Integrand size = 19, antiderivative size = 153 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{a^5 d}-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {488 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
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Time = 0.45 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2845, 3057, 12, 3855} \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{a^5 d}-\frac {488 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac {173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {34 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac {13 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]
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Rule 12
Rule 2845
Rule 3057
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {\int \frac {(9 a-4 a \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx}{9 a^2} \\ & = -\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {\int \frac {\left (63 a^2-39 a^2 \cos (c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{63 a^4} \\ & = -\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (315 a^3-204 a^3 \cos (c+d x)\right ) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6} \\ & = -\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\left (945 a^4-519 a^4 \cos (c+d x)\right ) \sec (c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8} \\ & = -\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {488 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac {\int 945 a^5 \sec (c+d x) \, dx}{945 a^{10}} \\ & = -\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {488 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac {\int \sec (c+d x) \, dx}{a^5} \\ & = \frac {\text {arctanh}(\sin (c+d x))}{a^5 d}-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}-\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {488 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.38 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (80640 \cos ^9\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec \left (\frac {c}{2}\right ) \left (35973 \sin \left (\frac {d x}{2}\right )-25515 \sin \left (c+\frac {d x}{2}\right )+29757 \sin \left (c+\frac {3 d x}{2}\right )-11235 \sin \left (2 c+\frac {3 d x}{2}\right )+14733 \sin \left (2 c+\frac {5 d x}{2}\right )-2835 \sin \left (3 c+\frac {5 d x}{2}\right )+4077 \sin \left (3 c+\frac {7 d x}{2}\right )-315 \sin \left (4 c+\frac {7 d x}{2}\right )+488 \sin \left (4 c+\frac {9 d x}{2}\right )\right )\right )}{2520 a^5 d (1+\cos (c+d x))^5} \]
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Time = 1.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d \,a^{5}}\) | \(101\) |
default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d \,a^{5}}\) | \(101\) |
parallelrisch | \(\frac {-35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-270 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1008 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2730 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-5040 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+5040 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{5040 a^{5} d}\) | \(101\) |
norman | \(\frac {-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}-\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d a}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d a}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d a}}{a^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5} d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5} d}\) | \(139\) |
risch | \(-\frac {2 i \left (315 \,{\mathrm e}^{8 i \left (d x +c \right )}+2835 \,{\mathrm e}^{7 i \left (d x +c \right )}+11235 \,{\mathrm e}^{6 i \left (d x +c \right )}+25515 \,{\mathrm e}^{5 i \left (d x +c \right )}+35973 \,{\mathrm e}^{4 i \left (d x +c \right )}+29757 \,{\mathrm e}^{3 i \left (d x +c \right )}+14733 \,{\mathrm e}^{2 i \left (d x +c \right )}+4077 \,{\mathrm e}^{i \left (d x +c \right )}+488\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{5} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{5} d}\) | \(155\) |
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Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.61 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {315 \, {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (488 \, \cos \left (d x + c\right )^{4} + 2125 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2740 \, \cos \left (d x + c\right ) + 863\right )} \sin \left (d x + c\right )}{630 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
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\[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\int \frac {\sec {\left (c + d x \right )}}{\cos ^{5}{\left (c + d x \right )} + 5 \cos ^{4}{\left (c + d x \right )} + 10 \cos ^{3}{\left (c + d x \right )} + 10 \cos ^{2}{\left (c + d x \right )} + 5 \cos {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
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Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.04 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {5040 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac {5040 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{5}}}{5040 \, d} \]
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Time = 0.44 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.82 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {5040 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac {5040 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]
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Time = 14.54 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.65 \[ \int \frac {\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^5}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5\,a^5}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^5}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{144\,a^5}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^5}+\frac {31\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^5}}{d} \]
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