Optimal. Leaf size=184 \[ \frac {146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac {268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac {130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac {118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac {14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5} \]
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Rubi [A] time = 0.41, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2765, 2977, 2968, 3019, 2750, 2648} \[ -\frac {118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}+\frac {146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac {268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac {130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac {\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac {14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2750
Rule 2765
Rule 2968
Rule 2977
Rule 3019
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {\int \frac {\cos ^3(c+d x) (4 a-10 a \cos (c+d x))}{(a+a \cos (c+d x))^5} \, dx}{11 a^2}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos ^2(c+d x) \left (42 a^2-76 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx}{99 a^4}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos (c+d x) \left (236 a^3-414 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{693 a^6}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {\int \frac {236 a^3 \cos (c+d x)-414 a^3 \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{693 a^6}\\ &=\frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}+\frac {\int \frac {-1950 a^4+2070 a^4 \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{3465 a^8}\\ &=\frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {268 \sin (c+d x)}{693 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac {146 \int \frac {1}{a+a \cos (c+d x)} \, dx}{693 a^5}\\ &=\frac {130 \sin (c+d x)}{693 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^4(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {14 \cos ^3(c+d x) \sin (c+d x)}{99 a d (a+a \cos (c+d x))^5}-\frac {118 \cos ^2(c+d x) \sin (c+d x)}{693 a^2 d (a+a \cos (c+d x))^4}-\frac {268 \sin (c+d x)}{693 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac {146 \sin (c+d x)}{693 d \left (a^6+a^6 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 164, normalized size = 0.89 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (-33726 \sin \left (c+\frac {d x}{2}\right )+25080 \sin \left (c+\frac {3 d x}{2}\right )-23100 \sin \left (2 c+\frac {3 d x}{2}\right )+12540 \sin \left (2 c+\frac {5 d x}{2}\right )-11550 \sin \left (3 c+\frac {5 d x}{2}\right )+4565 \sin \left (3 c+\frac {7 d x}{2}\right )-3465 \sin \left (4 c+\frac {7 d x}{2}\right )+913 \sin \left (4 c+\frac {9 d x}{2}\right )-693 \sin \left (5 c+\frac {9 d x}{2}\right )+146 \sin \left (5 c+\frac {11 d x}{2}\right )+33726 \sin \left (\frac {d x}{2}\right )\right ) \sec ^{11}\left (\frac {1}{2} (c+d x)\right )}{709632 a^6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 147, normalized size = 0.80 \[ \frac {{\left (146 \, \cos \left (d x + c\right )^{5} + 183 \, \cos \left (d x + c\right )^{4} + 184 \, \cos \left (d x + c\right )^{3} + 124 \, \cos \left (d x + c\right )^{2} + 48 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{693 \, {\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 85, normalized size = 0.46 \[ -\frac {63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{22176 \, a^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 84, normalized size = 0.46 \[ \frac {-\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 127, normalized size = 0.69 \[ \frac {\frac {693 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1386 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{22176 \, a^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 75, normalized size = 0.41 \[ \frac {\frac {495\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {495\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{16}+\frac {275\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {55\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{8}+\frac {73\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{16}}{22176\,a^6\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 39.01, size = 129, normalized size = 0.70 \[ \begin {cases} - \frac {\tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{352 a^{6} d} + \frac {5 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{288 a^{6} d} - \frac {5 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{112 a^{6} d} + \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{6} d} - \frac {5 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{96 a^{6} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{6}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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