Optimal. Leaf size=43 \[ -\frac{i \left (a^3+i a^3 \tan (c+d x)\right )^4}{8 d (a-i a \tan (c+d x))^4} \]
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Rubi [A] time = 0.0422143, antiderivative size = 43, normalized size of antiderivative = 1., 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, 37} \[ -\frac{i \left (a^3+i a^3 \tan (c+d x)\right )^4}{8 d (a-i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 37
Rubi steps
\begin{align*} \int \cos ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{\left (i a^9\right ) \operatorname{Subst}\left (\int \frac{(a+x)^3}{(a-x)^5} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i \left (a^3+i a^3 \tan (c+d x)\right )^4}{8 d (a-i a \tan (c+d x))^4}\\ \end{align*}
Mathematica [A] time = 0.295275, size = 31, normalized size = 0.72 \[ -\frac{i a^8 (\cos (c+d x)+i \sin (c+d x))^8}{8 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.077, size = 451, normalized size = 10.5 \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 = 2.41403, size = 185, normalized size = 4.3 \begin{align*} -\frac{384 \, a^{8} \tan \left (d x + c\right )^{7} - 1536 i \, a^{8} \tan \left (d x + c\right )^{6} - 2688 \, a^{8} \tan \left (d x + c\right )^{5} + 3072 i \, a^{8} \tan \left (d x + c\right )^{4} + 2688 \, a^{8} \tan \left (d x + c\right )^{3} - 1536 i \, a^{8} \tan \left (d x + c\right )^{2} - 384 \, a^{8} \tan \left (d x + c\right )}{384 \,{\left (\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \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 = 1.42455, size = 46, normalized size = 1.07 \begin{align*} -\frac{i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.948915, size = 37, normalized size = 0.86 \begin{align*} \begin{cases} - \frac{i a^{8} e^{8 i c} e^{8 i d x}}{8 d} & \text{for}\: 8 d \neq 0 \\a^{8} x e^{8 i c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.31477, size = 514, normalized size = 11.95 \begin{align*} \frac{-3840 i \, a^{8} e^{\left (36 i \, d x + 22 i \, c\right )} - 53760 i \, a^{8} e^{\left (34 i \, d x + 20 i \, c\right )} - 349440 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} - 1397760 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} - 3843840 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} - 7687680 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} - 11531520 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} - 13178880 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} - 11531520 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} - 7687680 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} - 3843840 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} - 349440 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} - 53760 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} - 3840 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} - 1397760 i \, a^{8} e^{\left (14 i \, d x\right )}}{30720 \,{\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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