\(\int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx\) [855]

Optimal result

Integrand size = 18, antiderivative size = 57 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\frac {2 \sqrt [4]{\cos ^2(x)} \sqrt {\sin (x)} \left (3 \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\sin ^2(x)\right )+\sqrt {\cos ^2(x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\sin ^2(x)\right ) \sin (x)\right )}{3 \cos ^{\frac {3}{2}}(x)} \] Input:

Integrate[(Cos[x] + Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]]),x]
 

Output:

(2*(Cos[x]^2)^(1/4)*Sqrt[Sin[x]]*(3*Cos[x]*Hypergeometric2F1[1/4, 1/4, 5/4 
, Sin[x]^2] + Sqrt[Cos[x]^2]*Hypergeometric2F1[3/4, 3/4, 7/4, Sin[x]^2]*Si 
n[x]))/(3*Cos[x]^(3/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(57)=114\).

Time = 0.50 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.26, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3586, 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.

\(\displaystyle \int \frac {\sin (x)+\cos (x)}{\sqrt {\sin (x)} \sqrt {\cos (x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (x)+\cos (x)}{\sqrt {\sin (x)} \sqrt {\cos (x)}}dx\)

\(\Big \downarrow \) 3586

\(\displaystyle \int \left (\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}+\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(Cos[x] + Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]]),x]
 

Output:

ArcTan[1 - (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/Sqrt[2] - ArcTan[1 + (Sqrt 
[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/Sqrt[2] - ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]] 
)/Sqrt[Cos[x]]]/Sqrt[2] + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/ 
Sqrt[2] - Log[1 + Cot[x] - (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/(2*Sqrt[2] 
) + Log[1 + Cot[x] + (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/(2*Sqrt[2]) + Lo 
g[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2]) - Log[1 + 
(Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2])
 

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 3586

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) 
+ (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(p_.), x_Symbol] :> In 
t[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c + d*x] 
)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(41)=82\).

Time = 14.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {\sqrt {\cos \left (x \right )}\, \left (\arctan \left (\frac {-\sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )-\arctan \left (\frac {\sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )\right ) \left (\cos \left (x \right )-1\right ) \sqrt {2}}{\sin \left (x \right )^{\frac {3}{2}} \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}}\)	\(96\)
parts	\(\frac {\left (1-\cos \left (x \right )\right ) \sqrt {\cos \left (x \right )}\, \left (\ln \left (\frac {-\csc \left (x \right ) \left (1-\cos \left (x \right )\right )^{2}+2 \sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+2-2 \cos \left (x \right )+\sin \left (x \right )}{1-\cos \left (x \right )}\right )+2 \arctan \left (\frac {\sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )-1}\right )-\ln \left (-\frac {\csc \left (x \right ) \left (1-\cos \left (x \right )\right )^{2}+2 \sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}-2+2 \cos \left (x \right )-\sin \left (x \right )}{1-\cos \left (x \right )}\right )+2 \arctan \left (\frac {\sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )\right ) \sqrt {2}}{4 \sin \left (x \right )^{\frac {3}{2}} \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}}+\frac {\sqrt {\cos \left (x \right )}\, \left (\ln \left (-\frac {2 \sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}-\cos \left (x \right ) \cot \left (x \right )+2 \cot \left (x \right )-2 \cos \left (x \right )+\sin \left (x \right )-\csc \left (x \right )+2}{\cos \left (x \right )-1}\right )-2 \arctan \left (\frac {\sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )-1}\right )-\ln \left (\frac {2 \sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right ) \cot \left (x \right )-2 \cot \left (x \right )+2 \cos \left (x \right )-\sin \left (x \right )+\csc \left (x \right )-2}{\cos \left (x \right )-1}\right )-2 \arctan \left (\frac {\sin \left (x \right ) \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )\right ) \left (\cos \left (x \right )-1\right ) \sqrt {2}}{4 \sin \left (x \right )^{\frac {3}{2}} \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (1+\cos \left (x \right )\right )^{2}}}}\)	\(408\)

