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 = 0.07, antiderivative size = 82, 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} \[ -\frac {i (a+i a \tan (c+d x))^{13}}{13 a^5 d}+\frac {i (a+i a \tan (c+d x))^{12}}{3 a^4 d}-\frac {4 i (a+i a \tan (c+d x))^{11}}{11 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
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*}
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Mathematica [B] time = 7.04, 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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fricas [B] time = 0.47, size = 307, normalized size = 3.74 \[ \frac {1171456 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} + 2928640 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} + 5271552 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 7028736 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 7028736 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 5271552 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 2928640 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 1171456 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 319488 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 53248 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 4096 i \, a^{8}}{429 \, {\left (d e^{\left (26 i \, d x + 26 i \, c\right )} + 13 \, d e^{\left (24 i \, d x + 24 i \, c\right )} + 78 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 286 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 715 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 1287 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 1716 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 1716 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 1287 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 715 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 286 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 78 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 13 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.33, size = 173, normalized size = 2.11 \[ \frac {33 \, a^{8} \tan \left (d x + c\right )^{13} - 286 i \, a^{8} \tan \left (d x + c\right )^{12} - 1014 \, a^{8} \tan \left (d x + c\right )^{11} + 1716 i \, a^{8} \tan \left (d x + c\right )^{10} + 715 \, a^{8} \tan \left (d x + c\right )^{9} + 2574 i \, a^{8} \tan \left (d x + c\right )^{8} + 5148 \, a^{8} \tan \left (d x + c\right )^{7} - 3432 i \, a^{8} \tan \left (d x + c\right )^{6} + 1287 \, a^{8} \tan \left (d x + c\right )^{5} - 4290 i \, a^{8} \tan \left (d x + c\right )^{4} - 3718 \, a^{8} \tan \left (d x + c\right )^{3} + 1716 i \, a^{8} \tan \left (d x + c\right )^{2} + 429 \, a^{8} \tan \left (d x + c\right )}{429 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 475, normalized size = 5.79 \[ \frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{13 \cos \left (d x +c \right )^{13}}+\frac {4 \left (\sin ^{9}\left (d x +c \right )\right )}{143 \cos \left (d x +c \right )^{11}}+\frac {8 \left (\sin ^{9}\left (d x +c \right )\right )}{1287 \cos \left (d x +c \right )^{9}}\right )+\frac {4 i a^{8}}{3 \cos \left (d x +c \right )^{6}}-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}+\frac {8 \left (\sin ^{7}\left (d x +c \right )\right )}{693 \cos \left (d x +c \right )^{7}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{12}}+\frac {\sin ^{8}\left (d x +c \right )}{30 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{120 \cos \left (d x +c \right )^{8}}\right )-a^{8} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 173, normalized size = 2.11 \[ \frac {495 \, a^{8} \tan \left (d x + c\right )^{13} - 4290 i \, a^{8} \tan \left (d x + c\right )^{12} - 15210 \, a^{8} \tan \left (d x + c\right )^{11} + 25740 i \, a^{8} \tan \left (d x + c\right )^{10} + 10725 \, a^{8} \tan \left (d x + c\right )^{9} + 38610 i \, a^{8} \tan \left (d x + c\right )^{8} + 77220 \, a^{8} \tan \left (d x + c\right )^{7} - 51480 i \, a^{8} \tan \left (d x + c\right )^{6} + 19305 \, a^{8} \tan \left (d x + c\right )^{5} - 64350 i \, a^{8} \tan \left (d x + c\right )^{4} - 55770 \, a^{8} \tan \left (d x + c\right )^{3} + 25740 i \, a^{8} \tan \left (d x + c\right )^{2} + 6435 \, a^{8} \tan \left (d x + c\right )}{6435 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 190, normalized size = 2.32 \[ \frac {a^8\,\sin \left (c+d\,x\right )\,\left (2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (-184\,{\sin \left (c+d\,x\right )}^2-184\,{\sin \left (2\,c+2\,d\,x\right )}^2+\frac {\sin \left (2\,c+2\,d\,x\right )\,9867{}\mathrm {i}}{256}-184\,{\sin \left (3\,c+3\,d\,x\right )}^2-184\,{\sin \left (4\,c+4\,d\,x\right )}^2+\frac {\sin \left (4\,c+4\,d\,x\right )\,69069{}\mathrm {i}}{1024}-28\,{\sin \left (5\,c+5\,d\,x\right )}^2-2\,{\sin \left (6\,c+6\,d\,x\right )}^2+\frac {\sin \left (6\,c+6\,d\,x\right )\,42757{}\mathrm {i}}{512}+\frac {\sin \left (8\,c+8\,d\,x\right )\,23023{}\mathrm {i}}{256}+\frac {\sin \left (10\,c+10\,d\,x\right )\,7007{}\mathrm {i}}{512}+\frac {\sin \left (12\,c+12\,d\,x\right )\,1001{}\mathrm {i}}{1024}+429\right )}{429\,d\,{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \sec ^{6}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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