Integrand size = 11, antiderivative size = 31 \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=-\frac {1}{2} \arcsin (\cos (x)-\sin (x))-\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right ) \] Output:
-1/2*arcsin(cos(x)-sin(x))-1/2*ln(cos(x)+sin(x)+sin(2*x)^(1/2))
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\frac {1}{2} \left (-\arcsin (\cos (x)-\sin (x))-\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\right ) \] Input:
Integrate[Sin[x]/Sqrt[Sin[2*x]],x]
Output:
(-ArcSin[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]])/2
Time = 0.18 (sec) , antiderivative size = 31, 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 = {3042, 4794}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x)}{\sqrt {\sin (2 x)}}dx\) |
\(\Big \downarrow \) 4794 |
\(\displaystyle -\frac {1}{2} \arcsin (\cos (x)-\sin (x))-\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )\) |
Input:
Int[Sin[x]/Sqrt[Sin[2*x]],x]
Output:
-1/2*ArcSin[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/2
Int[sin[(a_.) + (b_.)*(x_)]/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Sim p[-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]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.35 (sec) , antiderivative size = 266, normalized size of antiderivative = 8.58
method | result | size |
default | \(-\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan \left (\frac {x}{2}\right )^{2}-1}}\, \left (\tan \left (\frac {x}{2}\right )^{2}-1\right ) \left (2 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticE}\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {x}{2}\right )^{2}-\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticF}\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {x}{2}\right )^{2}+2 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticE}\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticF}\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )+2 \tan \left (\frac {x}{2}\right )^{4}-2 \tan \left (\frac {x}{2}\right )^{2}\right )}{2 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\, \left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )}}\) | \(266\) |
Input:
int(sin(x)/sin(2*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-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)*(-tan(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*ta n(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)
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.42 \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\frac {1}{4} \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) - \frac {1}{4} \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) + \frac {1}{8} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \] Input:
integrate(sin(x)/sin(2*x)^(1/2),x, algorithm="fricas")
Output:
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)*s in(x)) - cos(x) - sin(x))/(cos(x) - sin(x))) + 1/8*log(-32*cos(x)^4 + 4*sq rt(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)
Timed out. \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\text {Timed out} \] Input:
integrate(sin(x)/sin(2*x)**(1/2),x)
Output:
Timed out
\[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\int { \frac {\sin \left (x\right )}{\sqrt {\sin \left (2 \, x\right )}} \,d x } \] Input:
integrate(sin(x)/sin(2*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sin(x)/sqrt(sin(2*x)), x)
\[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\int { \frac {\sin \left (x\right )}{\sqrt {\sin \left (2 \, x\right )}} \,d x } \] Input:
integrate(sin(x)/sin(2*x)^(1/2),x, algorithm="giac")
Output:
integrate(sin(x)/sqrt(sin(2*x)), x)
Timed out. \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\int \frac {\sin \left (x\right )}{\sqrt {\sin \left (2\,x\right )}} \,d x \] Input:
int(sin(x)/sin(2*x)^(1/2),x)
Output:
int(sin(x)/sin(2*x)^(1/2), x)
\[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx=\int \frac {\sqrt {\sin \left (2 x \right )}\, \sin \left (x \right )}{\sin \left (2 x \right )}d x \] Input:
int(sin(x)/sin(2*x)^(1/2),x)
Output:
int((sqrt(sin(2*x))*sin(x))/sin(2*x),x)