Optimal. Leaf size=154 \[ \frac {(12 A-215 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac {(6 A-55 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {B x}{a^4}+\frac {(A-B) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {(3 A-10 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.50, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2977, 2968, 3019, 2735, 2648} \[ \frac {(12 A-215 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac {(6 A-55 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {B x}{a^4}+\frac {(A-B) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {(3 A-10 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rule 2968
Rule 2977
Rule 3019
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx &=\frac {(A-B) \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) (3 a (A-B)+7 a B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A-10 B) \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 (3 A-10 B)+35 a^2 B \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A-10 B) \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 (3 A-10 B) \cos (c+d x)+35 a^2 B \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(6 A-55 B) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A-10 B) \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 (6 A-55 B)-105 a^3 B \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac {B x}{a^4}-\frac {(6 A-55 B) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A-10 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(12 A-215 B) \int \frac {1}{a+a \cos (c+d x)} \, dx}{105 a^3}\\ &=\frac {B x}{a^4}-\frac {(6 A-55 B) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A-10 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(12 A-215 B) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.83, size = 329, normalized size = 2.14 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (-1260 A \sin \left (c+\frac {d x}{2}\right )+882 A \sin \left (c+\frac {3 d x}{2}\right )-630 A \sin \left (2 c+\frac {3 d x}{2}\right )+294 A \sin \left (2 c+\frac {5 d x}{2}\right )-210 A \sin \left (3 c+\frac {5 d x}{2}\right )+72 A \sin \left (3 c+\frac {7 d x}{2}\right )+1260 A \sin \left (\frac {d x}{2}\right )+8260 B \sin \left (c+\frac {d x}{2}\right )-7140 B \sin \left (c+\frac {3 d x}{2}\right )+3780 B \sin \left (2 c+\frac {3 d x}{2}\right )-2800 B \sin \left (2 c+\frac {5 d x}{2}\right )+840 B \sin \left (3 c+\frac {5 d x}{2}\right )-520 B \sin \left (3 c+\frac {7 d x}{2}\right )+3675 B d x \cos \left (c+\frac {d x}{2}\right )+2205 B d x \cos \left (c+\frac {3 d x}{2}\right )+2205 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+735 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+735 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+105 B d x \cos \left (3 c+\frac {7 d x}{2}\right )+105 B d x \cos \left (4 c+\frac {7 d x}{2}\right )-9940 B \sin \left (\frac {d x}{2}\right )+3675 B d x \cos \left (\frac {d x}{2}\right )\right )}{13440 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 180, normalized size = 1.17 \[ \frac {105 \, B d x \cos \left (d x + c\right )^{4} + 420 \, B d x \cos \left (d x + c\right )^{3} + 630 \, B d x \cos \left (d x + c\right )^{2} + 420 \, B d x \cos \left (d x + c\right ) + 105 \, B d x + {\left (4 \, {\left (9 \, A - 65 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (39 \, A - 620 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (24 \, A - 535 \, B\right )} \cos \left (d x + c\right ) + 6 \, A - 160 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 155, normalized size = 1.01 \[ \frac {\frac {840 \, {\left (d x + c\right )} B}{a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 177, normalized size = 1.15 \[ -\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}+\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {3 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{8 d \,a^{4}}+\frac {11 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {15 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 201, normalized size = 1.31 \[ -\frac {5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 162, normalized size = 1.05 \[ \frac {B\,x}{a^4}+\frac {\left (\frac {12\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}-\frac {52\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {16\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}-\frac {23\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {9\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}-\frac {5\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.27, size = 192, normalized size = 1.25 \[ \begin {cases} - \frac {A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {B x}{a^{4}} + \frac {B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {11 B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} - \frac {15 B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\relax (c )}\right ) \cos ^{3}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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