Optimal. Leaf size=60 \[ -\frac {i \sec ^3(c+d x)}{3 a d}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac {\tan (c+d x) \sec (c+d x)}{2 a d} \]
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Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3501, 3768, 3770} \[ -\frac {i \sec ^3(c+d x)}{3 a d}+\frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac {\tan (c+d x) \sec (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3501
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac {i \sec ^3(c+d x)}{3 a d}+\frac {\int \sec ^3(c+d x) \, dx}{a}\\ &=-\frac {i \sec ^3(c+d x)}{3 a d}+\frac {\sec (c+d x) \tan (c+d x)}{2 a d}+\frac {\int \sec (c+d x) \, dx}{2 a}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac {i \sec ^3(c+d x)}{3 a d}+\frac {\sec (c+d x) \tan (c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 50, normalized size = 0.83 \[ \frac {12 \tanh ^{-1}\left (\cos (c) \tan \left (\frac {d x}{2}\right )+\sin (c)\right )+(3 \sin (2 (c+d x))-4 i) \sec ^3(c+d x)}{12 a d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 174, normalized size = 2.90 \[ \frac {3 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 6 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 16 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 6 i \, e^{\left (i \, d x + i \, c\right )}}{6 \, {\left (a d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 99, normalized size = 1.65 \[ \frac {\frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a} - \frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 258, normalized size = 4.30 \[ \frac {i}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {1}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {i}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {i}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}-\frac {i}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {i}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {i}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 186, normalized size = 3.10 \[ \frac {\frac {4 \, {\left (\frac {3 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 i \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 2\right )}}{6 i \, a - \frac {18 i \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 i \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6 i \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.38, size = 116, normalized size = 1.93 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,2{}\mathrm {i}}{a}+\frac {2{}\mathrm {i}}{3\,a}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ - \frac {i \int \frac {\sec ^{5}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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