right )\right )}}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )} \sqrt{-\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} - 1}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(tan(2*x)*tan(x)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\tan{\left (x \right )} \tan{\left (2 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((tan(x)*tan(2*x))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.25037, size = 115, normalized size = 6.76 \[ \frac{1}{4} \, \sqrt{2}{\left ({\left (\sqrt{2}{\rm ln}\left (\sqrt{2} + \sqrt{-\tan \left (x\right )^{2} + 1}\right ) - \sqrt{2}{\rm ln}\left (\sqrt{2} - \sqrt{-\tan \left (x\right )^{2} + 1}\right )\right )}{\rm sign}\left (\tan \left (x\right )^{2} - 1\right ){\rm sign}\left (\tan \left (x\right )\right ) +{\left (\sqrt{2}{\rm ln}\left (\sqrt{2} + 1\right ) - \sqrt{2}{\rm ln}\left (\sqrt{2} - 1\right )\right )}{\rm sign}\left (\tan \left (x\right )\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(tan(2*x)*tan(x)),x, algorithm="giac")
[Out]