Optimal. Leaf size=57 \[ -\frac{5 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{x}{a^2}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.0845487, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2758, 2735, 2648} \[ -\frac{5 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{x}{a^2}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2758
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos ^2(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{\int \frac{-2 a+3 a \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=\frac{x}{a^2}+\frac{\sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{5 \int \frac{1}{a+a \cos (c+d x)} \, dx}{3 a}\\ &=\frac{x}{a^2}+\frac{\sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{5 \sin (c+d x)}{3 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.228174, size = 105, normalized size = 1.84 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left (6 d x \cos ^3\left (\frac{1}{2} (c+d x)\right )+\tan \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )-10 \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{3 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 56, normalized size = 1. \begin{align*}{\frac{1}{6\,{a}^{2}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{3}{2\,{a}^{2}d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74071, size = 97, normalized size = 1.7 \begin{align*} -\frac{\frac{\frac{9 \, \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}}}{a^{2}} - \frac{12 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80739, size = 198, normalized size = 3.47 \begin{align*} \frac{3 \, d x \cos \left (d x + c\right )^{2} + 6 \, d x \cos \left (d x + c\right ) + 3 \, d x -{\left (5 \, \cos \left (d x + c\right ) + 4\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 = 3.12045, size = 56, normalized size = 0.98 \begin{align*} \begin{cases} \frac{x}{a^{2}} + \frac{\tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d} - \frac{3 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \cos ^{2}{\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 = 1.40093, size = 68, normalized size = 1.19 \begin{align*} \frac{\frac{6 \,{\left (d x + c\right )}}{a^{2}} + \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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