3.402 \(\int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx\)

Optimal. Leaf size=31 \[ -\frac {1}{2} \sin ^{-1}(\cos (x)-\sin (x))-\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4306} \[ -\frac {1}{2} \sin ^{-1}(\cos (x)-\sin (x))-\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4306

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*} \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx &=-\frac {1}{2} \sin ^{-1}(\cos (x)-\sin (x))-\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 31, normalized size = 1.00 \[ \frac {1}{2} \left (-\sin ^{-1}(\cos (x)-\sin (x))-\log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )\right ) \]

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

________________________________________________________________________________________

fricas [B]  time = 0.92, size = 137, normalized size = 4.42 \[ \frac {1}{4} \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} {\left (\cos \relax (x) - \sin \relax (x)\right )} + \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) \sin \relax (x) - 1}\right ) - \frac {1}{4} \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} - \cos \relax (x) - \sin \relax (x)}{\cos \relax (x) - \sin \relax (x)}\right ) + \frac {1}{8} \, \log \left (-32 \, \cos \relax (x)^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \relax (x)^{3} - {\left (4 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x) - 5 \, \cos \relax (x)\right )} \sqrt {\cos \relax (x) \sin \relax (x)} + 32 \, \cos \relax (x)^{2} + 16 \, \cos \relax (x) \sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)/sqrt(sin(2*x)), x)

________________________________________________________________________________________

maple [C]  time = 0.18, size = 266, normalized size = 8.58

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

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(x)/sqrt(sin(2*x)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sin \relax (x)}{\sqrt {\sin \left (2\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sin(x)/sin(2*x)^(1/2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________