Optimal. Leaf size=109 \[ -\frac {2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d}+\frac {12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac {3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac {i (a+i a \tan (c+d x))^{15}}{15 a^7 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 109, 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 = {3568, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^3 (a+x)^{11} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \text {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*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(245\) vs. \(2(109)=218\).
time = 3.58, size = 245, normalized size = 2.25 \begin {gather*} \frac {a^8 \sec (c) \sec ^{15}(c+d x) (6435 i \cos (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 \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))}{10920 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 610 vs. \(2 (93 ) = 186\).
time = 0.32, size = 611, normalized size = 5.61 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 186 vs. \(2 (85) = 170\).
time = 0.29, size = 186, normalized size = 1.71 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 345 vs. \(2 (85) = 170\).
time = 0.38, size = 345, normalized size = 3.17 \begin {gather*} -\frac {8192 \, {\left (-1365 i \, a^{8} e^{\left (22 i \, d x + 22 i \, c\right )} - 3003 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} - 5005 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 6435 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 6435 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 5005 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 3003 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 1365 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 455 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 105 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 15 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{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 {gather*}
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 \begin {gather*} a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int \sec ^{8}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 186 vs. \(2 (85) = 170\).
time = 1.28, size = 186, normalized size = 1.71 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.83, size = 153, normalized size = 1.40 \begin {gather*} \frac {a^8\,\left (\frac {\sin \left (9\,c+9\,d\,x\right )}{12}+\frac {\sin \left (11\,c+11\,d\,x\right )}{52}+\frac {\sin \left (13\,c+13\,d\,x\right )}{364}+\frac {\sin \left (15\,c+15\,d\,x\right )}{5460}+\frac {\cos \left (c+d\,x\right )\,297{}\mathrm {i}}{7168}+\frac {\cos \left (3\,c+3\,d\,x\right )\,33{}\mathrm {i}}{1024}+\frac {\cos \left (5\,c+5\,d\,x\right )\,99{}\mathrm {i}}{5120}+\frac {\cos \left (7\,c+7\,d\,x\right )\,9{}\mathrm {i}}{1024}-\frac {\cos \left (9\,c+9\,d\,x\right )\,247{}\mathrm {i}}{3072}-\frac {\cos \left (11\,c+11\,d\,x\right )\,19{}\mathrm {i}}{1024}-\frac {\cos \left (13\,c+13\,d\,x\right )\,19{}\mathrm {i}}{7168}-\frac {\cos \left (15\,c+15\,d\,x\right )\,19{}\mathrm {i}}{107520}\right )}{d\,{\cos \left (c+d\,x\right )}^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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