\(\int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx\) [402]

Optimal result

Integrand size = 11, antiderivative size = 31 \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=-\frac {1}{2} \arcsin (\cos (x)-\sin (x))-\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4391} \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=-\frac {1}{2} \arcsin (\cos (x)-\sin (x))-\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \]

[In]

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

[Out]

-1/2*ArcSin[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/2

Rule 4391

Int[sin[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} \arcsin (\cos (x)-\sin (x))-\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\frac {1}{2} \left (-\arcsin (\cos (x)-\sin (x))-\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.38 (sec) , antiderivative size = 266, normalized size of antiderivative = 8.58

method	result	size
default	\(-\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (2 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )+2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )\right )}{2 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}}\)	\(266\)

[In]

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

[Out]

-1/2*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)*(2*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-t
an(1/2*x))^(1/2)*EllipticE((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)^2-(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+
2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*tan(1/2*x)^2+2*(1+tan(1/2*x))^(1/2)*(
-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticE((1+tan(1/2*x))^(1/2),1/2*2^(1/2))-(1+tan(1/2*x))^(1/2)*(-
2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))+2*tan(1/2*x)^4-2*tan(1/2
*x)^2)/(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)/(1+tan(1/2*x)^2)/(tan(1/2*x)^3-tan(1/2*x))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.42 \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\frac {1}{4} \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) - \frac {1}{4} \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) + \frac {1}{8} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]

[In]

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

[Out]

1/4*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))
- 1/4*arctan(-(2*sqrt(2)*sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos(x) - sin(x))) + 1/8*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. \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

integrate(sin(x)/sqrt(sin(2*x)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sin(x)/sqrt(sin(2*x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(sin(x)/sin(2*x)^(1/2), x)