Optimal. Leaf size=138 \[ \frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.137988, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2750, 2650, 2648} \[ \frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cos (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{(3 A+4 B) \int \frac{1}{(a+a \cos (c+d x))^3} \, dx}{7 a}\\ &=\frac{(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(2 (3 A+4 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{(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{(2 (3 A+4 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{(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.324019, size = 109, normalized size = 0.79 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left ((3 A+4 B) \left (21 \sin \left (c+\frac{3 d x}{2}\right )+7 \sin \left (2 c+\frac{5 d x}{2}\right )+\sin \left (3 c+\frac{7 d x}{2}\right )\right )+35 (3 A+2 B) \sin \left (\frac{d x}{2}\right )-70 B \sin \left (c+\frac{d x}{2}\right )\right )}{210 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, 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{3\,A-B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3\,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 = 1.00371, size = 236, normalized size = 1.71 \begin{align*} \frac{\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}} + \frac{3 \, A{\left (\frac{35 \, \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{5 \, \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 = 1.40174, size = 311, normalized size = 2.25 \begin{align*} \frac{{\left (2 \,{\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{2} + 13 \,{\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 36 \, A + 13 \, 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 = 8.08858, size = 177, normalized size = 1.28 \begin{align*} \begin{cases} \frac{A \tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{56 a^{4} d} + \frac{3 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 )}}{8 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 )}{\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 = 1.18987, 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} + 63 \, 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} + 105 \, 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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