Optimal. Leaf size=27 \[ -\frac {i a^{15}}{7 d (a-i a \tan (c+d x))^7} \]
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Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 32} \[ -\frac {i a^{15}}{7 d (a-i a \tan (c+d x))^7} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \cos ^{14}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^{15}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^8} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^{15}}{7 d (a-i a \tan (c+d x))^7}\\ \end {align*}
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Mathematica [B] time = 2.01, size = 116, normalized size = 4.30 \[ \frac {a^8 (-14 i \sin (2 (c+d x))-14 i \sin (4 (c+d x))-6 i \sin (6 (c+d x))+56 \cos (2 (c+d x))+28 \cos (4 (c+d x))+8 \cos (6 (c+d x))+35) (\sin (8 (c+2 d x))-i \cos (8 (c+2 d x)))}{896 d (\cos (d x)+i \sin (d x))^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 104, normalized size = 3.85 \[ \frac {-i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 7 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 21 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 35 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 35 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 21 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 7 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )}}{896 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 17.00, size = 465, normalized size = 17.22 \[ \frac {-3840 i \, a^{8} e^{\left (42 i \, d x + 28 i \, c\right )} - 80640 i \, a^{8} e^{\left (40 i \, d x + 26 i \, c\right )} - 806400 i \, a^{8} e^{\left (38 i \, d x + 24 i \, c\right )} - 5107200 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} - 22982400 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} - 78140160 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} - 208373760 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} - 446511360 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} - 781347840 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} - 1128341760 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} - 1353031680 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} - 1350585600 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} - 1121003520 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} - 769870080 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} - 196842240 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} - 70452480 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} - 19138560 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} - 3709440 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} - 456960 i \, a^{8} e^{\left (4 i \, d x - 10 i \, c\right )} - 26880 i \, a^{8} e^{\left (2 i \, d x - 12 i \, c\right )} - 433336320 i \, a^{8} e^{\left (14 i \, d x\right )}}{3440640 \, {\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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.77, size = 689, normalized size = 25.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 172, normalized size = 6.37 \[ -\frac {30720 \, a^{8} \tan \left (d x + c\right )^{7} - 215040 i \, a^{8} \tan \left (d x + c\right )^{6} - 645120 \, a^{8} \tan \left (d x + c\right )^{5} + 1075200 i \, a^{8} \tan \left (d x + c\right )^{4} + 1075200 \, a^{8} \tan \left (d x + c\right )^{3} - 645120 i \, a^{8} \tan \left (d x + c\right )^{2} - 215040 \, a^{8} \tan \left (d x + c\right ) + 30720 i \, a^{8}}{215040 \, {\left (\tan \left (d x + c\right )^{14} + 7 \, \tan \left (d x + c\right )^{12} + 21 \, \tan \left (d x + c\right )^{10} + 35 \, \tan \left (d x + c\right )^{8} + 35 \, \tan \left (d x + c\right )^{6} + 21 \, \tan \left (d x + c\right )^{4} + 7 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 105, normalized size = 3.89 \[ -\frac {a^8\,{\cos \left (c+d\,x\right )}^8\,\left (\mathrm {tan}\left (c+d\,x\right )-7{}\mathrm {i}\right )}{7\,d}+\frac {64\,a^8\,{\cos \left (c+d\,x\right )}^{14}\,\left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{7\,d}+\frac {8\,a^8\,{\cos \left (c+d\,x\right )}^{10}\,\left (3\,\mathrm {tan}\left (c+d\,x\right )-7{}\mathrm {i}\right )}{7\,d}-\frac {16\,a^8\,{\cos \left (c+d\,x\right )}^{12}\,\left (5\,\mathrm {tan}\left (c+d\,x\right )-7{}\mathrm {i}\right )}{7\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.38, size = 280, normalized size = 10.37 \[ \begin {cases} - \frac {4398046511104 i a^{8} d^{6} e^{14 i c} e^{14 i d x} + 30786325577728 i a^{8} d^{6} e^{12 i c} e^{12 i d x} + 92358976733184 i a^{8} d^{6} e^{10 i c} e^{10 i d x} + 153931627888640 i a^{8} d^{6} e^{8 i c} e^{8 i d x} + 153931627888640 i a^{8} d^{6} e^{6 i c} e^{6 i d x} + 92358976733184 i a^{8} d^{6} e^{4 i c} e^{4 i d x} + 30786325577728 i a^{8} d^{6} e^{2 i c} e^{2 i d x}}{3940649673949184 d^{7}} & \text {for}\: 3940649673949184 d^{7} \neq 0 \\x \left (\frac {a^{8} e^{14 i c}}{64} + \frac {3 a^{8} e^{12 i c}}{32} + \frac {15 a^{8} e^{10 i c}}{64} + \frac {5 a^{8} e^{8 i c}}{16} + \frac {15 a^{8} e^{6 i c}}{64} + \frac {3 a^{8} e^{4 i c}}{32} + \frac {a^{8} e^{2 i c}}{64}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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