Optimal. Leaf size=130 \[ -\frac {35 i a^5 \sec (c+d x)}{2 d}-\frac {35 a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d}-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3496, 3498, 3486, 3770} \[ -\frac {35 i a^5 \sec (c+d x)}{2 d}-\frac {35 a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3496
Rule 3498
Rule 3770
Rubi steps
\begin {align*} \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}-\left (7 a^2\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}-\frac {1}{3} \left (35 a^3\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d}-\frac {1}{2} \left (35 a^4\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac {35 i a^5 \sec (c+d x)}{2 d}-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d}-\frac {1}{2} \left (35 a^5\right ) \int \sec (c+d x) \, dx\\ &=-\frac {35 a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {35 i a^5 \sec (c+d x)}{2 d}-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 151, normalized size = 1.16 \[ \frac {a^5 \cos ^2(c+d x) (\tan (c+d x)-i)^5 \left ((\cos (4 c-d x)-i \sin (4 c-d x)) (-i (49 \sin (c+d x)+57 \sin (3 (c+d x)))+511 \cos (c+d x)+153 \cos (3 (c+d x)))-840 i (\cos (5 c)-i \sin (5 c)) \cos ^3(c+d x) \tanh ^{-1}\left (\cos (c) \tan \left (\frac {d x}{2}\right )+\sin (c)\right )\right )}{24 d (\cos (d x)+i \sin (d x))^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 216, normalized size = 1.66 \[ \frac {-96 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 462 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 560 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 210 i \, a^{5} e^{\left (i \, d x + i \, c\right )} - 105 \, {\left (a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 105 \, {\left (a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{6 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.49, size = 510, normalized size = 3.92 \[ \frac {8295 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 24885 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 24885 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 18585 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 55755 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 55755 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 8295 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 24885 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 24885 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 18585 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 55755 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 55755 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 24576 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 118272 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 143360 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 53760 i \, a^{5} e^{\left (i \, d x + i \, c\right )} + 8295 \, a^{5} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 18585 \, a^{5} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 8295 \, a^{5} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 18585 \, a^{5} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right )}{1536 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 214, normalized size = 1.65 \[ \frac {i a^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}-\frac {34 i a^{5} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3 d}-\frac {i a^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}-\frac {i a^{5} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{d}-\frac {83 i a^{5} \cos \left (d x +c \right )}{3 d}+\frac {5 a^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {5 a^{5} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {37 a^{5} \sin \left (d x +c \right )}{2 d}-\frac {35 a^{5} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}-\frac {10 i a^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 173, normalized size = 1.33 \[ -\frac {15 \, a^{5} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} + 120 i \, a^{5} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 4 i \, a^{5} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} + 60 \, a^{5} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 60 i \, a^{5} \cos \left (d x + c\right ) - 12 \, a^{5} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.14, size = 222, normalized size = 1.71 \[ -\frac {35\,a^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {37\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,27{}\mathrm {i}-118\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,48{}\mathrm {i}+139\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,55{}\mathrm {i}}{3}-\frac {166\,a^5}{3}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,1{}\mathrm {i}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,3{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 201, normalized size = 1.55 \[ \frac {35 a^{5} \left (\frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{2} - \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{2}\right )}{d} + \frac {- 87 a^{5} e^{5 i c} e^{5 i d x} - 136 a^{5} e^{3 i c} e^{3 i d x} - 57 a^{5} e^{i c} e^{i d x}}{- 3 i d e^{6 i c} e^{6 i d x} - 9 i d e^{4 i c} e^{4 i d x} - 9 i d e^{2 i c} e^{2 i d x} - 3 i d} + \begin {cases} - \frac {16 i a^{5} e^{i c} e^{i d x}}{d} & \text {for}\: d \neq 0 \\16 a^{5} x e^{i c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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