Integrand size = 31, antiderivative size = 232 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(21 A-8 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac {(21 A-8 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))}-\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3} \]
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Time = 1.02 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3057, 2827, 3853, 3855, 3852, 8} \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(21 A-8 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac {(21 A-8 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac {4 (216 A-83 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac {(129 A-52 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac {(A-B) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rule 8
Rule 2827
Rule 3057
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(a (9 A-2 B)-5 a (A-B) \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (a^2 (73 A-24 B)-28 a^2 (2 A-B) \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (a^3 (477 A-176 B)-3 a^3 (129 A-52 B) \cos (c+d x)\right ) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \left (105 a^4 (21 A-8 B)-8 a^4 (216 A-83 B) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{105 a^8} \\ & = -\frac {(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(8 (216 A-83 B)) \int \sec ^2(c+d x) \, dx}{105 a^4}+\frac {(21 A-8 B) \int \sec ^3(c+d x) \, dx}{a^4} \\ & = \frac {(21 A-8 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {(21 A-8 B) \int \sec (c+d x) \, dx}{2 a^4}+\frac {(8 (216 A-83 B)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = \frac {(21 A-8 B) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac {(21 A-8 B) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(129 A-52 B) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {4 (216 A-83 B) \sec (c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(798\) vs. \(2(232)=464\).
Time = 8.17 (sec) , antiderivative size = 798, normalized size of antiderivative = 3.44 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {8 (21 A-8 B) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^4}+\frac {8 (21 A-8 B) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^4}+\frac {\cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (73206 A \sin \left (\frac {d x}{2}\right )-38668 B \sin \left (\frac {d x}{2}\right )-166668 A \sin \left (\frac {3 d x}{2}\right )+64384 B \sin \left (\frac {3 d x}{2}\right )+183162 A \sin \left (c-\frac {d x}{2}\right )-70896 B \sin \left (c-\frac {d x}{2}\right )-100842 A \sin \left (c+\frac {d x}{2}\right )+50316 B \sin \left (c+\frac {d x}{2}\right )+155526 A \sin \left (2 c+\frac {d x}{2}\right )-59248 B \sin \left (2 c+\frac {d x}{2}\right )+37380 A \sin \left (c+\frac {3 d x}{2}\right )-22820 B \sin \left (c+\frac {3 d x}{2}\right )-101148 A \sin \left (2 c+\frac {3 d x}{2}\right )+48004 B \sin \left (2 c+\frac {3 d x}{2}\right )+102900 A \sin \left (3 c+\frac {3 d x}{2}\right )-39200 B \sin \left (3 c+\frac {3 d x}{2}\right )-119364 A \sin \left (c+\frac {5 d x}{2}\right )+46032 B \sin \left (c+\frac {5 d x}{2}\right )+8820 A \sin \left (2 c+\frac {5 d x}{2}\right )-8750 B \sin \left (2 c+\frac {5 d x}{2}\right )-78204 A \sin \left (3 c+\frac {5 d x}{2}\right )+35742 B \sin \left (3 c+\frac {5 d x}{2}\right )+49980 A \sin \left (4 c+\frac {5 d x}{2}\right )-19040 B \sin \left (4 c+\frac {5 d x}{2}\right )-64053 A \sin \left (2 c+\frac {7 d x}{2}\right )+24664 B \sin \left (2 c+\frac {7 d x}{2}\right )-3885 A \sin \left (3 c+\frac {7 d x}{2}\right )-1050 B \sin \left (3 c+\frac {7 d x}{2}\right )-44733 A \sin \left (4 c+\frac {7 d x}{2}\right )+19834 B \sin \left (4 c+\frac {7 d x}{2}\right )+15435 A \sin \left (5 c+\frac {7 d x}{2}\right )-5880 B \sin \left (5 c+\frac {7 d x}{2}\right )-21987 A \sin \left (3 c+\frac {9 d x}{2}\right )+8456 B \sin \left (3 c+\frac {9 d x}{2}\right )-3675 A \sin \left (4 c+\frac {9 d x}{2}\right )+630 B \sin \left (4 c+\frac {9 d x}{2}\right )-16107 A \sin \left (5 c+\frac {9 d x}{2}\right )+6986 B \sin \left (5 c+\frac {9 d x}{2}\right )+2205 A \sin \left (6 c+\frac {9 d x}{2}\right )-840 B \sin \left (6 c+\frac {9 d x}{2}\right )-3456 A \sin \left (4 c+\frac {11 d x}{2}\right )+1328 B \sin \left (4 c+\frac {11 d x}{2}\right )-840 A \sin \left (5 c+\frac {11 d x}{2}\right )+210 B \sin \left (5 c+\frac {11 d x}{2}\right )-2616 A \sin \left (6 c+\frac {11 d x}{2}\right )+1118 B \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{6720 d (a+a \cos (c+d x))^4} \]
