3.68 \(\int \frac{\sin (x)}{\cos (x)+\sin (x)} \, dx\)

Optimal. Leaf size=16 \[ \frac{x}{2}-\frac{1}{2} \log (\sin (x)+\cos (x)) \]

[Out]

x/2 - Log[Cos[x] + Sin[x]]/2

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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]

x/2 - Log[Cos[x] + Sin[x]]/2

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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]

x/2 - log(sin(x) + cos(x))/2

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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]

x/2 - Log[Cos[x] + Sin[x]]/2

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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]

1/4*ln(1+tan(x)^2)-1/2*ln(1+tan(x))+1/2*x

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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]

arctan(sin(x)/(cos(x) + 1)) - 1/2*log(-2*sin(x)/(cos(x) + 1) + sin(x)^2/(cos(x)
+ 1)^2 - 1) + 1/2*log(sin(x)^2/(cos(x) + 1)^2 + 1)

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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]

1/2*x - 1/4*log(2*cos(x)*sin(x) + 1)

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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]

x/2 - log(sin(x) + cos(x))/2

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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]

1/2*x + 1/4*ln(tan(x)^2 + 1) - 1/2*ln(abs(tan(x) + 1))