Optimal. Leaf size=29 \[ \frac {x}{a}-\frac {\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.03, antiderivative size = 29, 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 = {2735, 2648} \[ \frac {x}{a}-\frac {\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx &=\frac {x}{a}-\int \frac {1}{a+a \cos (c+d x)} \, dx\\ &=\frac {x}{a}-\frac {\sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 57, normalized size = 1.97 \[ \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (d x \cos \left (\frac {1}{2} (c+d x)\right )-\sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )\right )}{a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 37, normalized size = 1.28 \[ \frac {d x \cos \left (d x + c\right ) + d x - \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 28, normalized size = 0.97 \[ \frac {\frac {d x + c}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 1.28 \[ -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 49, normalized size = 1.69 \[ \frac {\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 23, normalized size = 0.79 \[ \frac {x}{a}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 27, normalized size = 0.93 \[ \begin {cases} \frac {x}{a} - \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{a \cos {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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