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 = 0.0476216, antiderivative size = 55, normalized size of antiderivative = 1., 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 3487
Rule 43
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*}
Mathematica [A] time = 1.16954, size = 77, normalized size = 1.4 \[ \frac{a^8 (-16 i \sin (2 (c+d x))-10 i \sin (4 (c+d x))+64 \cos (2 (c+d x))+20 \cos (4 (c+d x))+45) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{960 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.092, size = 639, normalized size = 11.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52452, size = 219, normalized size = 3.98 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68441, size = 227, normalized size = 4.13 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.61318, size = 199, normalized size = 3.62 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.54573, size = 590, normalized size = 10.73 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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