\(\int \sqrt {\cot (2 x) \tan (x)} \, dx\) [457]

Optimal result

Integrand size = 11, antiderivative size = 32 \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=-\frac {\arcsin (\tan (x))}{\sqrt {2}}+\arctan \left (\frac {\sqrt {2} \tan (x)}{\sqrt {1-\tan ^2(x)}}\right ) \]

[Out]

arctan(2^(1/2)*tan(x)/(1-tan(x)^2)^(1/2))-1/2*arcsin(tan(x))*2^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {12, 399, 222, 385, 209} \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=\arctan \left (\frac {\sqrt {2} \tan (x)}{\sqrt {1-\tan ^2(x)}}\right )-\frac {\arcsin (\tan (x))}{\sqrt {2}} \]

[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 209

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

Rule 222

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 385

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 399

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {2} \left (1+x^2\right )} \, dx,x,\tan (x)\right ) \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {1-x^2}}{1+x^2} \, dx,x,\tan (x)\right )}{\sqrt {2}} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tan (x)\right )}{\sqrt {2}}+\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx,x,\tan (x)\right ) \\ & = -\frac {\arcsin (\tan (x))}{\sqrt {2}}+\sqrt {2} \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\tan (x)}{\sqrt {1-\tan ^2(x)}}\right ) \\ & = -\frac {\arcsin (\tan (x))}{\sqrt {2}}+\arctan \left (\frac {\sqrt {2} \tan (x)}{\sqrt {1-\tan ^2(x)}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=\frac {\left (\sqrt {2} \arcsin \left (\sqrt {2} \sin (x)\right )-\arctan \left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )\right ) \cos (x) \sqrt {\cot (2 x) \tan (x)}}{\sqrt {\cos (2 x)}} \]

[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]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs. \(2(26)=52\).

Time = 6.69 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.47

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {2-\left (\sec ^{2}\left (x \right )\right )}\, \left (2 \sqrt {2}\, \arctan \left (\frac {\sin \left (x \right ) \sqrt {2}}{\left (\cos \left (x \right )+1\right ) \sqrt {\frac {2 \left (\cos ^{2}\left (x \right )\right )-1}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )-\arctan \left (\frac {1+2 \sin \left (x \right )}{\left (\cos \left (x \right )+1\right ) \sqrt {\frac {2 \left (\cos ^{2}\left (x \right )\right )-1}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )-\arctan \left (\frac {2 \sin \left (x \right )-1}{\left (\cos \left (x \right )+1\right ) \sqrt {\frac {2 \left (\cos ^{2}\left (x \right )\right )-1}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )\right ) \cos \left (x \right )}{4 \left (\cos \left (x \right )+1\right ) \sqrt {\frac {2 \left (\cos ^{2}\left (x \right )\right )-1}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(143\)

[In]

int((cot(2*x)/cot(x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.59 \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (3 \, \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}}}{4 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {2} \cos \left (2 \, x\right )^{2} + \sqrt {2} \cos \left (2 \, x\right ) - \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}}}{4 \, \cos \left (2 \, x\right ) \sin \left (2 \, x\right )}\right ) \]

[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)))

Sympy [F]

\[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=\int \sqrt {\frac {\cot {\left (2 x \right )}}{\cot {\left (x \right )}}}\, dx \]

[In]

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

[Out]

Integral(sqrt(cot(2*x)/cot(x)), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 507, normalized size of antiderivative = 15.84 \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} \arctan \left ({\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ), {\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 ) + \cos \left (2 \, x\right )\right ) - 2 \, \arctan \left (\frac {{\left ({\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{4} + 16 \, \cos \left (2 \, x\right )^{4} + 16 \, \sin \left (2 \, x\right )^{4} + 8 \, {\left (\cos \left (2 \, x\right )^{2} - \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} {\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2} - 64 \, \cos \left (2 \, x\right )^{3} + 32 \, {\left (\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (2 \, x\right )^{2} + 96 \, \cos \left (2 \, x\right )^{2} - 64 \, \cos \left (2 \, x\right ) + 16\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\frac {8 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}{{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2}}, \frac {{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2} + 4 \, \cos \left (2 \, x\right )^{2} - 4 \, \sin \left (2 \, x\right )^{2} - 8 \, \cos \left (2 \, x\right ) + 4}{{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2}}\right )\right ) + 2 \, \sin \left (2 \, x\right )}{{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}}, \frac {{\left ({\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{4} + 16 \, \cos \left (2 \, x\right )^{4} + 16 \, \sin \left (2 \, x\right )^{4} + 8 \, {\left (\cos \left (2 \, x\right )^{2} - \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} {\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2} - 64 \, \cos \left (2 \, x\right )^{3} + 32 \, {\left (\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (2 \, x\right )^{2} + 96 \, \cos \left (2 \, x\right )^{2} - 64 \, \cos \left (2 \, x\right ) + 16\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\frac {8 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}{{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2}}, \frac {{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2} + 4 \, \cos \left (2 \, x\right )^{2} - 4 \, \sin \left (2 \, x\right )^{2} - 8 \, \cos \left (2 \, x\right ) + 4}{{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}^{2}}\right )\right ) + 2 \, \cos \left (2 \, x\right ) - 2}{{\left | 2 \, e^{\left (2 i \, x\right )} + 2 \right |}}\right )\right )} \]

