3.239 \(\int \frac{\cos (x)}{\sin (x)+\sin ^2(x)} \, dx\)

Optimal. Leaf size=11 \[ \log (\sin (x))-\log (\sin (x)+1) \]

[Out]

Log[Sin[x]] - Log[1 + Sin[x]]

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Rubi [A]  time = 0.0352887, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \log (\sin (x))-\log (\sin (x)+1) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]/(Sin[x] + Sin[x]^2),x]

[Out]

Log[Sin[x]] - Log[1 + Sin[x]]

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Rubi in Sympy [A]  time = 3.53724, size = 10, normalized size = 0.91 \[ - \log{\left (\sin{\left (x \right )} + 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)/(sin(x)+sin(x)**2),x)

[Out]

-log(sin(x) + 1) + log(sin(x))

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Mathematica [A]  time = 0.0125043, size = 20, normalized size = 1.82 \[ \log (\sin (x))-2 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]/(Sin[x] + Sin[x]^2),x]

[Out]

-2*Log[Cos[x/2] + Sin[x/2]] + Log[Sin[x]]

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Maple [A]  time = 0.044, size = 12, normalized size = 1.1 \[ \ln \left ( \sin \left ( x \right ) \right ) -\ln \left ( 1+\sin \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)/(sin(x)+sin(x)^2),x)

[Out]

ln(sin(x))-ln(1+sin(x))

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Maxima [A]  time = 1.39643, size = 15, normalized size = 1.36 \[ -\log \left (\sin \left (x\right ) + 1\right ) + \log \left (\sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/(sin(x)^2 + sin(x)),x, algorithm="maxima")

[Out]

-log(sin(x) + 1) + log(sin(x))

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Fricas [A]  time = 0.229957, size = 18, normalized size = 1.64 \[ \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - \log \left (\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/(sin(x)^2 + sin(x)),x, algorithm="fricas")

[Out]

log(1/2*sin(x)) - log(sin(x) + 1)

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Sympy [A]  time = 0.243918, size = 10, normalized size = 0.91 \[ - \log{\left (\sin{\left (x \right )} + 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/(sin(x)+sin(x)**2),x)

[Out]

-log(sin(x) + 1) + log(sin(x))

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GIAC/XCAS [A]  time = 0.210021, size = 16, normalized size = 1.45 \[ -{\rm ln}\left (\sin \left (x\right ) + 1\right ) +{\rm ln}\left ({\left | \sin \left (x\right ) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/(sin(x)^2 + sin(x)),x, algorithm="giac")

[Out]

-ln(sin(x) + 1) + ln(abs(sin(x)))