Optimal. Leaf size=16 \[ \frac {\log (\sin (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2667, 31} \[ \frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\log (1+\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 16, normalized size = 1.00 \[ \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 19, normalized size = 1.19 \[ \frac {\log \left ({\left | a \sin \left (d x + c\right ) + a \right |}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 19, normalized size = 1.19 \[ \frac {\ln \left (a +a \sin \left (d x +c \right )\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 18, normalized size = 1.12 \[ \frac {\log \left (a \sin \left (d x + c\right ) + a\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 16, normalized size = 1.00 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 24, normalized size = 1.50 \[ \begin {cases} \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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