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]]]

________________________________________________________________________________________

Rubi [A]  time = 0.0190383, 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.

[In]

Int[Sqrt[Tan[x]*Tan[2*x]],x]

[Out]

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

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 3774

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(a + x^2), x], x, (b*C
ot[c + d*x])/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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*}

Mathematica [B]  time = 0.0692801, 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]*Tan[2*x]])/Sqrt[2])

________________________________________________________________________________________

Maple [B]  time = 0.144, size = 88, normalized size = 5.2 \begin{align*} -{\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 ( \cos \left ( x \right ) +1 \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 ( \cos \left ( x \right ) +1 \right ) ^{2}}}}}}} \right ) } \end{align*}

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)/(cos(x)+1)^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)/(cos(x)+1)^2)^(1/2))/(cos(x)-1)

________________________________________________________________________________________

Maxima [B]  time = 1.48664, size = 350, normalized size = 20.59 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(x)*tan(2*x))^(1/2),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(c
os(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)*sin(1/2*arctan2(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 + s
qrt(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(si
n(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))))

________________________________________________________________________________________

Fricas [B]  time = 2.87459, size = 150, normalized size = 8.82 \begin{align*} \frac{1}{2} \, \log \left (-\frac{\tan \left (x\right )^{3} - 2 \, \sqrt{2}{\left (\tan \left (x\right )^{2} - 1\right )} \sqrt{-\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} - 1}} - 3 \, \tan \left (x\right )}{\tan \left (x\right )^{3} + \tan \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(x)*tan(2*x))^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\tan{\left (x \right )} \tan{\left (2 x \right )}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [B]  time = 1.21842, size = 115, normalized size = 6.76 \begin{align*} \frac{1}{4} \, \sqrt{2}{\left ({\left (\sqrt{2} \log \left (\sqrt{2} + \sqrt{-\tan \left (x\right )^{2} + 1}\right ) - \sqrt{2} \log \left (\sqrt{2} - \sqrt{-\tan \left (x\right )^{2} + 1}\right )\right )} \mathrm{sgn}\left (\tan \left (x\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (x\right )\right ) +{\left (\sqrt{2} \log \left (\sqrt{2} + 1\right ) - \sqrt{2} \log \left (\sqrt{2} - 1\right )\right )} \mathrm{sgn}\left (\tan \left (x\right )\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(x)*tan(2*x))^(1/2),x, algorithm="giac")

[Out]

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