[In]

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

[Out]

1/4*sqrt(2)*(sqrt(2)*arctan2((cos(4*x)^2 + sin(4*x)^2 + 2*cos(4*x) + 1)^(1/4)*sin(1/2*arctan2(sin(4*x), cos(4*
x) + 1)) + sin(2*x), (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))
+ cos(2*x)) - 2*arctan2(((abs(2*e^(2*I*x) + 2)^4 + 16*cos(2*x)^4 + 16*sin(2*x)^4 + 8*(cos(2*x)^2 - sin(2*x)^2
- 2*cos(2*x) + 1)*abs(2*e^(2*I*x) + 2)^2 - 64*cos(2*x)^3 + 32*(cos(2*x)^2 - 2*cos(2*x) + 1)*sin(2*x)^2 + 96*c
os(2*x)^2 - 64*cos(2*x) + 16)^(1/4)*sin(1/2*arctan2(8*(cos(2*x) - 1)*sin(2*x)/abs(2*e^(2*I*x) + 2)^2, (abs(2*e
^(2*I*x) + 2)^2 + 4*cos(2*x)^2 - 4*sin(2*x)^2 - 8*cos(2*x) + 4)/abs(2*e^(2*I*x) + 2)^2)) + 2*sin(2*x))/abs(2*e
^(2*I*x) + 2), ((abs(2*e^(2*I*x) + 2)^4 + 16*cos(2*x)^4 + 16*sin(2*x)^4 + 8*(cos(2*x)^2 - sin(2*x)^2 - 2*cos(2
*x) + 1)*abs(2*e^(2*I*x) + 2)^2 - 64*cos(2*x)^3 + 32*(cos(2*x)^2 - 2*cos(2*x) + 1)*sin(2*x)^2 + 96*cos(2*x)^2
- 64*cos(2*x) + 16)^(1/4)*cos(1/2*arctan2(8*(cos(2*x) - 1)*sin(2*x)/abs(2*e^(2*I*x) + 2)^2, (abs(2*e^(2*I*x) +
2)^2 + 4*cos(2*x)^2 - 4*sin(2*x)^2 - 8*cos(2*x) + 4)/abs(2*e^(2*I*x) + 2)^2)) + 2*cos(2*x) - 2)/abs(2*e^(2*I*
x) + 2)))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.31 \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=\frac {1}{2} \, {\left (\pi - \sqrt {2} \arctan \left (-i\right ) - \sqrt {2} \arctan \left (\sqrt {2}\right ) - i \, \log \left (2 \, \sqrt {2} + 3\right )\right )} \mathrm {sgn}\left (\sin \left (2 \, x\right )\right ) - \frac {\sqrt {2} {\left (-i \, \sqrt {2} \log \left (2 i \, \sqrt {2} + 3 i\right ) - 2 \, \arctan \left (-i\right )\right )} \mathrm {sgn}\left (\cos \left (x\right )\right ) + 2 \, {\left (\sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (\frac {3 \, {\left (2 \, \sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1} - 1\right )}}{4 \, \cos \left (x\right )^{2} - 3} - 1\right )}\right ) + \arcsin \left (4 \, \cos \left (x\right )^{2} - 3\right )\right )} \mathrm {sgn}\left (\cos \left (x\right )\right )}{4 \, \mathrm {sgn}\left (\cos \left (x\right )\right ) \mathrm {sgn}\left (\sin \left (2 \, x\right )\right )} \]

[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)*(-
I*sqrt(2)*log(2*I*sqrt(2) + 3*I) - 2*arctan(-I))*sgn(cos(x)) + 2*(sqrt(2)*arctan(1/4*sqrt(2)*(3*(2*sqrt(2)*sqr
t(-2*cos(x)^4 + 3*cos(x)^2 - 1) - 1)/(4*cos(x)^2 - 3) - 1)) + arcsin(4*cos(x)^2 - 3))*sgn(cos(x)))/(sgn(cos(x)
)*sgn(sin(2*x)))

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\cot (2 x) \tan (x)} \, dx=\int \sqrt {\frac {\mathrm {cot}\left (2\,x\right )}{\mathrm {cot}\left (x\right )}} \,d x \]

[In]

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

[Out]

int((cot(2*x)/cot(x))^(1/2), x)