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Time = 1.54 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {-70560 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {8 B}{21}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+70560 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {8 B}{21}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-11619 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {23540 A}{3873}-\frac {3040 B}{1291}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {3992 A}{1291}-\frac {13864 B}{11619}\right ) \cos \left (3 d x +3 c \right )+\left (A -\frac {4472 B}{11619}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {192 A}{1291}-\frac {664 B}{11619}\right ) \cos \left (5 d x +5 c \right )+\left (\frac {34168 A}{3873}-\frac {39952 B}{11619}\right ) \cos \left (d x +c \right )+\frac {19387 A}{3873}-\frac {22888 B}{11619}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 d \,a^{4} \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(192\) |
derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {9 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-111 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-84 A +32 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 A -32 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(234\) |
default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {9 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-111 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-84 A +32 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 A -32 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(234\) |
norman | \(\frac {-\frac {\left (A -B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {\left (29 A -22 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}-\frac {\left (167 A -65 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (171 A -62 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {\left (1161 A -643 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{840 a d}-\frac {\left (2529 A -1052 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{210 a d}+\frac {\left (2913 A -1069 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} a^{3}}-\frac {\left (21 A -8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{4} d}+\frac {\left (21 A -8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{4} d}\) | \(269\) |
risch | \(-\frac {i \left (2205 A \,{\mathrm e}^{10 i \left (d x +c \right )}-840 B \,{\mathrm e}^{10 i \left (d x +c \right )}+15435 A \,{\mathrm e}^{9 i \left (d x +c \right )}-5880 B \,{\mathrm e}^{9 i \left (d x +c \right )}+49980 A \,{\mathrm e}^{8 i \left (d x +c \right )}-19040 B \,{\mathrm e}^{8 i \left (d x +c \right )}+102900 A \,{\mathrm e}^{7 i \left (d x +c \right )}-39200 B \,{\mathrm e}^{7 i \left (d x +c \right )}+155526 A \,{\mathrm e}^{6 i \left (d x +c \right )}-59248 B \,{\mathrm e}^{6 i \left (d x +c \right )}+183162 A \,{\mathrm e}^{5 i \left (d x +c \right )}-70896 B \,{\mathrm e}^{5 i \left (d x +c \right )}+166668 A \,{\mathrm e}^{4 i \left (d x +c \right )}-64384 B \,{\mathrm e}^{4 i \left (d x +c \right )}+119364 A \,{\mathrm e}^{3 i \left (d x +c \right )}-46032 B \,{\mathrm e}^{3 i \left (d x +c \right )}+64053 A \,{\mathrm e}^{2 i \left (d x +c \right )}-24664 B \,{\mathrm e}^{2 i \left (d x +c \right )}+21987 A \,{\mathrm e}^{i \left (d x +c \right )}-8456 B \,{\mathrm e}^{i \left (d x +c \right )}+3456 A -1328 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {21 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}-\frac {21 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{4} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}\) | \(372\) |
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Time = 0.32 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, {\left ({\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A - 8 \, B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (216 \, A - 83 \, B\right )} \cos \left (d x + c\right )^{5} + {\left (11619 \, A - 4472 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3411 \, A - 1318 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (1509 \, A - 592 \, B\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (2 \, A - B\right )} \cos \left (d x + c\right ) - 105 \, A\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
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\[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{3}{\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Time = 0.21 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.81 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {3 \, A {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \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}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (21 \, A - 8 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (21 \, A - 8 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (9\,A-2\,B\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A-2\,B\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,A}{2\,a^4}+\frac {5\,\left (A-B\right )}{4\,a^4}+\frac {3\,\left (6\,A-4\,B\right )}{4\,a^4}+\frac {3\,\left (15\,A-5\,B\right )}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^4}+\frac {6\,A-4\,B}{8\,a^4}+\frac {15\,A-5\,B}{24\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A-B\right )}{40\,a^4}+\frac {6\,A-4\,B}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (21\,A-8\,B\right )}{a^4\,d} \]
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