Integrand size = 18, antiderivative size = 57 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right ) \]
-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))*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\frac {2 \sqrt [4]{\cos ^2(x)} \sqrt {\sin (x)} \left (3 \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\sin ^2(x)\right )+\sqrt {\cos ^2(x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\sin ^2(x)\right ) \sin (x)\right )}{3 \cos ^{\frac {3}{2}}(x)} \]
(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))
Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(57)=114\).
Time = 0.46 (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 |
\(\displaystyle \frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}+\frac {\log \left (\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (x)-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\cot (x)+\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{2 \sqrt {2}}\) |
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])
3.10.12.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(41)=82\).
Time = 24.51 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.68
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (\arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )+\arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )-1}\right )\right ) \left (\cos \left (x \right )-1\right ) \sqrt {\cos \left (x \right )}}{\sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )^{\frac {3}{2}}}\) | \(96\) |
parts | \(-\frac {\sqrt {\cos \left (x \right )}\, \left (\cos \left (x \right )-1\right ) \left (\ln \left (-2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )+2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )-\ln \left (2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )+2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\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 {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}+\frac {\left (\cos \left (x \right )-1\right ) \sqrt {\cos \left (x \right )}\, \left (\ln \left (-2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )-2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+\cos \left (x \right )-1}{\cos \left (x \right )-1}\right )-\ln \left (2 \cot \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \csc \left (x \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \cot \left (x \right )+2\right )-2 \arctan \left (\frac {\sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\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 {\cos \left (x \right ) \sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\) | \(392\) |
-2^(1/2)*(arctan((2^(1/2)*sin(x)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)+cos(x) -1)/(cos(x)-1))+arctan((2^(1/2)*sin(x)*(cos(x)*sin(x)/(cos(x)+1)^2)^(1/2)- cos(x)+1)/(cos(x)-1)))*(cos(x)-1)*cos(x)^(1/2)/(cos(x)*sin(x)/(cos(x)+1)^2 )^(1/2)/sin(x)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (41) = 82\).
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (32 \, \sqrt {2} \cos \left (x\right )^{4} - 32 \, \sqrt {2} \cos \left (x\right )^{2} + 16 \, \sqrt {2} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {2}\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}{8 \, {\left (4 \, \cos \left (x\right )^{5} - 3 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{4} - 5 \, \cos \left (x\right )^{2}\right )} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) \]
-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)))
\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int \frac {\sin {\left (x \right )} + \cos {\left (x \right )}}{\sqrt {\sin {\left (x \right )}} \sqrt {\cos {\left (x \right )}}}\, dx \]
\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int { \frac {\cos \left (x\right ) + \sin \left (x\right )}{\sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}} \,d x } \]
\[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=\int { \frac {\cos \left (x\right ) + \sin \left (x\right )}{\sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}} \,d x } \]
Time = 28.44 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx=-\frac {2\,\sqrt {\cos \left (x\right )}\,{\sin \left (x\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{3/4}}-\frac {2\,{\cos \left (x\right )}^{3/2}\,\sqrt {\sin \left (x\right )}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {3}{4};\ \frac {7}{4};\ {\cos \left (x\right )}^2\right )}{3\,{\left ({\sin \left (x\right )}^2\right )}^{1/4}} \]