Optimal. Leaf size=19 \[ \frac {x \sqrt {\sin (2 x)+1}}{\sin (x)+\cos (x)} \]
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Rubi [B] time = 1.71, antiderivative size = 72, normalized size of antiderivative = 3.79, number of steps used = 17, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4401, 6719, 1075, 628, 635, 203, 260, 12, 1023} \[ \frac {2 \cos ^2\left (\frac {x}{2}\right ) \tan ^{-1}\left (\tan \left (\frac {x}{2}\right )\right ) \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 260
Rule 628
Rule 635
Rule 1023
Rule 1075
Rule 4401
Rule 6719
Rubi steps
\begin {align*} \int \frac {\cos (x)+\sin (x)}{\sqrt {1+\sin (2 x)}} \, dx &=\int \left (\frac {\cos (x)}{\sqrt {1+\sin (2 x)}}+\frac {\sin (x)}{\sqrt {1+\sin (2 x)}}\right ) \, dx\\ &=\int \frac {\cos (x)}{\sqrt {1+\sin (2 x)}} \, dx+\int \frac {\sin (x)}{\sqrt {1+\sin (2 x)}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {2 x}{\left (1+x^2\right )^2 \sqrt {\frac {\left (-1-2 x+x^2\right )^2}{\left (1+x^2\right )^2}}} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+2 \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right )^2 \sqrt {\frac {\left (-1-2 x+x^2\right )^2}{\left (1+x^2\right )^2}}} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2 \sqrt {\frac {\left (-1-2 x+x^2\right )^2}{\left (1+x^2\right )^2}}} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {\left (2 \cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right ) \left (-1-2 x+x^2\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}\\ &=\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {-4+4 x}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{4 \sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}+\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {4-4 x}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{4 \sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}+\frac {\left (4 \cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (-1-2 x+x^2\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}\\ &=\frac {\cos ^2\left (\frac {x}{2}\right ) \log \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}{2 \sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}}+\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {-2-2 x}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{2 \sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}+\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {-2+2 x}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{2 \sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}-\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}+\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}\\ &=\frac {x \cos ^2\left (\frac {x}{2}\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}{2 \sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}}+\frac {\cos ^2\left (\frac {x}{2}\right ) \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}}-\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}-\frac {\left (\cos ^2\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (-1-2 \tan \left (\frac {x}{2}\right )+\tan ^2\left (\frac {x}{2}\right )\right )^2}}\\ &=\frac {x \cos ^2\left (\frac {x}{2}\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}{\sqrt {\cos ^4\left (\frac {x}{2}\right ) \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.89 \[ \frac {x (\sin (x)+\cos (x))}{\sqrt {\sin (2 x)+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 3, normalized size = 0.16 \[ -x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 42, normalized size = 2.21 \[ \frac {2 \, \pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor - x}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 2 \, \tan \left (\frac {1}{2} \, x\right ) - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.42, size = 12372, normalized size = 651.16 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 329, normalized size = 17.32 \[ \frac {1}{16} \, \sqrt {2} {\left (2 \, \sqrt {2} \arctan \left (\sin \left (2 \, x\right ) + 1, \cos \left (2 \, x\right )\right ) + \sqrt {2} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) + 1\right ) + 4 \, {\left (\cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) \sin \left (4 \, x\right ) + \sin \left (4 \, x\right )^{2} - 4 \, \cos \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2}\right )}^{\frac {1}{4}} {\left (\cos \left (\frac {1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right ) \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) \sin \left (\frac {1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right )\right )}\right )} + \frac {1}{16} \, \sqrt {2} {\left (2 \, \sqrt {2} \arctan \left (\sin \left (2 \, x\right ) + 1, \cos \left (2 \, x\right )\right ) - \sqrt {2} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) + 1\right ) - 4 \, {\left (\cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) \sin \left (4 \, x\right ) + \sin \left (4 \, x\right )^{2} - 4 \, \cos \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2}\right )}^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) \cos \left (\frac {1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right ) - \sin \left (2 \, x\right ) \sin \left (\frac {1}{2} \, \arctan \left (\cos \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right ), 2 \, \cos \left (2 \, x\right ) + \sin \left (4 \, x\right )\right )\right )\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\cos \relax (x)+\sin \relax (x)}{\sqrt {\sin \left (2\,x\right )+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\relax (x )} + \cos {\relax (x )}}{\sqrt {\sin {\left (2 x \right )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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