3.5.5 \(\int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx\) [405]

3.5.5.1 Optimal result

Integrand size = 16, antiderivative size = 47 \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=-\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\frac {1}{2} \cos (x) \sqrt {\sin (2 x)}+\frac {1}{2} \sin (x) \sqrt {\sin (2 x)} \]

output

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

3.5.5.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\frac {1}{2} \left (-\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\cos (x) \sqrt {\sin (2 x)}+\sin (x) \sqrt {\sin (2 x)}\right ) \]

input

Integrate[(Cos[x] - Sin[x])*Sqrt[Sin[2*x]],x]
 

output

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

3.5.5.3 Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 4901, 2009}

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.

\(\displaystyle \int \sqrt {\sin (2 x)} (\cos (x)-\sin (x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {\sin (2 x)} (\cos (x)-\sin (x))dx\)

\(\Big \downarrow \) 4901

\(\displaystyle \int \left (\sqrt {\sin (2 x)} \cos (x)-\sin (x) \sqrt {\sin (2 x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \sin (x) \sqrt {\sin (2 x)}+\frac {1}{2} \sqrt {\sin (2 x)} \cos (x)-\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )\)

input

Int[(Cos[x] - Sin[x])*Sqrt[Sin[2*x]],x]
 

output

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

3.5.5.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4901

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; 
 !InertTrigFreeQ[u]
 

3.5.5.4 Maple [C] (verified)

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

Time = 5.27 (sec) , antiderivative size = 396, normalized size of antiderivative = 8.43

method	result	size
parts	\(\frac {2 \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 {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{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 )}\, 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 ) \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}+2 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}}-\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 (\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 )+\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 ^{3}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\)	\(396\)
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 (4 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{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 )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-3 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{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 ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+4 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{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 )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )-3 \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 ) \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}+4 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-2 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+4 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\)	\(442\)

input

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

output

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)) 
*(tan(1/2*x)-1)*tan(1/2*x))^(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))-(1+ta 
n(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*EllipticF((1+t 
an(1/2*x))^(1/2),1/2*2^(1/2))*((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^( 
1/2)+2*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*tan(1/2*x)^2)/(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))*(tan(1/2*x) 
-1)*tan(1/2*x))^(1/2)-(-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)*Ellipt 
icF((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)
 

3.5.5.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (35) = 70\).

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.62 \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\frac {1}{2} \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \frac {1}{8} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]

input

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

output

1/2*sqrt(2)*sqrt(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)*s 
in(x)) + 32*cos(x)^2 + 16*cos(x)*sin(x) + 1)
 

3.5.5.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.5.5.7 Maxima [F]

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

input

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

output

integrate((cos(x) - sin(x))*sqrt(sin(2*x)), x)
 

3.5.5.8 Giac [F]

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

input

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

output

integrate((cos(x) - sin(x))*sqrt(sin(2*x)), x)
 

3.5.5.9 Mupad [F(-1)]

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

input

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

output

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

3.5.5.10 Reduce [F]

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

input

int(sqrt(sin(2*x))*(cos(x) - sin(x)),x)
 

output

int(sqrt(sin(2*x))*cos(x),x) - int(sqrt(sin(2*x))*sin(x),x)