Integrand size = 24, antiderivative size = 120 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {5 a^4 \sin (c+d x)}{21 d}-\frac {10 a^4 \sin ^3(c+d x)}{63 d}+\frac {a^4 \sin ^5(c+d x)}{21 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d} \]
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Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3577, 2713} \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {a^4 \sin ^5(c+d x)}{21 d}-\frac {10 a^4 \sin ^3(c+d x)}{63 d}+\frac {5 a^4 \sin (c+d x)}{21 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d} \]
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Rule 2713
Rule 3577
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}+\frac {1}{3} a^2 \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx \\ & = -\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}+\frac {1}{21} \left (5 a^4\right ) \int \cos ^5(c+d x) \, dx \\ & = -\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac {\left (5 a^4\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 d} \\ & = \frac {5 a^4 \sin (c+d x)}{21 d}-\frac {10 a^4 \sin ^3(c+d x)}{63 d}+\frac {a^4 \sin ^5(c+d x)}{21 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac {2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.80 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {a^4 (-i \cos (4 (c+d x))+\sin (4 (c+d x))) \left (168 \cos (c+d x) \sqrt {\cos ^2(c+d x)}+4 \left (16+45 \sqrt {\cos ^2(c+d x)}\right ) \cos (3 (c+d x))+64 \cos (5 (c+d x))-28 \sqrt {\cos ^2(c+d x)} \cos (5 (c+d x))-42 i \sqrt {\cos ^2(c+d x)} \sin (c+d x)-64 i \sin (3 (c+d x))-135 i \sqrt {\cos ^2(c+d x)} \sin (3 (c+d x))-64 i \sin (5 (c+d x))+35 i \sqrt {\cos ^2(c+d x)} \sin (5 (c+d x))\right )}{1008 d \sqrt {\cos ^2(c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (106 ) = 212\).
Time = 0.82 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.94
\[\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )-4 i a^{4} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-6 a^{4} \left (-\frac {\left (\cos ^{8}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {4 i a^{4} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{4} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\]
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Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.75 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {{\left (-7 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 126 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 210 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 315 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 63 i \, a^{4}\right )} e^{\left (-i \, d x - i \, c\right )}}{2016 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (107) = 214\).
Time = 0.39 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.90 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=\begin {cases} \frac {\left (- 176160768 i a^{4} d^{5} e^{10 i c} e^{9 i d x} - 1132462080 i a^{4} d^{5} e^{8 i c} e^{7 i d x} - 3170893824 i a^{4} d^{5} e^{6 i c} e^{5 i d x} - 5284823040 i a^{4} d^{5} e^{4 i c} e^{3 i d x} - 7927234560 i a^{4} d^{5} e^{2 i c} e^{i d x} + 1585446912 i a^{4} d^{5} e^{- i d x}\right ) e^{- i c}}{50734301184 d^{6}} & \text {for}\: d^{6} e^{i c} \neq 0 \\\frac {x \left (a^{4} e^{10 i c} + 5 a^{4} e^{8 i c} + 10 a^{4} e^{6 i c} + 10 a^{4} e^{4 i c} + 5 a^{4} e^{2 i c} + a^{4}\right ) e^{- i c}}{32} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.51 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=-\frac {140 i \, a^{4} \cos \left (d x + c\right )^{9} + 20 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{4} - {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{4} - 6 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{4} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{4}}{315 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (102) = 204\).
Time = 0.81 (sec) , antiderivative size = 1409, normalized size of antiderivative = 11.74 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=\text {Too large to display} \]
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Time = 6.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx=\frac {2\,a^4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {89\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}-\frac {55\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}+\frac {55\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{4}-\frac {355\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {35\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,21{}\mathrm {i}}{2}+\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,21{}\mathrm {i}}{2}-\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,87{}\mathrm {i}}{4}+\frac {\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,7{}\mathrm {i}}{4}\right )}{63\,d\,\left (\cos \left (4\,c+4\,d\,x\right )-\sin \left (4\,c+4\,d\,x\right )\,1{}\mathrm {i}\right )} \]
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