Optimal. Leaf size=156 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a^8 d}-\frac {2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^3(c+d x)}{3 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7} \]
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Rubi [A] time = 0.22, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3500, 3770} \[ \frac {\tanh ^{-1}(\sin (c+d x))}{a^8 d}-\frac {2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^3(c+d x)}{3 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}-\frac {2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac {2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac {\int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{a^2}\\ &=\frac {2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac {2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {\int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{a^4}\\ &=\frac {2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac {2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^3(c+d x)}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac {\int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^6}\\ &=\frac {2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac {2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^3(c+d x)}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac {2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {\int \sec (c+d x) \, dx}{a^8}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{a^8 d}+\frac {2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac {2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^3(c+d x)}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac {2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 304, normalized size = 1.95 \[ \frac {\sec ^8(c+d x) \left (\cos \left (\frac {9}{2} (c+d x)\right )+i \sin \left (\frac {9}{2} (c+d x)\right )\right ) \left (-70 \sin \left (\frac {1}{2} (c+d x)\right )-42 \sin \left (\frac {3}{2} (c+d x)\right )+210 \sin \left (\frac {5}{2} (c+d x)\right )+30 \sin \left (\frac {7}{2} (c+d x)\right )+70 i \cos \left (\frac {1}{2} (c+d x)\right )-42 i \cos \left (\frac {3}{2} (c+d x)\right )-210 i \cos \left (\frac {5}{2} (c+d x)\right )+30 i \cos \left (\frac {7}{2} (c+d x)\right )-105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-105 i \sin \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 i \sin \left (\frac {7}{2} (c+d x)\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{105 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 98, normalized size = 0.63 \[ \frac {{\left (105 \, e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 210 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 70 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 42 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 30 i\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{105 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.74, size = 123, normalized size = 0.79 \[ \frac {\frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} - \frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} - \frac {2 \, {\left (-840 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3920 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2352 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1064 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 152\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 176, normalized size = 1.13 \[ -\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{8} d}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{8} d}+\frac {128 i}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{6}}+\frac {16 i}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {128 i}{a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}-\frac {256}{7 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{7}}+\frac {896}{5 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}-\frac {160}{3 a^{8} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 185, normalized size = 1.19 \[ \frac {-210 i \, \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) - 210 i \, \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) + 60 i \, \cos \left (7 \, d x + 7 \, c\right ) - 84 i \, \cos \left (5 \, d x + 5 \, c\right ) + 140 i \, \cos \left (3 \, d x + 3 \, c\right ) - 420 i \, \cos \left (d x + c\right ) + 105 \, \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - 105 \, \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 60 \, \sin \left (7 \, d x + 7 \, c\right ) - 84 \, \sin \left (5 \, d x + 5 \, c\right ) + 140 \, \sin \left (3 \, d x + 3 \, c\right ) - 420 \, \sin \left (d x + c\right )}{210 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.94, size = 207, normalized size = 1.33 \[ \frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^8\,d}+\frac {\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^8}-\frac {224\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,a^8}+\frac {304\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15\,a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,224{}\mathrm {i}}{5\,a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,80{}\mathrm {i}}{3\,a^8}-\frac {304{}\mathrm {i}}{105\,a^8}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,1{}\mathrm {i}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,21{}\mathrm {i}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,35{}\mathrm {i}-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,7{}\mathrm {i}+1\right )} \]
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 \[ \frac {\int \frac {\sec ^{9}{\left (c + d x \right )}}{\tan ^{8}{\left (c + d x \right )} - 8 i \tan ^{7}{\left (c + d x \right )} - 28 \tan ^{6}{\left (c + d x \right )} + 56 i \tan ^{5}{\left (c + d x \right )} + 70 \tan ^{4}{\left (c + d x \right )} - 56 i \tan ^{3}{\left (c + d x \right )} - 28 \tan ^{2}{\left (c + d x \right )} + 8 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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