Integrand size = 24, antiderivative size = 205 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {1155 a^8 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {1155 i a^8 \sec (c+d x)}{8 d}+\frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac {33 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{4 d}+\frac {77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac {385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d} \]
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Time = 0.26 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3577, 3579, 3567, 3855} \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {1155 a^8 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {1155 i a^8 \sec (c+d x)}{8 d}+\frac {385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}+\frac {77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}+\frac {33 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{4 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d} \]
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Rule 3567
Rule 3577
Rule 3579
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}-\frac {1}{3} \left (11 a^2\right ) \int \cos (c+d x) (a+i a \tan (c+d x))^6 \, dx \\ & = \frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\left (33 a^4\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^4 \, dx \\ & = \frac {33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac {1}{4} \left (231 a^5\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx \\ & = \frac {33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac {77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac {1}{4} \left (385 a^6\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx \\ & = \frac {33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac {77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac {385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}+\frac {1}{8} \left (1155 a^7\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx \\ & = \frac {1155 i a^8 \sec (c+d x)}{8 d}+\frac {33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac {77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac {385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}+\frac {1}{8} \left (1155 a^8\right ) \int \sec (c+d x) \, dx \\ & = \frac {1155 a^8 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {1155 i a^8 \sec (c+d x)}{8 d}+\frac {33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac {77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac {385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1540\) vs. \(2(205)=410\).
Time = 8.30 (sec) , antiderivative size = 1540, normalized size of antiderivative = 7.51 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {1155 \cos (8 c) \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}{8 d (\cos (d x)+i \sin (d x))^8}+\frac {1155 \cos (8 c) \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}{8 d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (3 d x) \cos ^8(c+d x) \left (-\frac {32}{3} i \cos (5 c)-\frac {32}{3} \sin (5 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos (d x) \cos ^8(c+d x) (160 i \cos (7 c)+160 \sin (7 c)) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {1155 i \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sin (8 c) (a+i a \tan (c+d x))^8}{8 d (\cos (d x)+i \sin (d x))^8}-\frac {1155 i \cos ^8(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sin (8 c) (a+i a \tan (c+d x))^8}{8 d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \sec (c) \left (\frac {236}{3} i \cos (8 c)+\frac {236}{3} \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) (-160 \cos (7 c)+160 i \sin (7 c)) \sin (d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \left (\frac {32}{3} \cos (5 c)-\frac {32}{3} i \sin (5 c)\right ) \sin (3 d x) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8}+\frac {\cos ^8(c+d x) \left (\frac {1}{16} \cos (8 c)-\frac {1}{16} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4}-\frac {i \cos ^8(c+d x) \left (\frac {4}{3} \cos (8 c)-\frac {4}{3} i \sin (8 c)\right ) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^8(c+d x) \left ((-375-32 i) \cos \left (\frac {c}{2}\right )+(375-32 i) \sin \left (\frac {c}{2}\right )\right ) \left (\frac {1}{48} \cos (8 c)-\frac {1}{48} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {i \cos ^8(c+d x) \left (\frac {236}{3} \cos (8 c)-\frac {236}{3} i \sin (8 c)\right ) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {\cos ^8(c+d x) \left (-\frac {1}{16} \cos (8 c)+\frac {1}{16} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4}+\frac {i \cos ^8(c+d x) \left (\frac {4}{3} \cos (8 c)-\frac {4}{3} i \sin (8 c)\right ) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^8(c+d x) \left ((375-32 i) \cos \left (\frac {c}{2}\right )+(375+32 i) \sin \left (\frac {c}{2}\right )\right ) \left (\frac {1}{48} \cos (8 c)-\frac {1}{48} i \sin (8 c)\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}-\frac {i \cos ^8(c+d x) \left (\frac {236}{3} \cos (8 c)-\frac {236}{3} i \sin (8 c)\right ) \sin \left (\frac {d x}{2}\right ) (a+i a \tan (c+d x))^8}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8 \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]
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Time = 253.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {32 i a^{8} {\mathrm e}^{3 i \left (d x +c \right )}}{3 d}+\frac {160 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {i a^{8} \left (2295 \,{\mathrm e}^{7 i \left (d x +c \right )}+5855 \,{\mathrm e}^{5 i \left (d x +c \right )}+5153 \,{\mathrm e}^{3 i \left (d x +c \right )}+1545 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {1155 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {1155 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(147\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {56 i a^{8} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-\frac {8 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right )}{3}-\frac {28 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3}+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+\frac {a^{8} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(409\) |
default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {56 i a^{8} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-\frac {8 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right )}{3}-\frac {28 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3}+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+\frac {a^{8} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(409\) |
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Time = 0.25 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.39 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-256 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} + 2816 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} + 18414 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} + 33726 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} + 25410 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} + 6930 i \, a^{8} e^{\left (i \, d x + i \, c\right )} + 3465 \, {\left (a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3465 \, {\left (a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{24 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Time = 0.46 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.35 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {1155 a^{8} \left (- \frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{8} + \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{8}\right )}{d} + \frac {2295 i a^{8} e^{7 i c} e^{7 i d x} + 5855 i a^{8} e^{5 i c} e^{5 i d x} + 5153 i a^{8} e^{3 i c} e^{3 i d x} + 1545 i a^{8} e^{i c} e^{i d x}}{12 d e^{8 i c} e^{8 i d x} + 48 d e^{6 i c} e^{6 i d x} + 72 d e^{4 i c} e^{4 i d x} + 48 d e^{2 i c} e^{2 i d x} + 12 d} + \begin {cases} \frac {- 32 i a^{8} d e^{3 i c} e^{3 i d x} + 480 i a^{8} d e^{i c} e^{i d x}}{3 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (32 a^{8} e^{3 i c} - 160 a^{8} e^{i c}\right ) & \text {otherwise} \end {cases} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (169) = 338\).
Time = 0.29 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.72 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {128 i \, a^{8} \cos \left (d x + c\right )^{3} + 448 \, a^{8} \sin \left (d x + c\right )^{3} + 896 i \, {\left (\cos \left (d x + c\right )^{3} - \frac {3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{8} + 128 i \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{8} + 896 i \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{8} + {\left (16 \, \sin \left (d x + c\right )^{3} - \frac {6 \, {\left (13 \, \sin \left (d x + c\right )^{3} - 11 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, \sin \left (d x + c\right )\right )} a^{8} + 112 \, {\left (4 \, \sin \left (d x + c\right )^{3} - \frac {6 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, \sin \left (d x + c\right )\right )} a^{8} + 560 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{8} + 16 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{8}}{48 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2835 vs. \(2 (169) = 338\).
Time = 1.35 (sec) , antiderivative size = 2835, normalized size of antiderivative = 13.83 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]
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Time = 8.75 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.67 \[ \int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {\frac {1147\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{4}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,3505{}\mathrm {i}}{4}-\frac {5639\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,3585{}\mathrm {i}+\frac {25993\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{6}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,33847{}\mathrm {i}}{6}-4575\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,12041{}\mathrm {i}}{3}+\frac {27565\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{12}+\frac {a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,4293{}\mathrm {i}}{4}-\frac {1360\,a^8}{3}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,3{}\mathrm {i}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,13{}\mathrm {i}-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,22{}\mathrm {i}+22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,18{}\mathrm {i}-13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,7{}\mathrm {i}+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )}+\frac {1155\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d} \]
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