Optimal. Leaf size=84 \[ -\frac {2 i \sec ^3(c+d x)}{3 a^2 d}+\frac {5 \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac {\tan (c+d x) \sec ^3(c+d x)}{4 a^2 d}+\frac {5 \tan (c+d x) \sec (c+d x)}{8 a^2 d} \]
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Rubi [A] time = 0.19, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3092, 3090, 3768, 3770, 2606, 30, 2611} \[ -\frac {2 i \sec ^3(c+d x)}{3 a^2 d}+\frac {5 \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac {\tan (c+d x) \sec ^3(c+d x)}{4 a^2 d}+\frac {5 \tan (c+d x) \sec (c+d x)}{8 a^2 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 3090
Rule 3092
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx &=-\frac {\int \sec ^5(c+d x) (i a \cos (c+d x)+a \sin (c+d x))^2 \, dx}{a^4}\\ &=-\frac {\int \left (-a^2 \sec ^3(c+d x)+2 i a^2 \sec ^3(c+d x) \tan (c+d x)+a^2 \sec ^3(c+d x) \tan ^2(c+d x)\right ) \, dx}{a^4}\\ &=-\frac {(2 i) \int \sec ^3(c+d x) \tan (c+d x) \, dx}{a^2}+\frac {\int \sec ^3(c+d x) \, dx}{a^2}-\frac {\int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{a^2}\\ &=\frac {\sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{4 a^2 d}+\frac {\int \sec ^3(c+d x) \, dx}{4 a^2}+\frac {\int \sec (c+d x) \, dx}{2 a^2}-\frac {(2 i) \operatorname {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac {2 i \sec ^3(c+d x)}{3 a^2 d}+\frac {5 \sec (c+d x) \tan (c+d x)}{8 a^2 d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{4 a^2 d}+\frac {\int \sec (c+d x) \, dx}{8 a^2}\\ &=\frac {5 \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac {2 i \sec ^3(c+d x)}{3 a^2 d}+\frac {5 \sec (c+d x) \tan (c+d x)}{8 a^2 d}-\frac {\sec ^3(c+d x) \tan (c+d x)}{4 a^2 d}\\ \end {align*}
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Mathematica [B] time = 1.00, size = 215, normalized size = 2.56 \[ -\frac {\sec ^4(c+d x) \left (18 \sin (c+d x)-30 \sin (3 (c+d x))+128 i \cos (c+d x)+45 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+15 \cos (4 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-45 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{192 a^2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 230, normalized size = 2.74 \[ \frac {15 \, {\left (e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 15 \, {\left (e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 30 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 110 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 146 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 30 i \, e^{\left (i \, d x + i \, c\right )}}{24 \, {\left (a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 151, normalized size = 1.80 \[ \frac {\frac {15 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2}} - \frac {15 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{2}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 16 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a^{2}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 342, normalized size = 4.07 \[ \frac {3}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {i}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 i}{3 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {i}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{2} d}+\frac {3}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {i}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {i}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 i}{3 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {1}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 295, normalized size = 3.51 \[ \frac {\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 i \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {33 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {48 i \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {33 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {48 i \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {9 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 16 i\right )}}{a^{2} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {15 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} - \frac {15 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 136, normalized size = 1.62 \[ \frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a^2\,d}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,4{}\mathrm {i}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,4{}\mathrm {i}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,4{}\mathrm {i}}{3}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {4}{3}{}\mathrm {i}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^4} \]
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 \[ \frac {\int \frac {\sec ^{5}{\left (c + d x \right )}}{- \sin ^{2}{\left (c + d x \right )} + 2 i \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} + \cos ^{2}{\left (c + d x \right )}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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