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

Optimal. Leaf size=32 \[ \tan ^{-1}\left (\frac{\sqrt{2} \tan (x)}{\sqrt{1-\tan ^2(x)}}\right )-\frac{\sin ^{-1}(\tan (x))}{\sqrt{2}} \]

[Out]

-(ArcSin[Tan[x]]/Sqrt[2]) + ArcTan[(Sqrt[2]*Tan[x])/Sqrt[1 - Tan[x]^2]]

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Rubi [A]  time = 0.0385622, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {12, 402, 216, 377, 203} \[ \tan ^{-1}\left (\frac{\sqrt{2} \tan (x)}{\sqrt{1-\tan ^2(x)}}\right )-\frac{\sin ^{-1}(\tan (x))}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcSin[Tan[x]]/Sqrt[2]) + ArcTan[(Sqrt[2]*Tan[x])/Sqrt[1 - Tan[x]^2]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 216

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

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

Rubi steps

\begin{align*} \int \sqrt{\cot (2 x) \tan (x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{\sqrt{2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{1+x^2} \, dx,x,\tan (x)\right )}{\sqrt{2}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\tan (x)\right )}{\sqrt{2}}+\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=-\frac{\sin ^{-1}(\tan (x))}{\sqrt{2}}+\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{\tan (x)}{\sqrt{1-\tan ^2(x)}}\right )\\ &=-\frac{\sin ^{-1}(\tan (x))}{\sqrt{2}}+\tan ^{-1}\left (\frac{\sqrt{2} \tan (x)}{\sqrt{1-\tan ^2(x)}}\right )\\ \end{align*}

Mathematica [A]  time = 0.067457, size = 52, normalized size = 1.62 \[ \frac{\cos (x) \sqrt{\tan (x) \cot (2 x)} \left (\sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin (x)\right )-\tan ^{-1}\left (\frac{\sin (x)}{\sqrt{\cos (2 x)}}\right )\right )}{\sqrt{\cos (2 x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cot[2*x]*Tan[x]],x]

[Out]

((Sqrt[2]*ArcSin[Sqrt[2]*Sin[x]] - ArcTan[Sin[x]/Sqrt[Cos[2*x]]])*Cos[x]*Sqrt[Cot[2*x]*Tan[x]])/Sqrt[Cos[2*x]]

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Maple [C]  time = 0.327, size = 242, normalized size = 7.6 \begin{align*}{\frac{\sqrt{2} \left ( 2+\sqrt{2} \right ) \cos \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{2}}{2\,\sqrt{3+2\,\sqrt{2}} \left ( 1+\sqrt{2} \right ) \left ( \cos \left ( x \right ) -1 \right ) \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) } \left ( 4\,{\it EllipticPi} \left ({\frac{\sqrt{3+2\,\sqrt{2}} \left ( \cos \left ( x \right ) -1 \right ) }{\sin \left ( x \right ) }},- \left ( 3+2\,\sqrt{2} \right ) ^{-1},{\frac{\sqrt{3-2\,\sqrt{2}}}{\sqrt{3+2\,\sqrt{2}}}} \right ) -2\,{\it EllipticPi} \left ({\frac{\sqrt{3+2\,\sqrt{2}} \left ( \cos \left ( x \right ) -1 \right ) }{\sin \left ( x \right ) }}, \left ( 3+2\,\sqrt{2} \right ) ^{-1},{\frac{\sqrt{3-2\,\sqrt{2}}}{\sqrt{3+2\,\sqrt{2}}}} \right ) -{\it EllipticF} \left ({\frac{ \left ( \cos \left ( x \right ) -1 \right ) \left ( 1+\sqrt{2} \right ) }{\sin \left ( x \right ) }},3-2\,\sqrt{2} \right ) \right ) \sqrt{-2\,{\frac{\cos \left ( x \right ) \sqrt{2}-2\,\cos \left ( x \right ) -\sqrt{2}+1}{\cos \left ( x \right ) +1}}}\sqrt{{\frac{\cos \left ( x \right ) \sqrt{2}+2\,\cos \left ( x \right ) -\sqrt{2}-1}{\cos \left ( x \right ) +1}}}\sqrt{{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(2*x)/cot(x))^(1/2),x)

