Optimal. Leaf size=22 \[ \frac {\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2648} \[ \frac {\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin {align*} \int \frac {1}{a+a \cos (c+d x)} \, dx &=\frac {\sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.77 \[ \frac {\tan \left (\frac {1}{2} (c+d x)\right )}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 22, normalized size = 1.00 \[ \frac {\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 16, normalized size = 0.73 \[ \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 0.77 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 23, normalized size = 1.05 \[ \frac {\sin \left (d x + c\right )}{a d {\left (\cos \left (d x + c\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 16, normalized size = 0.73 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 20, normalized size = 0.91 \[ \begin {cases} \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x}{a \cos {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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