Input:

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

Output:

cos(x)^(1/2)*(arctan((-sin(x)*2^(1/2)*(sin(x)*cos(x)/(1+cos(x))^2)^(1/2)+c 
os(x)-1)/(cos(x)-1))-arctan((sin(x)*2^(1/2)*(sin(x)*cos(x)/(1+cos(x))^2)^( 
1/2)+cos(x)-1)/(cos(x)-1)))*(cos(x)-1)*2^(1/2)/sin(x)^(3/2)/(sin(x)*cos(x) 
/(1+cos(x))^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (41) = 82\).

Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (32 \, \sqrt {2} \cos \left (x\right )^{4} - 32 \, \sqrt {2} \cos \left (x\right )^{2} + 16 \, \sqrt {2} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {2}\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}{8 \, {\left (4 \, \cos \left (x\right )^{5} - 3 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{4} - 5 \, \cos \left (x\right )^{2}\right )} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) \] Input:

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

Output:

-1/4*sqrt(2)*arctan(-1/8*(32*sqrt(2)*cos(x)^4 - 32*sqrt(2)*cos(x)^2 + 16*s 
qrt(2)*cos(x)*sin(x) - sqrt(2))*sqrt(cos(x))*sqrt(sin(x))/(4*cos(x)^5 - 3* 
cos(x)^3 - (4*cos(x)^4 - 5*cos(x)^2)*sin(x) - cos(x)))
 

Sympy [F]

\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int \frac {\sin {\left (x \right )} + \cos {\left (x \right )}}{\sqrt {\sin {\left (x \right )}} \sqrt {\cos {\left (x \right )}}}\, dx \] Input:

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

Output:

Integral((sin(x) + cos(x))/(sqrt(sin(x))*sqrt(cos(x))), x)
 

Maxima [F]

\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int { \frac {\cos \left (x\right ) + \sin \left (x\right )}{\sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}} \,d x } \] Input:

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

Output:

integrate((cos(x) + sin(x))/(sqrt(cos(x))*sqrt(sin(x))), x)
 

Giac [F]

\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int { \frac {\cos \left (x\right ) + \sin \left (x\right )}{\sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}} \,d x } \] Input:

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

Output:

integrate((cos(x) + sin(x))/(sqrt(cos(x))*sqrt(sin(x))), x)
 

Mupad [B] (verification not implemented)

Time = 19.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\frac {2\,\sqrt {\cos \left (x\right )}\,{\sin \left (x\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{3/4}}-\frac {2\,{\cos \left (x\right )}^{3/2}\,\sqrt {\sin \left (x\right )}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {3}{4};\ \frac {7}{4};\ {\cos \left (x\right )}^2\right )}{3\,{\left ({\sin \left (x\right )}^2\right )}^{1/4}} \] Input:

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

Output:

- (2*cos(x)^(1/2)*sin(x)^(3/2)*hypergeom([1/4, 1/4], 5/4, cos(x)^2))/(sin( 
x)^2)^(3/4) - (2*cos(x)^(3/2)*sin(x)^(1/2)*hypergeom([3/4, 3/4], 7/4, cos( 
x)^2))/(3*(sin(x)^2)^(1/4))
 

Reduce [F]

\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int \frac {\sqrt {\sin \left (x \right )}\, \sqrt {\cos \left (x \right )}}{\cos \left (x \right )}d x +\int \frac {\sqrt {\sin \left (x \right )}\, \sqrt {\cos \left (x \right )}}{\sin \left (x \right )}d x \] Input:

int((cos(x)+sin(x))/cos(x)^(1/2)/sin(x)^(1/2),x)
 

Output:

int((sqrt(sin(x))*sqrt(cos(x)))/cos(x),x) + int((sqrt(sin(x))*sqrt(cos(x)) 
)/sin(x),x)