Optimal. Leaf size=16 \[ \frac{x}{2}-\frac{1}{2} \log (\sin (x)+\cos (x)) \]
[Out]
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Rubi [A] time = 0.0580929, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x}{2}-\frac{1}{2} \log (\sin (x)+\cos (x)) \]
Antiderivative was successfully verified.
[In] Int[Sin[x]/(Cos[x] + Sin[x]),x]
[Out]
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Rubi in Sympy [A] time = 4.64384, size = 12, normalized size = 0.75 \[ \frac{x}{2} - \frac{\log{\left (\sin{\left (x \right )} + \cos{\left (x \right )} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(sin(x)/(cos(x)+sin(x)),x)
[Out]
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Mathematica [A] time = 0.0108244, size = 16, normalized size = 1. \[ \frac{x}{2}-\frac{1}{2} \log (\sin (x)+\cos (x)) \]
Antiderivative was successfully verified.
[In] Integrate[Sin[x]/(Cos[x] + Sin[x]),x]
[Out]
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Maple [A] time = 0.036, size = 21, normalized size = 1.3 \[{\frac{\ln \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }{4}}-{\frac{\ln \left ( 1+\tan \left ( x \right ) \right ) }{2}}+{\frac{x}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(sin(x)/(cos(x)+sin(x)),x)
[Out]
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Maxima [A] time = 1.50212, size = 72, normalized size = 4.5 \[ \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) - \frac{1}{2} \, \log \left (-\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right ) + \frac{1}{2} \, \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(x)/(cos(x) + sin(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226001, size = 20, normalized size = 1.25 \[ \frac{1}{2} \, x - \frac{1}{4} \, \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(x)/(cos(x) + sin(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.157405, size = 12, normalized size = 0.75 \[ \frac{x}{2} - \frac{\log{\left (\sin{\left (x \right )} + \cos{\left (x \right )} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(x)/(cos(x)+sin(x)),x)
[Out]
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GIAC/XCAS [A] time = 0.232057, size = 28, normalized size = 1.75 \[ \frac{1}{2} \, x + \frac{1}{4} \,{\rm ln}\left (\tan \left (x\right )^{2} + 1\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | \tan \left (x\right ) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(x)/(cos(x) + sin(x)),x, algorithm="giac")
[Out]