Optimal. Leaf size=55 \[ \frac {i a^{13}}{5 d (a-i a \tan (c+d x))^5}-\frac {i a^{14}}{3 d (a-i a \tan (c+d x))^6} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac {i a^{13}}{5 d (a-i a \tan (c+d x))^5}-\frac {i a^{14}}{3 d (a-i a \tan (c+d x))^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \cos ^{12}(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^{13}\right ) \operatorname {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 ) \operatorname {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*}
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Mathematica [A] time = 1.47, size = 77, normalized size = 1.40 \[ \frac {a^8 (-16 i \sin (2 (c+d x))-10 i \sin (4 (c+d x))+64 \cos (2 (c+d x))+20 \cos (4 (c+d x))+45) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 76, normalized size = 1.38 \[ \frac {-5 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 24 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 40 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 15 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 15.23, size = 437, normalized size = 7.95 \[ \frac {-8960 i \, a^{8} e^{\left (40 i \, d x + 26 i \, c\right )} - 168448 i \, a^{8} e^{\left (38 i \, d x + 24 i \, c\right )} - 1498112 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} - 8375808 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} - 32992512 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} - 97241088 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} - 222267136 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} - 402881024 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} - 587082496 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} - 692916224 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} - 663959296 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} - 515260928 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} - 321414912 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} - 60947712 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} - 17479168 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} - 3530240 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} - 448000 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} - 26880 i \, a^{8} e^{\left (4 i \, d x - 10 i \, c\right )} - 158957568 i \, a^{8} e^{\left (14 i \, d x\right )}}{1720320 \, {\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 = 0.76, size = 639, normalized size = 11.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 162, normalized size = 2.95 \[ -\frac {3072 \, a^{8} \tan \left (d x + c\right )^{7} - 20480 i \, a^{8} \tan \left (d x + c\right )^{6} - 58368 \, a^{8} \tan \left (d x + c\right )^{5} + 92160 i \, a^{8} \tan \left (d x + c\right )^{4} + 87040 \, a^{8} \tan \left (d x + c\right )^{3} - 49152 i \, a^{8} \tan \left (d x + c\right )^{2} - 15360 \, a^{8} \tan \left (d x + c\right ) + 2048 i \, a^{8}}{15360 \, {\left (\tan \left (d x + c\right )^{12} + 6 \, \tan \left (d x + c\right )^{10} + 15 \, \tan \left (d x + c\right )^{8} + 20 \, \tan \left (d x + c\right )^{6} + 15 \, \tan \left (d x + c\right )^{4} + 6 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.40, size = 82, normalized size = 1.49 \[ -\frac {a^8\,\left (3\,\mathrm {tan}\left (c+d\,x\right )-2{}\mathrm {i}\right )}{15\,d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^6+{\mathrm {tan}\left (c+d\,x\right )}^5\,6{}\mathrm {i}-15\,{\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,20{}\mathrm {i}+15\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,6{}\mathrm {i}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 199, normalized size = 3.62 \[ \begin {cases} - \frac {3932160 i a^{8} d^{4} e^{12 i c} e^{12 i d x} + 18874368 i a^{8} d^{4} e^{10 i c} e^{10 i d x} + 35389440 i a^{8} d^{4} e^{8 i c} e^{8 i d x} + 31457280 i a^{8} d^{4} e^{6 i c} e^{6 i d x} + 11796480 i a^{8} d^{4} e^{4 i c} e^{4 i d x}}{754974720 d^{5}} & \text {for}\: 754974720 d^{5} \neq 0 \\x \left (\frac {a^{8} e^{12 i c}}{16} + \frac {a^{8} e^{10 i c}}{4} + \frac {3 a^{8} e^{8 i c}}{8} + \frac {a^{8} e^{6 i c}}{4} + \frac {a^{8} e^{4 i c}}{16}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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