Integrand size = 11, antiderivative size = 45 \[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=-\frac {1}{4} \arcsin (\cos (x)-\sin (x))+\frac {1}{4} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )-\frac {1}{2} \cos (x) \sqrt {\sin (2 x)} \] Output:
-1/4*arcsin(cos(x)-sin(x))+1/4*ln(cos(x)+sin(x)+sin(2*x)^(1/2))-1/2*cos(x) *sin(2*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=\frac {1}{4} \left (-\arcsin (\cos (x)-\sin (x))+\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )-2 \cos (x) \sqrt {\sin (2 x)}\right ) \] 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*Cos[ x]*Sqrt[Sin[2*x]])/4
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4790, 3042, 4793}
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.
\(\displaystyle \int \sin (x) \sqrt {\sin (2 x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (x) \sqrt {\sin (2 x)}dx\) |
\(\Big \downarrow \) 4790 |
\(\displaystyle \frac {1}{2} \int \frac {\cos (x)}{\sqrt {\sin (2 x)}}dx-\frac {1}{2} \sqrt {\sin (2 x)} \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {\cos (x)}{\sqrt {\sin (2 x)}}dx-\frac {1}{2} \sqrt {\sin (2 x)} \cos (x)\) |
\(\Big \downarrow \) 4793 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )-\frac {1}{2} \arcsin (\cos (x)-\sin (x))\right )-\frac {1}{2} \sqrt {\sin (2 x)} \cos (x)\) |
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)/2 - (Cos[x]*Sqrt[Sin[2*x]])/2
Int[sin[(a_.) + (b_.)*(x_)]*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[-2*Cos[a + b*x]*((g*Sin[c + d*x])^p/(d*(2*p + 1))), x] + Simp[2*p* (g/(2*p + 1)) Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[ {a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[p, 0] && IntegerQ[2*p]
Int[cos[(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 = 2.84 (sec) , antiderivative size = 171, normalized size of antiderivative = 3.80
method | result | size |
default | \(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan \left (\frac {x}{2}\right )^{2}-1}}\, \left (\tan \left (\frac {x}{2}\right )^{2}-1\right ) \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticF}\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {x}{2}\right )^{2}+\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \operatorname {EllipticF}\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )+2 \tan \left (\frac {x}{2}\right )^{3}-2 \tan \left (\frac {x}{2}\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\, \sqrt {\tan \left (\frac {x}{2}\right )^{3}-\tan \left (\frac {x}{2}\right )}\, \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}\) | \(171\) |
Input:
int(sin(x)*sin(2*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)*((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+(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)^3-2*tan(1/2*x))/(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)/(tan(1/2*x)^3-tan(1 /2*x))^(1/2)/(1+tan(1/2*x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (35) = 70\).
Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.36 \[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} \cos \left (x\right ) + \frac {1}{8} \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) - \frac {1}{8} \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac {1}{16} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \] Input:
integrate(sin(x)*sin(2*x)^(1/2),x, algorithm="fricas")
Output:
-1/2*sqrt(2)*sqrt(cos(x)*sin(x))*cos(x) + 1/8*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)) - 1/8*arctan(-(2*sqrt(2)*sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos(x ) - sin(x))) - 1/16*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)
Timed out. \[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=\text {Timed out} \] Input:
integrate(sin(x)*sin(2*x)**(1/2),x)
Output:
Timed out
\[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=\int { \sqrt {\sin \left (2 \, x\right )} \sin \left (x\right ) \,d x } \] Input:
integrate(sin(x)*sin(2*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(sin(2*x))*sin(x), x)
\[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=\int { \sqrt {\sin \left (2 \, x\right )} \sin \left (x\right ) \,d x } \] Input:
integrate(sin(x)*sin(2*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(sin(2*x))*sin(x), x)
Timed out. \[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=\int \sqrt {\sin \left (2\,x\right )}\,\sin \left (x\right ) \,d x \] Input:
int(sin(2*x)^(1/2)*sin(x),x)
Output:
int(sin(2*x)^(1/2)*sin(x), x)
\[ \int \sin (x) \sqrt {\sin (2 x)} \, dx=\int \sqrt {\sin \left (2 x \right )}\, \sin \left (x \right )d x \] Input:
int(sin(x)*sin(2*x)^(1/2),x)
Output:
int(sqrt(sin(2*x))*sin(x),x)