Optimal. Leaf size=40 \[ \frac {\sec ^3(x)}{3}+\frac {1}{8} i \tanh ^{-1}(\sin (x))-\frac {1}{4} i \tan (x) \sec ^3(x)+\frac {1}{8} i \tan (x) \sec (x) \]
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Rubi [A] time = 0.17, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3518, 3108, 3107, 2606, 30, 2611, 3768, 3770} \[ \frac {\sec ^3(x)}{3}+\frac {1}{8} i \tanh ^{-1}(\sin (x))-\frac {1}{4} i \tan (x) \sec ^3(x)+\frac {1}{8} i \tan (x) \sec (x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 3107
Rule 3108
Rule 3518
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^5(x)}{i+\cot (x)} \, dx &=-\int \frac {\sec ^4(x) \tan (x)}{-\cos (x)-i \sin (x)} \, dx\\ &=i \int \sec ^4(x) (-i \cos (x)-\sin (x)) \tan (x) \, dx\\ &=i \int \left (-i \sec ^3(x) \tan (x)-\sec ^3(x) \tan ^2(x)\right ) \, dx\\ &=-\left (i \int \sec ^3(x) \tan ^2(x) \, dx\right )+\int \sec ^3(x) \tan (x) \, dx\\ &=-\frac {1}{4} i \sec ^3(x) \tan (x)+\frac {1}{4} i \int \sec ^3(x) \, dx+\operatorname {Subst}\left (\int x^2 \, dx,x,\sec (x)\right )\\ &=\frac {\sec ^3(x)}{3}+\frac {1}{8} i \sec (x) \tan (x)-\frac {1}{4} i \sec ^3(x) \tan (x)+\frac {1}{8} i \int \sec (x) \, dx\\ &=\frac {1}{8} i \tanh ^{-1}(\sin (x))+\frac {\sec ^3(x)}{3}+\frac {1}{8} i \sec (x) \tan (x)-\frac {1}{4} i \sec ^3(x) \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.60, size = 61, normalized size = 1.52 \[ -\frac {1}{48} i \left (6 \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )+\sec ^3(x) (-3 (\cos (2 x)-3) \tan (x)+16 i)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 121, normalized size = 3.02 \[ \frac {{\left (3 i \, e^{\left (8 i \, x\right )} + 12 i \, e^{\left (6 i \, x\right )} + 18 i \, e^{\left (4 i \, x\right )} + 12 i \, e^{\left (2 i \, x\right )} + 3 i\right )} \log \left (e^{\left (i \, x\right )} + i\right ) + {\left (-3 i \, e^{\left (8 i \, x\right )} - 12 i \, e^{\left (6 i \, x\right )} - 18 i \, e^{\left (4 i \, x\right )} - 12 i \, e^{\left (2 i \, x\right )} - 3 i\right )} \log \left (e^{\left (i \, x\right )} - i\right ) + 6 \, e^{\left (7 i \, x\right )} + 22 \, e^{\left (5 i \, x\right )} + 106 \, e^{\left (3 i \, x\right )} - 6 \, e^{\left (i \, x\right )}}{24 \, {\left (e^{\left (8 i \, x\right )} + 4 \, e^{\left (6 i \, x\right )} + 6 \, e^{\left (4 i \, x\right )} + 4 \, e^{\left (2 i \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 87, normalized size = 2.18 \[ -\frac {3 i \, \tan \left (\frac {1}{2} \, x\right )^{7} + 24 \, \tan \left (\frac {1}{2} \, x\right )^{6} + 21 i \, \tan \left (\frac {1}{2} \, x\right )^{5} - 24 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 21 i \, \tan \left (\frac {1}{2} \, x\right )^{3} + 8 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 3 i \, \tan \left (\frac {1}{2} \, x\right ) - 8}{12 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )}^{4}} + \frac {1}{8} i \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 1\right ) - \frac {1}{8} i \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.35, size = 170, normalized size = 4.25 \[ \frac {i}{4 \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {i}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {i \ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{8}-\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {i}{2 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )}-\frac {i}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )}-\frac {i}{8 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {i}{4 \left (\tan \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {3 i}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 \tan \left (\frac {x}{2}\right )+2}+\frac {3 i}{8 \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {i \ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 167, normalized size = 4.18 \[ -\frac {-\frac {3 i \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {8 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {21 i \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {24 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {21 i \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} - \frac {24 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {3 i \, \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}} + 8}{12 \, {\left (\frac {4 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {6 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {4 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {\sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} - 1\right )}} + \frac {1}{8} i \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) - \frac {1}{8} i \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 81, normalized size = 2.02 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,1{}\mathrm {i}}{4}-\frac {\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^7\,1{}\mathrm {i}}{4}+2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,7{}\mathrm {i}}{4}-2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,7{}\mathrm {i}}{4}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}}{4}-\frac {2}{3}}{{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-1\right )}^4} \]
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 \[ \int \frac {\sec ^{5}{\relax (x )}}{\cot {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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