Optimal. Leaf size=99 \[ \frac{2 (B-C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 B-3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 B-3 C)}{2 a} \]
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Rubi [A] time = 0.170956, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {3029, 2977, 2734} \[ \frac{2 (B-C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 B-3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 B-3 C)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2977
Rule 2734
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx &=\int \frac{\cos ^2(c+d x) (B+C \cos (c+d x))}{a+a \cos (c+d x)} \, dx\\ &=\frac{(B-C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int \cos (c+d x) (2 a (B-C)-a (2 B-3 C) \cos (c+d x)) \, dx}{a^2}\\ &=-\frac{(2 B-3 C) x}{2 a}+\frac{2 (B-C) \sin (c+d x)}{a d}-\frac{(2 B-3 C) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac{(B-C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.439461, size = 197, normalized size = 1.99 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-4 d x (2 B-3 C) \cos \left (c+\frac{d x}{2}\right )+4 B \sin \left (c+\frac{d x}{2}\right )+4 B \sin \left (c+\frac{3 d x}{2}\right )+4 B \sin \left (2 c+\frac{3 d x}{2}\right )-4 d x (2 B-3 C) \cos \left (\frac{d x}{2}\right )+20 B \sin \left (\frac{d x}{2}\right )-4 C \sin \left (c+\frac{d x}{2}\right )-3 C \sin \left (c+\frac{3 d x}{2}\right )-3 C \sin \left (2 c+\frac{3 d x}{2}\right )+C \sin \left (2 c+\frac{5 d x}{2}\right )+C \sin \left (3 c+\frac{5 d x}{2}\right )-20 C \sin \left (\frac{d x}{2}\right )\right )}{8 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 211, normalized size = 2.1 \begin{align*}{\frac{B}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-3\,{\frac{C \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}B}{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+2\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{ad}}+3\,{\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 = 1.90977, size = 304, normalized size = 3.07 \begin{align*} -\frac{C{\left (\frac{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac{3 \, \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 )}}\right )} + B{\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 )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59427, size = 204, normalized size = 2.06 \begin{align*} -\frac{{\left (2 \, B - 3 \, C\right )} d x \cos \left (d x + c\right ) +{\left (2 \, B - 3 \, C\right )} d x -{\left (C \cos \left (d x + c\right )^{2} +{\left (2 \, B - C\right )} \cos \left (d x + c\right ) + 4 \, B - 4 \, C\right )} \sin \left (d x + c\right )}{2 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.48015, size = 668, normalized size = 6.75 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.61571, size = 167, normalized size = 1.69 \begin{align*} -\frac{\frac{{\left (d x + c\right )}{\left (2 \, B - 3 \, C\right )}}{a} - \frac{2 \,{\left (B \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 )\right )}}{a} - \frac{2 \,{\left (2 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B \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 )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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