Optimal. Leaf size=82 \[ -\frac{i (a+i a \tan (c+d x))^{13}}{13 a^5 d}+\frac{i (a+i a \tan (c+d x))^{12}}{3 a^4 d}-\frac{4 i (a+i a \tan (c+d x))^{11}}{11 a^3 d} \]
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Rubi [A] time = 0.0705574, antiderivative size = 82, 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} \[ -\frac{i (a+i a \tan (c+d x))^{13}}{13 a^5 d}+\frac{i (a+i a \tan (c+d x))^{12}}{3 a^4 d}-\frac{4 i (a+i a \tan (c+d x))^{11}}{11 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^{10} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{10}-4 a (a+x)^{11}+(a+x)^{12}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{4 i (a+i a \tan (c+d x))^{11}}{11 a^3 d}+\frac{i (a+i a \tan (c+d x))^{12}}{3 a^4 d}-\frac{i (a+i a \tan (c+d x))^{13}}{13 a^5 d}\\ \end{align*}
Mathematica [B] time = 6.22252, size = 234, normalized size = 2.85 \[ \frac{a^8 \sec (c) \sec ^{13}(c+d x) (-1716 \sin (2 c+d x)+1287 \sin (2 c+3 d x)-1287 \sin (4 c+3 d x)+715 \sin (4 c+5 d x)-715 \sin (6 c+5 d x)+286 \sin (6 c+7 d x)-286 \sin (8 c+7 d x)+156 \sin (8 c+9 d x)+26 \sin (10 c+11 d x)+2 \sin (12 c+13 d x)+1716 i \cos (2 c+d x)+1287 i \cos (2 c+3 d x)+1287 i \cos (4 c+3 d x)+715 i \cos (4 c+5 d x)+715 i \cos (6 c+5 d x)+286 i \cos (6 c+7 d x)+286 i \cos (8 c+7 d x)+1716 \sin (d x)+1716 i \cos (d x))}{1716 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.097, size = 475, normalized size = 5.8 \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 = 1.1192, size = 234, normalized size = 2.85 \begin{align*} \frac{495 \, a^{8} \tan \left (d x + c\right )^{13} - 4290 i \, a^{8} \tan \left (d x + c\right )^{12} - 15210 \, a^{8} \tan \left (d x + c\right )^{11} + 25740 i \, a^{8} \tan \left (d x + c\right )^{10} + 10725 \, a^{8} \tan \left (d x + c\right )^{9} + 38610 i \, a^{8} \tan \left (d x + c\right )^{8} + 77220 \, a^{8} \tan \left (d x + c\right )^{7} - 51480 i \, a^{8} \tan \left (d x + c\right )^{6} + 19305 \, a^{8} \tan \left (d x + c\right )^{5} - 64350 i \, a^{8} \tan \left (d x + c\right )^{4} - 55770 \, a^{8} \tan \left (d x + c\right )^{3} + 25740 i \, a^{8} \tan \left (d x + c\right )^{2} + 6435 \, a^{8} \tan \left (d x + c\right )}{6435 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39586, size = 1041, normalized size = 12.7 \begin{align*} \frac{1171456 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} + 2928640 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} + 5271552 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 7028736 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 7028736 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 5271552 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 2928640 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 1171456 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 319488 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 53248 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 4096 i \, a^{8}}{429 \,{\left (d e^{\left (26 i \, d x + 26 i \, c\right )} + 13 \, d e^{\left (24 i \, d x + 24 i \, c\right )} + 78 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 286 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 715 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 1287 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 1716 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 1716 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 1287 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 715 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 286 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 78 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 13 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.76486, size = 234, normalized size = 2.85 \begin{align*} \frac{33 \, a^{8} \tan \left (d x + c\right )^{13} - 286 i \, a^{8} \tan \left (d x + c\right )^{12} - 1014 \, a^{8} \tan \left (d x + c\right )^{11} + 1716 i \, a^{8} \tan \left (d x + c\right )^{10} + 715 \, a^{8} \tan \left (d x + c\right )^{9} + 2574 i \, a^{8} \tan \left (d x + c\right )^{8} + 5148 \, a^{8} \tan \left (d x + c\right )^{7} - 3432 i \, a^{8} \tan \left (d x + c\right )^{6} + 1287 \, a^{8} \tan \left (d x + c\right )^{5} - 4290 i \, a^{8} \tan \left (d x + c\right )^{4} - 3718 \, a^{8} \tan \left (d x + c\right )^{3} + 1716 i \, a^{8} \tan \left (d x + c\right )^{2} + 429 \, a^{8} \tan \left (d x + c\right )}{429 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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