\(\int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx\) [289]

Optimal result

Integrand size = 13, antiderivative size = 89 \[ \int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)} (1+\tan (x))}\right )}{\sqrt {2}} \] Output:

-1/2*arctan(1-2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/2)+1/2*arctan(1+2^(1 
/2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/2)-1/2*arctanh(2^(1/2)*sin(x)^(1/2)/co 
s(x)^(1/2)/(1+tan(x)))*2^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx=\frac {2 \cos ^2(x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\sin ^2(x)\right ) \sin ^{\frac {3}{2}}(x)}{3 \cos ^{\frac {3}{2}}(x)} \] Input:

Integrate[Sqrt[Sin[x]]/Sqrt[Cos[x]],x]
 

Output:

(2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, Sin[x]^2]*Sin[x]^(3/2 
))/(3*Cos[x]^(3/2))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.51, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3054

\(\displaystyle 2 \int \frac {\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\)

\(\Big \downarrow \) 826

\(\displaystyle 2 \left (\frac {1}{2} \int \frac {\tan (x)+1}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\frac {1}{2} \int \frac {1}{\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (x)-1}d\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (x)-1}d\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\)

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[Sqrt[Sin[x]]/Sqrt[Cos[x]],x]
 

Output:

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])/2 + (Log[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]))/2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

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

rule 3054

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f)   Subst[Int[x^(k 
*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + 
 f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] 
&& LtQ[m, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(66)=132\).

Time = 47.00 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.27

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {\cos \left (x \right )}\, \left (\ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\cot \left (x \right ) \cos \left (x \right )-2 \cot \left (x \right )-\sin \left (x \right )+2 \cos \left (x \right )+\csc \left (x \right )-2}{-1+\cos \left (x \right )}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )-\ln \left (-\frac {2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )-\cot \left (x \right ) \cos \left (x \right )+2 \cot \left (x \right )+\sin \left (x \right )-2 \cos \left (x \right )-\csc \left (x \right )+2}{-1+\cos \left (x \right )}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )-\cos \left (x \right )+1}{-1+\cos \left (x \right )}\right )\right ) \left (-1+\cos \left (x \right )\right )}{4 \sin \left (x \right )^{\frac {3}{2}} \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(202\)

Input:

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

Output:

-1/4*2^(1/2)*cos(x)^(1/2)*(ln((2*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2 
)*sin(x)+cot(x)*cos(x)-2*cot(x)-sin(x)+2*cos(x)+csc(x)-2)/(-1+cos(x)))+2*a 
rctan((2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*sin(x)+cos(x)-1)/(-1+cos 
(x)))-ln(-(2*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*sin(x)-cot(x)*cos( 
x)+2*cot(x)+sin(x)-2*cos(x)-csc(x)+2)/(-1+cos(x)))+2*arctan((2^(1/2)*(sin( 
x)*cos(x)/(cos(x)+1)^2)^(1/2)*sin(x)-cos(x)+1)/(-1+cos(x))))*(-1+cos(x))/s 
in(x)^(3/2)/(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (66) = 132\).

Time = 0.11 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.47 \[ \int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx=-\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {2 \, \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - 2 \, \cos \left (x\right )}{2 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {2 \, \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} \sin \left (x\right ) - \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - 2 \, \cos \left (x\right )}{2 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cos \left (x\right ) - \sqrt {2} \sin \left (x\right )}{2 \, \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \] Input:

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

Output:

-1/8*sqrt(2)*arctan(1/2*(2*cos(x)^3 - 2*cos(x)^2*sin(x) + sqrt(2)*sqrt(cos 
(x))*sqrt(sin(x)) - 2*cos(x))/(cos(x)^3 + cos(x)^2*sin(x) - cos(x))) - 1/8 
*sqrt(2)*arctan(-1/2*(2*cos(x)^3 - 2*cos(x)^2*sin(x) - sqrt(2)*sqrt(cos(x) 
)*sqrt(sin(x)) - 2*cos(x))/(cos(x)^3 + cos(x)^2*sin(x) - cos(x))) + 1/4*sq 
rt(2)*arctan(-1/2*(sqrt(2)*cos(x) - sqrt(2)*sin(x))/(sqrt(cos(x))*sqrt(sin 
(x)))) - 1/8*sqrt(2)*log(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))* 
sqrt(sin(x)) + 4*cos(x)*sin(x) + 1) + 1/8*sqrt(2)*log(-2*(sqrt(2)*cos(x) + 
 sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 25.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx=-\frac {2\,\sqrt {\cos \left (x\right )}\,{\sin \left (x\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{3/4}} \] Input:

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

Reduce [F]

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

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

Output:

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