Optimal. Leaf size=55 \[ \frac{2 \sin (c+d x)}{3 d \left (a^2 \cos (c+d x)+a^2\right )}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.0384962, antiderivative size = 55, normalized size of antiderivative = 1., 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 = {2750, 2648} \[ \frac{2 \sin (c+d x)}{3 d \left (a^2 \cos (c+d x)+a^2\right )}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx &=-\frac{\sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{2 \int \frac{1}{a+a \cos (c+d x)} \, dx}{3 a}\\ &=-\frac{\sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{2 \sin (c+d x)}{3 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.112811, size = 60, normalized size = 1.09 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-3 \sin \left (c+\frac{d x}{2}\right )+2 \sin \left (c+\frac{3 d x}{2}\right )+3 \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right )}{12 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 32, normalized size = 0.6 \begin{align*}{\frac{1}{2\,{a}^{2}d} \left ( -{\frac{1}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17205, size = 63, normalized size = 1.15 \begin{align*} \frac{\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{6 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69925, size = 126, normalized size = 2.29 \begin{align*} \frac{{\left (2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.07279, size = 48, normalized size = 0.87 \begin{align*} \begin{cases} - \frac{\tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d} + \frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{\left (a \cos{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19589, size = 42, normalized size = 0.76 \begin{align*} -\frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{6 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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