Optimal. Leaf size=119 \[ -\frac {7 i \sec ^5(c+d x)}{15 a^3 d}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac {7 \tan (c+d x) \sec ^3(c+d x)}{12 a^3 d}+\frac {7 \tan (c+d x) \sec (c+d x)}{8 a^3 d}-\frac {2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.11, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3500, 3501, 3768, 3770} \[ -\frac {7 i \sec ^5(c+d x)}{15 a^3 d}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}+\frac {7 \tan (c+d x) \sec ^3(c+d x)}{12 a^3 d}+\frac {7 \tan (c+d x) \sec (c+d x)}{8 a^3 d}-\frac {2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3501
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac {7 \int \frac {\sec ^7(c+d x)}{a+i a \tan (c+d x)} \, dx}{3 a^2}\\ &=-\frac {7 i \sec ^5(c+d x)}{15 a^3 d}-\frac {2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac {7 \int \sec ^5(c+d x) \, dx}{3 a^3}\\ &=-\frac {7 i \sec ^5(c+d x)}{15 a^3 d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{12 a^3 d}-\frac {2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac {7 \int \sec ^3(c+d x) \, dx}{4 a^3}\\ &=-\frac {7 i \sec ^5(c+d x)}{15 a^3 d}+\frac {7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{12 a^3 d}-\frac {2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}+\frac {7 \int \sec (c+d x) \, dx}{8 a^3}\\ &=\frac {7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac {7 i \sec ^5(c+d x)}{15 a^3 d}+\frac {7 \sec (c+d x) \tan (c+d x)}{8 a^3 d}+\frac {7 \sec ^3(c+d x) \tan (c+d x)}{12 a^3 d}-\frac {2 i \sec ^7(c+d x)}{3 a d (a+i a \tan (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 113, normalized size = 0.95 \[ \frac {\sec ^8(c+d x) (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left (-150 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))+640 \cos (2 (c+d x))+1680 i \cos ^5(c+d x) \tanh ^{-1}\left (\cos (c) \tan \left (\frac {d x}{2}\right )+\sin (c)\right )+448\right )}{960 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 278, normalized size = 2.34 \[ \frac {105 \, {\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, {\left (e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 210 i \, e^{\left (9 i \, d x + 9 i \, c\right )} - 980 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 1792 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 1580 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, e^{\left (i \, d x + i \, c\right )}}{120 \, {\left (a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.94, size = 164, normalized size = 1.38 \[ \frac {\frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3}} - \frac {105 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{3}} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 400 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 390 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 136 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} a^{3}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 430, normalized size = 3.61 \[ -\frac {i}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {7 i}{12 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {13 i}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {5}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {i}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {3}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {i}{5 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{3} d}+\frac {11 i}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {i}{5 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {3}{4 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {11 i}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {7 i}{12 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {13 i}{8 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 341, normalized size = 2.87 \[ \frac {\frac {16 \, {\left (-\frac {15 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {390 i \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {400 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {390 i \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {360 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {15 i \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 136\right )}}{-120 i \, a^{3} + \frac {600 i \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1200 i \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1200 i \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {600 i \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {120 i \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {7 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} - \frac {7 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.08, size = 150, normalized size = 1.26 \[ \frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a^3\,d}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,6{}\mathrm {i}-\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,16{}\mathrm {i}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,20{}\mathrm {i}}{3}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,16{}\mathrm {i}}{3}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {34}{15}{}\mathrm {i}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^5} \]
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 \[ \frac {i \int \frac {\sec ^{9}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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