Optimal. Leaf size=48 \[ \frac{(A+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{C \sin (c+d x)}{a d}-\frac{C x}{a} \]
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Rubi [A] time = 0.0966013, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3024, 2735, 2648} \[ \frac{(A+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{C \sin (c+d x)}{a d}-\frac{C x}{a} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx &=\frac{C \sin (c+d x)}{a d}+\frac{\int \frac{a A-a C \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a}\\ &=-\frac{C x}{a}+\frac{C \sin (c+d x)}{a d}+(A+C) \int \frac{1}{a+a \cos (c+d x)} \, dx\\ &=-\frac{C x}{a}+\frac{C \sin (c+d x)}{a d}+\frac{(A+C) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.243367, size = 108, normalized size = 2.25 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (4 A \sin \left (\frac{d x}{2}\right )+C \sin \left (c+\frac{d x}{2}\right )+C \sin \left (c+\frac{3 d x}{2}\right )+C \sin \left (2 c+\frac{3 d x}{2}\right )-2 C d x \cos \left (c+\frac{d x}{2}\right )+5 C \sin \left (\frac{d x}{2}\right )-2 C d x \cos \left (\frac{d x}{2}\right )\right )}{4 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 88, normalized size = 1.8 \begin{align*}{\frac{A}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54586, size = 158, normalized size = 3.29 \begin{align*} -\frac{C{\left (\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{2 \, \sin \left (d x + c\right )}{{\left (a + \frac{a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - \frac{A \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3344, size = 132, normalized size = 2.75 \begin{align*} -\frac{C d x \cos \left (d x + c\right ) + C d x -{\left (C \cos \left (d x + c\right ) + A + 2 \, C\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.04413, size = 202, normalized size = 4.21 \begin{align*} \begin{cases} \frac{A \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} - \frac{C d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} - \frac{C d x}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{C \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{3 C \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} & \text{for}\: d \neq 0 \\\frac{x \left (A + C \cos ^{2}{\left (c \right )}\right )}{a \cos{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1696, size = 100, normalized size = 2.08 \begin{align*} -\frac{\frac{{\left (d x + c\right )} C}{a} - \frac{A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{2 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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