Optimal. Leaf size=183 \[ -\frac {63 \tanh ^{-1}(\sin (c+d x))}{2 a^8 d}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \tan (c+d x) \sec (c+d x)}{2 a^8 d}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7} \]
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Rubi [A] time = 0.20, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3500, 3768, 3770} \[ -\frac {63 \tanh ^{-1}(\sin (c+d x))}{2 a^8 d}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \tan (c+d x) \sec (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {9 \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{5 a^2}\\ &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {21 \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{5 a^4}\\ &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}-\frac {21 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^6}\\ &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \int \sec ^3(c+d x) \, dx}{a^8}\\ &=-\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \int \sec (c+d x) \, dx}{2 a^8}\\ &=-\frac {63 \tanh ^{-1}(\sin (c+d x))}{2 a^8 d}-\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 6.28, size = 1244, normalized size = 6.80 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 182, normalized size = 0.99 \[ -\frac {315 \, {\left (e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, e^{\left (7 i \, d x + 7 i \, c\right )} + e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 315 \, {\left (e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, e^{\left (7 i \, d x + 7 i \, c\right )} + e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 630 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 1050 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 336 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 48 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 16 i}{10 \, {\left (a^{8} d e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, a^{8} d e^{\left (7 i \, d x + 7 i \, c\right )} + a^{8} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.57, size = 165, normalized size = 0.90 \[ -\frac {\frac {315 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} - \frac {315 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} - \frac {10 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{8}} - \frac {4 \, {\left (160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 720 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 880 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 208\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{5}}}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 282, normalized size = 1.54 \[ \frac {1}{2 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8 i}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{2 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{8} d}+\frac {1}{2 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {8 i}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{2 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{8} d}-\frac {32 i}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {128 i}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}+\frac {256}{5 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}-\frac {64}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {64}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 541, normalized size = 2.96 \[ \frac {{\left (630 \, \cos \left (9 \, d x + 9 \, c\right ) + 1260 \, \cos \left (7 \, d x + 7 \, c\right ) + 630 \, \cos \left (5 \, d x + 5 \, c\right ) + 630 i \, \sin \left (9 \, d x + 9 \, c\right ) + 1260 i \, \sin \left (7 \, d x + 7 \, c\right ) + 630 i \, \sin \left (5 \, d x + 5 \, c\right )\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) + {\left (630 \, \cos \left (9 \, d x + 9 \, c\right ) + 1260 \, \cos \left (7 \, d x + 7 \, c\right ) + 630 \, \cos \left (5 \, d x + 5 \, c\right ) + 630 i \, \sin \left (9 \, d x + 9 \, c\right ) + 1260 i \, \sin \left (7 \, d x + 7 \, c\right ) + 630 i \, \sin \left (5 \, d x + 5 \, c\right )\right )} \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) - {\left (-315 i \, \cos \left (9 \, d x + 9 \, c\right ) - 630 i \, \cos \left (7 \, d x + 7 \, c\right ) - 315 i \, \cos \left (5 \, d x + 5 \, c\right ) + 315 \, \sin \left (9 \, d x + 9 \, c\right ) + 630 \, \sin \left (7 \, d x + 7 \, c\right ) + 315 \, \sin \left (5 \, d x + 5 \, c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - {\left (315 i \, \cos \left (9 \, d x + 9 \, c\right ) + 630 i \, \cos \left (7 \, d x + 7 \, c\right ) + 315 i \, \cos \left (5 \, d x + 5 \, c\right ) - 315 \, \sin \left (9 \, d x + 9 \, c\right ) - 630 \, \sin \left (7 \, d x + 7 \, c\right ) - 315 \, \sin \left (5 \, d x + 5 \, c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 1260 \, \cos \left (8 \, d x + 8 \, c\right ) + 2100 \, \cos \left (6 \, d x + 6 \, c\right ) + 672 \, \cos \left (4 \, d x + 4 \, c\right ) - 96 \, \cos \left (2 \, d x + 2 \, c\right ) + 1260 i \, \sin \left (8 \, d x + 8 \, c\right ) + 2100 i \, \sin \left (6 \, d x + 6 \, c\right ) + 672 i \, \sin \left (4 \, d x + 4 \, c\right ) - 96 i \, \sin \left (2 \, d x + 2 \, c\right ) + 32}{{\left (-20 i \, a^{8} \cos \left (9 \, d x + 9 \, c\right ) - 40 i \, a^{8} \cos \left (7 \, d x + 7 \, c\right ) - 20 i \, a^{8} \cos \left (5 \, d x + 5 \, c\right ) + 20 \, a^{8} \sin \left (9 \, d x + 9 \, c\right ) + 40 \, a^{8} \sin \left (7 \, d x + 7 \, c\right ) + 20 \, a^{8} \sin \left (5 \, d x + 5 \, c\right )\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.44, size = 284, normalized size = 1.55 \[ -\frac {63\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^8\,d}+\frac {\frac {1223\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{a^8}-\frac {1109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^8}+\frac {309\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{a^8}-\frac {431\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,4407{}\mathrm {i}}{5\,a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,7351{}\mathrm {i}}{5\,a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,761{}\mathrm {i}}{a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,65{}\mathrm {i}}{a^8}+\frac {496{}\mathrm {i}}{5\,a^8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,12{}\mathrm {i}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,26{}\mathrm {i}+26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,20{}\mathrm {i}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,5{}\mathrm {i}+1\right )} \]
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 \[ \frac {\int \frac {\sec ^{11}{\left (c + d x \right )}}{\tan ^{8}{\left (c + d x \right )} - 8 i \tan ^{7}{\left (c + d x \right )} - 28 \tan ^{6}{\left (c + d x \right )} + 56 i \tan ^{5}{\left (c + d x \right )} + 70 \tan ^{4}{\left (c + d x \right )} - 56 i \tan ^{3}{\left (c + d x \right )} - 28 \tan ^{2}{\left (c + d x \right )} + 8 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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