3.80 \(\int \sec (x) (1-\sin (x)) \, dx\)

Optimal. Leaf size=5 \[ \log (\sin (x)+1) \]

[Out]

Log[1 + Sin[x]]

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Rubi [A]  time = 0.0220001, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \log (\sin (x)+1) \]

Antiderivative was successfully verified.

[In]  Int[Sec[x]*(1 - Sin[x]),x]

[Out]

Log[1 + Sin[x]]

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Rubi in Sympy [A]  time = 1.55506, size = 5, normalized size = 1. \[ \log{\left (\sin{\left (x \right )} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-sin(x))/cos(x),x)

[Out]

log(sin(x) + 1)

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Mathematica [B]  time = 0.0111511, size = 36, normalized size = 7.2 \[ \log (\cos (x))-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sec[x]*(1 - Sin[x]),x]

[Out]

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

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Maple [A]  time = 0.036, size = 6, normalized size = 1.2 \[ \ln \left ( 1+\sin \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-sin(x))/cos(x),x)

[Out]

ln(1+sin(x))

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Maxima [A]  time = 1.35132, size = 7, normalized size = 1.4 \[ \log \left (\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sin(x) - 1)/cos(x),x, algorithm="maxima")

[Out]

log(sin(x) + 1)

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Fricas [A]  time = 0.223727, size = 7, normalized size = 1.4 \[ \log \left (\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sin(x) - 1)/cos(x),x, algorithm="fricas")

[Out]

log(sin(x) + 1)

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Sympy [A]  time = 0.481488, size = 19, normalized size = 3.8 \[ 2 \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} - \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-sin(x))/cos(x),x)

[Out]

2*log(tan(x/2) + 1) - log(tan(x/2)**2 + 1)

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GIAC/XCAS [A]  time = 0.214291, size = 7, normalized size = 1.4 \[{\rm ln}\left (\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(sin(x) - 1)/cos(x),x, algorithm="giac")

[Out]

ln(sin(x) + 1)