Optimal. Leaf size=109 \[ \frac{i (a+i a \tan (c+d x))^{15}}{15 a^7 d}-\frac{3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac{12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac{2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d} \]
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Rubi [A] time = 0.0808121, antiderivative size = 109, 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} \[ \frac{i (a+i a \tan (c+d x))^{15}}{15 a^7 d}-\frac{3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac{12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac{2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^{11} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{11}-12 a^2 (a+x)^{12}+6 a (a+x)^{13}-(a+x)^{14}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d}+\frac{12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac{3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac{i (a+i a \tan (c+d x))^{15}}{15 a^7 d}\\ \end{align*}
Mathematica [B] time = 8.61341, size = 245, normalized size = 2.25 \[ \frac{a^8 \sec (c) \sec ^{15}(c+d x) (-6435 \sin (2 c+d x)+5005 \sin (2 c+3 d x)-5005 \sin (4 c+3 d x)+3003 \sin (4 c+5 d x)-3003 \sin (6 c+5 d x)+1365 \sin (6 c+7 d x)-1365 \sin (8 c+7 d x)+910 \sin (8 c+9 d x)+210 \sin (10 c+11 d x)+30 \sin (12 c+13 d x)+2 \sin (14 c+15 d x)+6435 i \cos (2 c+d x)+5005 i \cos (2 c+3 d x)+5005 i \cos (4 c+3 d x)+3003 i \cos (4 c+5 d x)+3003 i \cos (6 c+5 d x)+1365 i \cos (6 c+7 d x)+1365 i \cos (8 c+7 d x)+6435 \sin (d x)+6435 i \cos (d x))}{10920 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.1, size = 611, normalized size = 5.6 \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 = 1.47601, size = 251, normalized size = 2.3 \begin{align*} \frac{3003 \, a^{8} \tan \left (d x + c\right )^{15} - 25740 i \, a^{8} \tan \left (d x + c\right )^{14} - 86625 \, a^{8} \tan \left (d x + c\right )^{13} + 120120 i \, a^{8} \tan \left (d x + c\right )^{12} - 45045 \, a^{8} \tan \left (d x + c\right )^{11} + 396396 i \, a^{8} \tan \left (d x + c\right )^{10} + 495495 \, a^{8} \tan \left (d x + c\right )^{9} + 637065 \, a^{8} \tan \left (d x + c\right )^{7} - 660660 i \, a^{8} \tan \left (d x + c\right )^{6} - 99099 \, a^{8} \tan \left (d x + c\right )^{5} - 360360 i \, a^{8} \tan \left (d x + c\right )^{4} - 375375 \, a^{8} \tan \left (d x + c\right )^{3} + 180180 i \, a^{8} \tan \left (d x + c\right )^{2} + 45045 \, a^{8} \tan \left (d x + c\right )}{45045 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44153, size = 1195, normalized size = 10.96 \begin{align*} \frac{11182080 i \, a^{8} e^{\left (22 i \, d x + 22 i \, c\right )} + 24600576 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} + 41000960 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} + 52715520 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 52715520 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 41000960 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 24600576 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 11182080 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 3727360 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 860160 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 122880 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 8192 i \, a^{8}}{1365 \,{\left (d e^{\left (30 i \, d x + 30 i \, c\right )} + 15 \, d e^{\left (28 i \, d x + 28 i \, c\right )} + 105 \, d e^{\left (26 i \, d x + 26 i \, c\right )} + 455 \, d e^{\left (24 i \, d x + 24 i \, c\right )} + 1365 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 3003 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 5005 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 6435 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 6435 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 5005 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 3003 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 1365 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 455 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 105 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 15 \, 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 = 1.81532, size = 251, normalized size = 2.3 \begin{align*} \frac{91 \, a^{8} \tan \left (d x + c\right )^{15} - 780 i \, a^{8} \tan \left (d x + c\right )^{14} - 2625 \, a^{8} \tan \left (d x + c\right )^{13} + 3640 i \, a^{8} \tan \left (d x + c\right )^{12} - 1365 \, a^{8} \tan \left (d x + c\right )^{11} + 12012 i \, a^{8} \tan \left (d x + c\right )^{10} + 15015 \, a^{8} \tan \left (d x + c\right )^{9} + 19305 \, a^{8} \tan \left (d x + c\right )^{7} - 20020 i \, a^{8} \tan \left (d x + c\right )^{6} - 3003 \, a^{8} \tan \left (d x + c\right )^{5} - 10920 i \, a^{8} \tan \left (d x + c\right )^{4} - 11375 \, a^{8} \tan \left (d x + c\right )^{3} + 5460 i \, a^{8} \tan \left (d x + c\right )^{2} + 1365 \, a^{8} \tan \left (d x + c\right )}{1365 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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