3.147 \(\int \sqrt{\sin (x) \tan (x)} \, dx\)

Optimal. Leaf size=13 \[ -2 \cot (x) \sqrt{\sin (x) \tan (x)} \]

[Out]

-2*Cot[x]*Sqrt[Sin[x]*Tan[x]]

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Rubi [A]  time = 0.0321317, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4400, 2589} \[ -2 \cot (x) \sqrt{\sin (x) \tan (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cot[x]*Sqrt[Sin[x]*Tan[x]]

Rule 4400

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rule 2589

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*(a*Sin[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*m), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]

Rubi steps

\begin{align*} \int \sqrt{\sin (x) \tan (x)} \, dx &=\frac{\sqrt{\sin (x) \tan (x)} \int \sqrt{\sin (x)} \sqrt{\tan (x)} \, dx}{\sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=-2 \cot (x) \sqrt{\sin (x) \tan (x)}\\ \end{align*}

Mathematica [A]  time = 0.0817373, size = 13, normalized size = 1. \[ -2 \cot (x) \sqrt{\sin (x) \tan (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sin[x]*Tan[x]],x]

[Out]

-2*Cot[x]*Sqrt[Sin[x]*Tan[x]]

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Maple [B]  time = 0.217, size = 177, normalized size = 13.6 \begin{align*}{\frac{\sqrt{4} \left ( -1+\cos \left ( x \right ) \right ) \cos \left ( x \right ) }{4\, \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( 4\,\cos \left ( x \right ) \sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}+4\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-\ln \left ( -{\frac{1}{ \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}- \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -2\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-1 \right ) } \right ) +\ln \left ( -2\,{\frac{1}{ \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}- \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -2\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-1 \right ) } \right ) \right ) \sqrt{-{\frac{-1+ \left ( \cos \left ( x \right ) \right ) ^{2}}{\cos \left ( x \right ) }}}{\frac{1}{\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(x)*tan(x))^(1/2),x)

[Out]

1/4*4^(1/2)*(-1+cos(x))*(4*cos(x)*(-cos(x)/(1+cos(x))^2)^(1/2)+4*(-cos(x)/(1+cos(x))^2)^(1/2)-ln(-(2*cos(x)^2*
(-cos(x)/(1+cos(x))^2)^(1/2)-cos(x)^2+2*cos(x)-2*(-cos(x)/(1+cos(x))^2)^(1/2)-1)/sin(x)^2)+ln(-2*(2*cos(x)^2*(
-cos(x)/(1+cos(x))^2)^(1/2)-cos(x)^2+2*cos(x)-2*(-cos(x)/(1+cos(x))^2)^(1/2)-1)/sin(x)^2))*cos(x)*(-(-1+cos(x)
^2)/cos(x))^(1/2)/sin(x)^3/(-cos(x)/(1+cos(x))^2)^(1/2)

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Maxima [B]  time = 1.53871, size = 77, normalized size = 5.92 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}}{\sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} \sqrt{-\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} \sqrt{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sin(x)^2/(cos(x) + 1)^2 - 1)/(sqrt(sin(x)/(cos(x) + 1) + 1)*sqrt(-sin(x)/(cos(x) + 1) + 1)*sqrt(sin(x)^2/(c
os(x) + 1)^2 + 1))

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Fricas [A]  time = 2.43167, size = 63, normalized size = 4.85 \begin{align*} -\frac{2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-(cos(x)^2 - 1)/cos(x))*cos(x)/sin(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sin(x)*tan(x))**(1/2),x)

[Out]

Integral(sqrt(sin(x)*tan(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(x)*tan(x)), x)