Integrand size = 21, antiderivative size = 176 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=-\frac {241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}+\frac {61 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^2}+\frac {61 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))}-\frac {\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac {9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4} \]
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Time = 0.42 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2844, 3056, 3047, 3098, 2829, 2729, 2727} \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac {61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac {241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac {9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac {4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5} \]
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Rule 2727
Rule 2729
Rule 2829
Rule 2844
Rule 3047
Rule 3056
Rule 3098
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 5.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.43 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\left (16+96 \cos (c+d x)+248 \cos ^2(c+d x)+368 \cos ^3(c+d x)+366 \cos ^4(c+d x)+61 \cos ^5(c+d x)\right ) \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^6} \]
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Time = 0.87 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.48
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\) | \(84\) |
default | \(\frac {\frac {\left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{11}-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d \,a^{6}}\) | \(84\) |
parallelrisch | \(\frac {105 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-385 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+330 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+462 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1155 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1155 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{36960 a^{6} d}\) | \(86\) |
risch | \(\frac {2 i \left (1155 \,{\mathrm e}^{9 i \left (d x +c \right )}+3465 \,{\mathrm e}^{8 i \left (d x +c \right )}+9240 \,{\mathrm e}^{7 i \left (d x +c \right )}+12936 \,{\mathrm e}^{6 i \left (d x +c \right )}+15246 \,{\mathrm e}^{5 i \left (d x +c \right )}+10890 \,{\mathrm e}^{4 i \left (d x +c \right )}+6600 \,{\mathrm e}^{3 i \left (d x +c \right )}+2200 \,{\mathrm e}^{2 i \left (d x +c \right )}+671 \,{\mathrm e}^{i \left (d x +c \right )}+61\right )}{1155 d \,a^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11}}\) | \(124\) |
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Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {{\left (61 \, \cos \left (d x + c\right )^{5} + 366 \, \cos \left (d x + c\right )^{4} + 368 \, \cos \left (d x + c\right )^{3} + 248 \, \cos \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{1155 \, {\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} d\right )}} \]
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Time = 9.67 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\begin {cases} \frac {\tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{352 a^{6} d} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{96 a^{6} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{112 a^{6} d} + \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{80 a^{6} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32 a^{6} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{6}} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\frac {1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {462 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{36960 \, a^{6} d} \]
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Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.48 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 462 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{36960 \, a^{6} d} \]
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Time = 14.79 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx=\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-1155\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+462\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+330\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-385\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\right )}{36960\,a^6\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]
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