Optimal. Leaf size=102 \[ \frac{(2 A+3 B) \sin (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(2 A+3 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.0788386, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2750, 2650, 2648} \[ \frac{(2 A+3 B) \sin (c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(2 A+3 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx &=\frac{(A-B) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{(2 A+3 B) \int \frac{1}{(a+a \cos (c+d x))^2} \, dx}{5 a}\\ &=\frac{(A-B) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{(2 A+3 B) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{(2 A+3 B) \int \frac{1}{a+a \cos (c+d x)} \, dx}{15 a^2}\\ &=\frac{(A-B) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{(2 A+3 B) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{(2 A+3 B) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.259017, size = 96, normalized size = 0.94 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left ((2 A+3 B) \left (5 \sin \left (c+\frac{3 d x}{2}\right )+\sin \left (2 c+\frac{5 d x}{2}\right )\right )+5 (4 A+3 B) \sin \left (\frac{d x}{2}\right )-15 B \sin \left (c+\frac{d x}{2}\right )\right )}{30 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 64, normalized size = 0.6 \begin{align*}{\frac{1}{4\,d{a}^{3}} \left ({\frac{A-B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{2\,A}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+A\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +B\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 = 1.0223, size = 155, normalized size = 1.52 \begin{align*} \frac{\frac{A{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac{3 \, B{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33754, size = 227, normalized size = 2.23 \begin{align*} \frac{{\left ({\left (2 \, A + 3 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (2 \, A + 3 \, B\right )} \cos \left (d x + c\right ) + 7 \, A + 3 \, B\right )} \sin \left (d x + c\right )}{15 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.66219, size = 114, normalized size = 1.12 \begin{align*} \begin{cases} \frac{A \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{20 a^{3} d} + \frac{A \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{3} d} + \frac{A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{4 a^{3} d} - \frac{B \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{20 a^{3} d} + \frac{B \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{4 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + B \cos{\left (c \right )}\right )}{\left (a \cos{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20609, size = 101, normalized size = 0.99 \begin{align*} \frac{3 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 10 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{60 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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