Optimal. Leaf size=57 \[ \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )-\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 0.21, antiderivative size = 243, normalized size of antiderivative = 4.26, number of steps used = 22, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3107, 2575, 297, 1162, 617, 204, 1165, 628, 2574} \[ \frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}+\frac {\log \left (\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (x)-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\cot (x)+\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}+1\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2574
Rule 2575
Rule 3107
Rubi steps
\begin {align*} \int \frac {\cos (x)+\sin (x)}{\sqrt {\cos (x)} \sqrt {\sin (x)}} \, dx &=\int \left (\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}+\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right ) \, dx\\ &=\int \frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}} \, dx+\int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )\right )+2 \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\\ &=\operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )-\operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )+\operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{2 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{2 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{2 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{2 \sqrt {2}}\\ &=-\frac {\log \left (1+\cot (x)-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{2 \sqrt {2}}+\frac {\log \left (1+\cot (x)+\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{2 \sqrt {2}}+\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{2 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}\\ &=\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}-\frac {\log \left (1+\cot (x)-\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{2 \sqrt {2}}+\frac {\log \left (1+\cot (x)+\frac {\sqrt {2} \sqrt {\cos (x)}}{\sqrt {\sin (x)}}\right )}{2 \sqrt {2}}+\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{2 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 68, normalized size = 1.19 \[ \frac {2 \sqrt {\sin (x)} \sqrt [4]{\cos ^2(x)} \left (\sin (x) \sqrt {\cos ^2(x)} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\sin ^2(x)\right )+3 \cos (x) \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\sin ^2(x)\right )\right )}{3 \cos ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.98, size = 85, normalized size = 1.49 \[ -\frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (32 \, \sqrt {2} \cos \relax (x)^{4} - 32 \, \sqrt {2} \cos \relax (x)^{2} + 16 \, \sqrt {2} \cos \relax (x) \sin \relax (x) - \sqrt {2}\right )} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)}}{8 \, {\left (4 \, \cos \relax (x)^{5} - 3 \, \cos \relax (x)^{3} - {\left (4 \, \cos \relax (x)^{4} - 5 \, \cos \relax (x)^{2}\right )} \sin \relax (x) - \cos \relax (x)\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x) + \sin \relax (x)}{\sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.26, size = 134, normalized size = 2.35 \[ \frac {\sqrt {\frac {1-\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {2}\, \sqrt {\frac {\cos \relax (x )-1+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \left (\sin ^{\frac {3}{2}}\relax (x )\right ) \left (i \EllipticPi \left (\sqrt {\frac {1-\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \EllipticPi \left (\sqrt {\frac {1-\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {\frac {1-\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {\cos \relax (x )}\, \left (-1+\cos \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x) + \sin \relax (x)}{\sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.62, size = 51, normalized size = 0.89 \[ -\frac {2\,\sqrt {\cos \relax (x)}\,{\sin \relax (x)}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \relax (x)}^2\right )}{{\left ({\sin \relax (x)}^2\right )}^{3/4}}-\frac {2\,{\cos \relax (x)}^{3/2}\,\sqrt {\sin \relax (x)}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {3}{4};\ \frac {7}{4};\ {\cos \relax (x)}^2\right )}{3\,{\left ({\sin \relax (x)}^2\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\relax (x )} + \cos {\relax (x )}}{\sqrt {\sin {\relax (x )}} \sqrt {\cos {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________