Optimal. Leaf size=11 \[ \frac {\log (a+b \sin (x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2668, 31} \[ \frac {\log (a+b \sin (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos (x)}{a+b \sin (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (x)\right )}{b}\\ &=\frac {\log (a+b \sin (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\log (a+b \sin (x))}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.97, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \sin \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 12, normalized size = 1.09 \[ \frac {\log \left ({\left | b \sin \relax (x) + a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 12, normalized size = 1.09 \[ \frac {\ln \left (a +b \sin \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \sin \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 11, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\sin \relax (x)\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 14, normalized size = 1.27 \[ \begin {cases} \frac {\log {\left (\frac {a}{b} + \sin {\relax (x )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {\sin {\relax (x )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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