Optimal. Leaf size=12 \[ \frac {\log (a-b \cos (x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2668, 31} \[ \frac {\log (a-b \cos (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a-b \cos (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,-b \cos (x)\right )}{b}\\ &=\frac {\log (a-b \cos (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.00 \[ \frac {\log (a-b \cos (x))}{b} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (x)}{a-b \cos (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.87, size = 12, normalized size = 1.00 \[ \frac {\log \left (-b \cos \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 14, normalized size = 1.17 \[ \frac {\log \left ({\left | b \cos \relax (x) - a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 13, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\ln \left (a -b \cos \relax (x )\right )}{b}\) | \(13\) |
default | \(\frac {\ln \left (a -b \cos \relax (x )\right )}{b}\) | \(13\) |
risch | \(-\frac {i x}{b}+\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right )}{b}\) | \(32\) |
norman | \(\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a -b \right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 13, normalized size = 1.08 \[ \frac {\log \left (b \cos \relax (x) - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 13, normalized size = 1.08 \[ \frac {\ln \left (b\,\cos \relax (x)-a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 15, normalized size = 1.25 \[ \begin {cases} \frac {\log {\left (- \frac {a}{b} + \cos {\relax (x )} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {\cos {\relax (x )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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