[Out]

1/2*2^(1/2)/(3+2*2^(1/2))^(1/2)/(1+2^(1/2))*(4*EllipticPi((3+2*2^(1/2))^(1/2)*(cos(x)-1)/sin(x),-1/(3+2*2^(1/2
)),(3-2*2^(1/2))^(1/2)/(3+2*2^(1/2))^(1/2))-2*EllipticPi((3+2*2^(1/2))^(1/2)*(cos(x)-1)/sin(x),1/(3+2*2^(1/2))
,(3-2*2^(1/2))^(1/2)/(3+2*2^(1/2))^(1/2))-EllipticF((cos(x)-1)*(1+2^(1/2))/sin(x),3-2*2^(1/2)))*(2+2^(1/2))*co
s(x)*(-2*(cos(x)*2^(1/2)-2*cos(x)-2^(1/2)+1)/(cos(x)+1))^(1/2)*((cos(x)*2^(1/2)+2*cos(x)-2^(1/2)-1)/(cos(x)+1)
)^(1/2)*sin(x)^2*((2*cos(x)^2-1)/cos(x)^2)^(1/2)/(cos(x)-1)/(2*cos(x)^2-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.67048, size = 328, normalized size = 10.25 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (3 \, \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )} \sqrt{\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}}}{4 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )}\right ) - \frac{1}{2} \, \arctan \left (\frac{\sqrt{2}{\left (2 \, \sqrt{2} \cos \left (2 \, x\right )^{2} + \sqrt{2} \cos \left (2 \, x\right ) - \sqrt{2}\right )} \sqrt{\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}}}{4 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((cot(2*x)/cot(x))**(1/2),x)

[Out]

Integral(sqrt(cot(2*x)/cot(x)), x)

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Giac [C]  time = 1.23145, size = 188, normalized size = 5.88 \begin{align*} \frac{1}{2} \,{\left (\pi - \sqrt{2} \arctan \left (-i\right ) - \sqrt{2} \arctan \left (\sqrt{2}\right ) - i \, \log \left (2 \, \sqrt{2} + 3\right )\right )} \mathrm{sgn}\left (\sin \left (2 \, x\right )\right ) - \frac{\sqrt{2}{\left ({\left (\sqrt{2} \arcsin \left (4 \, \cos \left (x\right )^{2} - 3\right ) + 2 \, \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (\frac{3 \,{\left (2 \, \sqrt{2} \sqrt{-2 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1} - 1\right )}}{4 \, \cos \left (x\right )^{2} - 3} - 1\right )}\right )\right )} \mathrm{sgn}\left (\cos \left (x\right )\right ) +{\left (-i \, \sqrt{2} \log \left (2 i \, \sqrt{2} + 3 i\right ) - 2 \, \arctan \left (-i\right )\right )} \mathrm{sgn}\left (\cos \left (x\right )\right )\right )}}{4 \, \mathrm{sgn}\left (\cos \left (x\right )\right ) \mathrm{sgn}\left (\sin \left (2 \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(pi - sqrt(2)*arctan(-I) - sqrt(2)*arctan(sqrt(2)) - I*log(2*sqrt(2) + 3))*sgn(sin(2*x)) - 1/4*sqrt(2)*((s
qrt(2)*arcsin(4*cos(x)^2 - 3) + 2*arctan(1/4*sqrt(2)*(3*(2*sqrt(2)*sqrt(-2*cos(x)^4 + 3*cos(x)^2 - 1) - 1)/(4*
cos(x)^2 - 3) - 1)))*sgn(cos(x)) + (-I*sqrt(2)*log(2*I*sqrt(2) + 3*I) - 2*arctan(-I))*sgn(cos(x)))/(sgn(cos(x)
)*sgn(sin(2*x)))