3.456 \(\int \sqrt{\tan (x) \tan (2 x)} \, dx\)

Optimal. Leaf size=17 \[ -\tanh ^{-1}\left (\frac{\tan (x)}{\sqrt{\tan (x) \tan (2 x)}}\right ) \]

[Out]

-ArcTanh[Tan[x]/Sqrt[Tan[x]*Tan[2*x]]]

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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]

-ArcTanh[Tan[2*x]/Sqrt[-1 + Sec[2*x]]]

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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]

Timed 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]

-((ArcTanh[(Sqrt[2]*Cos[x])/Sqrt[Cos[2*x]]]*Sqrt[Cos[2*x]]*Csc[x]*Sqrt[Tan[x]*Ta
n[2*x]])/Sqrt[2])

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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]

-1/2*4^(1/2)*((-cos(x)^2+1)/(2*cos(x)^2-1))^(1/2)*sin(x)*((2*cos(x)^2-1)/(1+cos(
x))^2)^(1/2)*arctanh(1/2*2^(1/2)*cos(x)*4^(1/2)*(cos(x)-1)/sin(x)^2/((2*cos(x)^2
-1)/(1+cos(x))^2)^(1/2))/(cos(x)-1)

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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]

1/4*log(4*sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*cos(1/2*arctan2(sin(4*x
), cos(4*x) + 1))^2 + 4*sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*sin(1/2*a
rctan2(sin(4*x), cos(4*x) + 1))^2 + 8*(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)
^(1/4)*cos(1/2*arctan2(sin(4*x), cos(4*x) + 1)) + 4) - 1/4*log(cos(2*x)^2 + sin(
2*x)^2 + sqrt(cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*(cos(1/2*arctan2(sin(4*x
), cos(4*x) + 1))^2 + sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))^2) + 2*(cos(4*x)^
2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*(cos(2*x)*cos(1/2*arctan2(sin(4*x), cos(4
*x) + 1)) + sin(2*x)*sin(1/2*arctan2(sin(4*x), cos(4*x) + 1))))

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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]

1/2*log(-1/2*sqrt(2)*(sqrt(2)*(tan(x)^2 - 3)*sqrt(-tan(x)^2/(tan(x)^2 - 1)) + 4*
tan(x))/((tan(x)^2 + 1)*sqrt(-tan(x)^2/(tan(x)^2 - 1))))

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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]

Integral(sqrt(tan(x)*tan(2*x)), x)

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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]

1/4*sqrt(2)*((sqrt(2)*ln(sqrt(2) + sqrt(-tan(x)^2 + 1)) - sqrt(2)*ln(sqrt(2) - s
qrt(-tan(x)^2 + 1)))*sign(tan(x)^2 - 1)*sign(tan(x)) + (sqrt(2)*ln(sqrt(2) + 1)
- sqrt(2)*ln(sqrt(2) - 1))*sign(tan(x)))