3.11 \(\int \frac{\sin (x)}{a-b \cos (x)} \, dx\)

Optimal. Leaf size=12 \[ \frac{\log (a-b \cos (x))}{b} \]

[Out]

Log[a - b*Cos[x]]/b

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Rubi [A]  time = 0.0388799, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\log (a-b \cos (x))}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]/(a - b*Cos[x]),x]

[Out]

Log[a - b*Cos[x]]/b

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Rubi in Sympy [A]  time = 2.2841, size = 8, normalized size = 0.67 \[ \frac{\log{\left (a - b \cos{\left (x \right )} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)/(a-b*cos(x)),x)

[Out]

log(a - b*cos(x))/b

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Mathematica [A]  time = 0.0067702, size = 12, normalized size = 1. \[ \frac{\log (a-b \cos (x))}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]/(a - b*Cos[x]),x]

[Out]

Log[a - b*Cos[x]]/b

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Maple [A]  time = 0.013, size = 13, normalized size = 1.1 \[{\frac{\ln \left ( a-b\cos \left ( x \right ) \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)/(a-b*cos(x)),x)

[Out]

ln(a-b*cos(x))/b

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Maxima [A]  time = 1.34029, size = 18, normalized size = 1.5 \[ \frac{\log \left (b \cos \left (x\right ) - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*cos(x) - a)/b

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Fricas [A]  time = 0.27593, size = 16, normalized size = 1.33 \[ \frac{\log \left (-b \cos \left (x\right ) + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(-b*cos(x) + a)/b

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Sympy [A]  time = 0.701188, size = 15, normalized size = 1.25 \[ \begin{cases} \frac{\log{\left (- \frac{a}{b} + \cos{\left (x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\- \frac{\cos{\left (x \right )}}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/(a-b*cos(x)),x)

[Out]

Piecewise((log(-a/b + cos(x))/b, Ne(b, 0)), (-cos(x)/a, True))

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GIAC/XCAS [A]  time = 0.199926, size = 19, normalized size = 1.58 \[ \frac{{\rm ln}\left ({\left | b \cos \left (x\right ) - a \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*cos(x) - a))/b