Optimal. Leaf size=176 \[ \frac {61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac {61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac {241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac {9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac {4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5} \]
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Rubi [A] time = 0.32, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2765, 2977, 2968, 3019, 2750, 2650, 2648} \[ \frac {61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac {61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac {241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac {9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac {4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2750
Rule 2765
Rule 2968
Rule 2977
Rule 3019
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {\int \frac {\cos ^2(c+d x) (3 a-9 a \cos (c+d x))}{(a+a \cos (c+d x))^5} \, dx}{11 a^2}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos (c+d x) \left (24 a^2-57 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx}{99 a^4}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}-\frac {\int \frac {24 a^2 \cos (c+d x)-57 a^2 \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx}{99 a^4}\\ &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac {9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac {\int \frac {-324 a^3+399 a^3 \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{693 a^6}\\ &=-\frac {241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac {9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac {61 \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{385 a^4}\\ &=-\frac {241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac {9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac {61 \sin (c+d x)}{1155 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac {61 \int \frac {1}{a+a \cos (c+d x)} \, dx}{1155 a^5}\\ &=-\frac {241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac {9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac {61 \sin (c+d x)}{1155 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac {61 \sin (c+d x)}{1155 d \left (a^6+a^6 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 151, normalized size = 0.86 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (-12936 \sin \left (c+\frac {d x}{2}\right )+10890 \sin \left (c+\frac {3 d x}{2}\right )-9240 \sin \left (2 c+\frac {3 d x}{2}\right )+6600 \sin \left (2 c+\frac {5 d x}{2}\right )-3465 \sin \left (3 c+\frac {5 d x}{2}\right )+2200 \sin \left (3 c+\frac {7 d x}{2}\right )-1155 \sin \left (4 c+\frac {7 d x}{2}\right )+671 \sin \left (4 c+\frac {9 d x}{2}\right )+61 \sin \left (5 c+\frac {11 d x}{2}\right )+15246 \sin \left (\frac {d x}{2}\right )\right ) \sec ^{11}\left (\frac {1}{2} (c+d x)\right )}{1182720 a^6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 147, normalized size = 0.84 \[ \frac {{\left (61 \, \cos \left (d x + c\right )^{5} + 366 \, \cos \left (d x + c\right )^{4} + 368 \, \cos \left (d x + c\right )^{3} + 248 \, \cos \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{1155 \, {\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 85, normalized size = 0.48 \[ \frac {105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 462 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{36960 \, a^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 84, normalized size = 0.48 \[ \frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 127, normalized size = 0.72 \[ \frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {462 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{36960 \, a^{6} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 151, normalized size = 0.86 \[ \frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+462\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+330\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-385\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\right )}{36960\,a^6\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.36, size = 124, normalized size = 0.70 \[ \begin {cases} \frac {\tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{352 a^{6} d} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{96 a^{6} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{112 a^{6} d} + \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{80 a^{6} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{6}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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