3.297 \(\int (1+\cos (x)) \csc (x) \, dx\)

Optimal. Leaf size=7 \[ \log (1-\cos (x)) \]

[Out]

Log[1 - Cos[x]]

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Rubi [A]  time = 0.021914, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \log (1-\cos (x)) \]

Antiderivative was successfully verified.

[In]  Int[(1 + Cos[x])*Csc[x],x]

[Out]

Log[1 - Cos[x]]

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Rubi in Sympy [A]  time = 1.37704, size = 5, normalized size = 0.71 \[ \log{\left (- \cos{\left (x \right )} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+cos(x))*csc(x),x)

[Out]

log(-cos(x) + 1)

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Mathematica [B]  time = 0.0090776, size = 20, normalized size = 2.86 \[ \log \left (\sin \left (\frac{x}{2}\right )\right )+\log (\sin (x))-\log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Cos[x])*Csc[x],x]

[Out]

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

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Maple [A]  time = 0.019, size = 6, normalized size = 0.9 \[ \ln \left ( \cos \left ( x \right ) -1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+cos(x))*csc(x),x)

[Out]

ln(cos(x)-1)

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Maxima [A]  time = 1.34635, size = 7, normalized size = 1. \[ \log \left (\cos \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 1)*csc(x),x, algorithm="maxima")

[Out]

log(cos(x) - 1)

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Fricas [A]  time = 0.227807, size = 9, normalized size = 1.29 \[ \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 1)*csc(x),x, algorithm="fricas")

[Out]

log(-1/2*cos(x) + 1/2)

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Sympy [A]  time = 2.26426, size = 12, normalized size = 1.71 \[ - \log{\left (\cot{\left (x \right )} + \csc{\left (x \right )} \right )} + \log{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+cos(x))*csc(x),x)

[Out]

-log(cot(x) + csc(x)) + log(sin(x))

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GIAC/XCAS [A]  time = 0.226328, size = 9, normalized size = 1.29 \[{\rm ln}\left (-\cos \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + 1)*csc(x),x, algorithm="giac")

[Out]

ln(-cos(x) + 1)