Optimal. Leaf size=164 \[ \frac{(16 A+12 B-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.478351, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {3041, 2977, 2968, 3019, 2735, 2648} \[ \frac{(16 A+12 B-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3041
Rule 2977
Rule 2968
Rule 3019
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{\cos ^2(c+d x) (a (4 A+3 B-3 C)+7 a C \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A+3 B-10 C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\cos (c+d x) \left (2 a^2 (4 A+3 B-10 C)+35 a^2 C \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A+3 B-10 C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{2 a^2 (4 A+3 B-10 C) \cos (c+d x)+35 a^2 C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A+3 B-10 C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{\int \frac{-2 a^3 (8 A+6 B-55 C)-105 a^3 C \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac{C x}{a^4}-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A+3 B-10 C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(16 A+12 B-215 C) \int \frac{1}{a+a \cos (c+d x)} \, dx}{105 a^3}\\ &=\frac{C x}{a^4}-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(4 A+3 B-10 C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(16 A+12 B-215 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 0.901815, size = 405, normalized size = 2.47 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) \left (-350 A \sin \left (c+\frac{d x}{2}\right )+336 A \sin \left (c+\frac{3 d x}{2}\right )-210 A \sin \left (2 c+\frac{3 d x}{2}\right )+182 A \sin \left (2 c+\frac{5 d x}{2}\right )+26 A \sin \left (3 c+\frac{7 d x}{2}\right )+560 A \sin \left (\frac{d x}{2}\right )-1260 B \sin \left (c+\frac{d x}{2}\right )+882 B \sin \left (c+\frac{3 d x}{2}\right )-630 B \sin \left (2 c+\frac{3 d x}{2}\right )+294 B \sin \left (2 c+\frac{5 d x}{2}\right )-210 B \sin \left (3 c+\frac{5 d x}{2}\right )+72 B \sin \left (3 c+\frac{7 d x}{2}\right )+1260 B \sin \left (\frac{d x}{2}\right )+8260 C \sin \left (c+\frac{d x}{2}\right )-7140 C \sin \left (c+\frac{3 d x}{2}\right )+3780 C \sin \left (2 c+\frac{3 d x}{2}\right )-2800 C \sin \left (2 c+\frac{5 d x}{2}\right )+840 C \sin \left (3 c+\frac{5 d x}{2}\right )-520 C \sin \left (3 c+\frac{7 d x}{2}\right )+3675 C d x \cos \left (c+\frac{d x}{2}\right )+2205 C d x \cos \left (c+\frac{3 d x}{2}\right )+2205 C d x \cos \left (2 c+\frac{3 d x}{2}\right )+735 C d x \cos \left (2 c+\frac{5 d x}{2}\right )+735 C d x \cos \left (3 c+\frac{5 d x}{2}\right )+105 C d x \cos \left (3 c+\frac{7 d x}{2}\right )+105 C d x \cos \left (4 c+\frac{7 d x}{2}\right )-9940 C \sin \left (\frac{d x}{2}\right )+3675 C d x \cos \left (\frac{d x}{2}\right )\right )}{13440 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 255, normalized size = 1.6 \begin{align*}{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{B}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{C}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3\,B}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{C}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{B}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{11\,C}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{15\,C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{d{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57301, size = 386, normalized size = 2.35 \begin{align*} -\frac{5 \, C{\left (\frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{77 \, \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{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{336 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \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{3 \, B{\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 = 2.00151, size = 512, normalized size = 3.12 \begin{align*} \frac{105 \, C d x \cos \left (d x + c\right )^{4} + 420 \, C d x \cos \left (d x + c\right )^{3} + 630 \, C d x \cos \left (d x + c\right )^{2} + 420 \, C d x \cos \left (d x + c\right ) + 105 \, C d x +{\left ({\left (13 \, A + 36 \, B - 260 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (52 \, A + 39 \, B - 620 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (32 \, A + 24 \, B - 535 \, C\right )} \cos \left (d x + c\right ) + 8 \, A + 6 \, B - 160 \, 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 = 35.8602, size = 279, normalized size = 1.7 \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{3 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 )}}{8 a^{4} d} + \frac{B \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{8 a^{4} d} + \frac{C x}{a^{4}} + \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 )}}{8 a^{4} d} + \frac{11 C \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{24 a^{4} d} - \frac{15 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 ) \cos ^{2}{\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 = 1.22266, size = 297, normalized size = 1.81 \begin{align*} \frac{\frac{840 \,{\left (d x + c\right )} C}{a^{4}} + \frac{15 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 21 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 63 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 105 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 35 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 105 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1575 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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