Optimal. Leaf size=138 \[ \frac{(8 A+13 B) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.21469, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2968, 3019, 2750, 2650, 2648} \[ \frac{(8 A+13 B) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3019
Rule 2750
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx &=\int \frac{A \cos (c+d x)+B \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx\\ &=-\frac{(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{\int \frac{-4 a (A-B)-7 a B \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(8 A+13 B) \int \frac{1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{(8 A+13 B) \int \frac{1}{a+a \cos (c+d x)} \, dx}{105 a^3}\\ &=-\frac{(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.358649, size = 163, normalized size = 1.18 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-35 (4 A+5 B) \sin \left (c+\frac{d x}{2}\right )+140 (A+2 B) \sin \left (\frac{d x}{2}\right )+168 A \sin \left (c+\frac{3 d x}{2}\right )+56 A \sin \left (2 c+\frac{5 d x}{2}\right )+8 A \sin \left (3 c+\frac{7 d x}{2}\right )+168 B \sin \left (c+\frac{3 d x}{2}\right )-105 B \sin \left (2 c+\frac{3 d x}{2}\right )+91 B \sin \left (2 c+\frac{5 d x}{2}\right )+13 B \sin \left (3 c+\frac{7 d x}{2}\right )\right )}{420 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 88, normalized size = 0.6 \begin{align*}{\frac{1}{8\,d{a}^{4}} \left ({\frac{-A+B}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{-A-B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{A-B}{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 = 1.00994, size = 235, normalized size = 1.7 \begin{align*} \frac{\frac{A{\left (\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac{B{\left (\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.297, size = 309, normalized size = 2.24 \begin{align*} \frac{{\left ({\left (8 \, A + 13 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (8 \, A + 13 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (13 \, A + 8 \, B\right )} \cos \left (d x + c\right ) + 13 \, A + 8 \, B\right )} \sin \left (d x + c\right )}{105 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.4166, size = 178, normalized size = 1.29 \begin{align*} \begin{cases} - \frac{A \tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{56 a^{4} d} - \frac{A \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{40 a^{4} d} + \frac{A \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{24 a^{4} d} + \frac{A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 a^{4} d} + \frac{B \tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{56 a^{4} d} - \frac{B \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{40 a^{4} d} - \frac{B \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{24 a^{4} d} + \frac{B \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + B \cos{\left (c \right )}\right ) \cos{\left (c \right )}}{\left (a \cos{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17708, size = 158, normalized size = 1.14 \begin{align*} -\frac{15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 21 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 21 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 35 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 105 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{840 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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