\(\int \csc (x) \sqrt {\sin (2 x)} \, dx\) [186]

Optimal result

Integrand size = 11, antiderivative size = 25 \[ \int \csc (x) \sqrt {\sin (2 x)} \, dx=-\arcsin (\cos (x)-\sin (x))+\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4393, 4390} \[ \int \csc (x) \sqrt {\sin (2 x)} \, dx=\log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )-\arcsin (\cos (x)-\sin (x)) \]

[In]

Int[Csc[x]*Sqrt[Sin[2*x]],x]

[Out]

-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]

Rule 4390

Int[cos[(a_.) + (b_.)*(x_)]/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-ArcSin[Cos[a + b*x] - Sin[a + b*
x]]/d, x] + Simp[Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d, x] /; FreeQ[{a, b, c, d}, x] && EqQ[
b*c - a*d, 0] && EqQ[d/b, 2]

Rule 4393

Int[((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_)/sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Dist[2*g, Int[Cos[a + b*x]*(g*S
in[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ
[p] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {\cos (x)}{\sqrt {\sin (2 x)}} \, dx \\ & = -\arcsin (\cos (x)-\sin (x))+\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \csc (x) \sqrt {\sin (2 x)} \, dx=-\arcsin (\cos (x)-\sin (x))+\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right ) \]

[In]

Integrate[Csc[x]*Sqrt[Sin[2*x]],x]

[Out]

-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.50 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.96

method	result	size
default	\(\frac {2 \sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan \left (\frac {x}{2}\right )^{2}-1}}\, \left (\tan \left (\frac {x}{2}\right )^{2}-1\right ) \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticF}\left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )}}\)	\(99\)

[In]

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

[Out]

2*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)/(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)+1)^(1/
2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)/(tan(1/2*x)^3-tan(1/2*x))^(1/2)*EllipticF((tan(1/2*x)+1)^(1/2),
1/2*2^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (23) = 46\).

Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 5.48 \[ \int \csc (x) \sqrt {\sin (2 x)} \, dx=\frac {1}{2} \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) - \frac {1}{2} \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac {1}{4} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]

[In]

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

[Out]

1/2*arctan(-(sqrt(2)*sqrt(cos(x)*sin(x))*(cos(x) - sin(x)) + cos(x)*sin(x))/(cos(x)^2 + 2*cos(x)*sin(x) - 1))
- 1/2*arctan(-(2*sqrt(2)*sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos(x) - sin(x))) - 1/4*log(-32*cos(x)^4 + 4*
sqrt(2)*(4*cos(x)^3 - (4*cos(x)^2 + 1)*sin(x) - 5*cos(x))*sqrt(cos(x)*sin(x)) + 32*cos(x)^2 + 16*cos(x)*sin(x)
+ 1)

Sympy [F(-1)]

Timed out. \[ \int \csc (x) \sqrt {\sin (2 x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \csc (x) \sqrt {\sin (2 x)} \, dx=\int { \csc \left (x\right ) \sqrt {\sin \left (2 \, x\right )} \,d x } \]

[In]

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

[Out]

integrate(csc(x)*sqrt(sin(2*x)), x)

Giac [F]

\[ \int \csc (x) \sqrt {\sin (2 x)} \, dx=\int { \csc \left (x\right ) \sqrt {\sin \left (2 \, x\right )} \,d x } \]

[In]

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

[Out]

integrate(csc(x)*sqrt(sin(2*x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(sin(2*x)^(1/2)/sin(x), x)