Integrand size = 24, antiderivative size = 55 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i a^{14}}{3 d (a-i a \tan (c+d x))^6}+\frac {i a^{13}}{5 d (a-i a \tan (c+d x))^5} \]
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Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {i a^{13}}{5 d (a-i a \tan (c+d x))^5}-\frac {i a^{14}}{3 d (a-i a \tan (c+d x))^6} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^{13}\right ) \text {Subst}\left (\int \frac {a+x}{(a-x)^7} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {\left (i a^{13}\right ) \text {Subst}\left (\int \left (\frac {2 a}{(a-x)^7}-\frac {1}{(a-x)^6}\right ) \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^{14}}{3 d (a-i a \tan (c+d x))^6}+\frac {i a^{13}}{5 d (a-i a \tan (c+d x))^5} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {a^8 (-2 i+3 \tan (c+d x))}{15 d (i+\tan (c+d x))^6} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (47 ) = 94\).
Time = 2.81 (sec) , antiderivative size = 639, normalized size of antiderivative = 11.62
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none
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.38 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-5 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 24 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 40 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 15 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )}}{960 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (42) = 84\).
Time = 0.52 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.58 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\begin {cases} \frac {- 3932160 i a^{8} d^{4} e^{12 i c} e^{12 i d x} - 18874368 i a^{8} d^{4} e^{10 i c} e^{10 i d x} - 35389440 i a^{8} d^{4} e^{8 i c} e^{8 i d x} - 31457280 i a^{8} d^{4} e^{6 i c} e^{6 i d x} - 11796480 i a^{8} d^{4} e^{4 i c} e^{4 i d x}}{754974720 d^{5}} & \text {for}\: d^{5} \neq 0 \\x \left (\frac {a^{8} e^{12 i c}}{16} + \frac {a^{8} e^{10 i c}}{4} + \frac {3 a^{8} e^{8 i c}}{8} + \frac {a^{8} e^{6 i c}}{4} + \frac {a^{8} e^{4 i c}}{16}\right ) & \text {otherwise} \end {cases} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (43) = 86\).
Time = 0.33 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.95 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {3 \, a^{8} \tan \left (d x + c\right )^{7} - 20 i \, a^{8} \tan \left (d x + c\right )^{6} - 57 \, a^{8} \tan \left (d x + c\right )^{5} + 90 i \, a^{8} \tan \left (d x + c\right )^{4} + 85 \, a^{8} \tan \left (d x + c\right )^{3} - 48 i \, a^{8} \tan \left (d x + c\right )^{2} - 15 \, a^{8} \tan \left (d x + c\right ) + 2 i \, a^{8}}{15 \, {\left (\tan \left (d x + c\right )^{12} + 6 \, \tan \left (d x + c\right )^{10} + 15 \, \tan \left (d x + c\right )^{8} + 20 \, \tan \left (d x + c\right )^{6} + 15 \, \tan \left (d x + c\right )^{4} + 6 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (43) = 86\).
Time = 1.27 (sec) , antiderivative size = 437, normalized size of antiderivative = 7.95 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {5 i \, a^{8} e^{\left (40 i \, d x + 26 i \, c\right )} + 94 i \, a^{8} e^{\left (38 i \, d x + 24 i \, c\right )} + 836 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} + 4674 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} + 18411 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} + 54264 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} + 124033 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} + 224822 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} + 327613 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} + 386672 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} + 370513 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} + 287534 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} + 179361 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} + 34011 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} + 9754 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} + 1970 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} + 250 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} + 15 i \, a^{8} e^{\left (4 i \, d x - 10 i \, c\right )} + 88704 i \, a^{8} e^{\left (14 i \, d x\right )}}{960 \, {\left (d e^{\left (28 i \, d x + 14 i \, c\right )} + 14 \, d e^{\left (26 i \, d x + 12 i \, c\right )} + 91 \, d e^{\left (24 i \, d x + 10 i \, c\right )} + 364 \, d e^{\left (22 i \, d x + 8 i \, c\right )} + 1001 \, d e^{\left (20 i \, d x + 6 i \, c\right )} + 2002 \, d e^{\left (18 i \, d x + 4 i \, c\right )} + 3003 \, d e^{\left (16 i \, d x + 2 i \, c\right )} + 3003 \, d e^{\left (12 i \, d x - 2 i \, c\right )} + 2002 \, d e^{\left (10 i \, d x - 4 i \, c\right )} + 1001 \, d e^{\left (8 i \, d x - 6 i \, c\right )} + 364 \, d e^{\left (6 i \, d x - 8 i \, c\right )} + 91 \, d e^{\left (4 i \, d x - 10 i \, c\right )} + 14 \, d e^{\left (2 i \, d x - 12 i \, c\right )} + 3432 \, d e^{\left (14 i \, d x\right )} + d e^{\left (-14 i \, c\right )}\right )}} \]
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Time = 3.84 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.49 \[ \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {a^8\,\left (3\,\mathrm {tan}\left (c+d\,x\right )-2{}\mathrm {i}\right )}{15\,d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^6+{\mathrm {tan}\left (c+d\,x\right )}^5\,6{}\mathrm {i}-15\,{\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,20{}\mathrm {i}+15\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,6{}\mathrm {i}-1\right )} \]
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