Optimal. Leaf size=148 \[ \frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.175611, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {3019, 2750, 2650, 2648} \[ \frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 3019
Rule 2750
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{\int \frac{-a (3 A+4 B-4 C)-7 a C \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(6 A+8 B+13 C) \int \frac{1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2}\\ &=\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{(6 A+8 B+13 C) \int \frac{1}{a+a \cos (c+d x)} \, dx}{105 a^3}\\ &=\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.457853, size = 208, normalized size = 1.41 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (70 (3 A+2 B+4 C) \sin \left (\frac{d x}{2}\right )+126 A \sin \left (c+\frac{3 d x}{2}\right )+42 A \sin \left (2 c+\frac{5 d x}{2}\right )+6 A \sin \left (3 c+\frac{7 d x}{2}\right )-35 (4 B+5 C) \sin \left (c+\frac{d x}{2}\right )+168 B \sin \left (c+\frac{3 d x}{2}\right )+56 B \sin \left (2 c+\frac{5 d x}{2}\right )+8 B \sin \left (3 c+\frac{7 d x}{2}\right )+168 C \sin \left (c+\frac{3 d x}{2}\right )-105 C \sin \left (2 c+\frac{3 d x}{2}\right )+91 C \sin \left (2 c+\frac{5 d x}{2}\right )+13 C \sin \left (3 c+\frac{7 d x}{2}\right )\right )}{420 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 106, normalized size = 0.7 \begin{align*}{\frac{1}{8\,d{a}^{4}} \left ({\frac{A-B+C}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3\,A-C-B}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3\,A+B-C}{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 ) +C\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 = 1.0392, size = 350, normalized size = 2.36 \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{C{\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 = 1.78031, size = 343, normalized size = 2.32 \begin{align*} \frac{{\left ({\left (6 \, A + 8 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (6 \, A + 8 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (39 \, A + 52 \, B + 32 \, C\right )} \cos \left (d x + c\right ) + 36 \, A + 13 \, B + 8 \, C\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 = 16.4384, size = 264, normalized size = 1.78 \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} + \frac{C \tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{56 a^{4} d} - \frac{C \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{40 a^{4} d} - \frac{C \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{24 a^{4} d} + \frac{C \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 )} + C \cos ^{2}{\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 = 1.17779, size = 231, normalized size = 1.56 \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} + 15 \, C \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} - 21 \, C \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} - 35 \, C \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 ) + 105 \, C \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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