Integrand size = 22, antiderivative size = 130 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {35 a^5 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {35 i a^5 \sec (c+d x)}{2 d}-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d} \]
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Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3577, 3579, 3567, 3855} \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {35 a^5 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {35 i a^5 \sec (c+d x)}{2 d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d}-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d} \]
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Rule 3567
Rule 3577
Rule 3579
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\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 \text {arctanh}(\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*}
Time = 1.78 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 \cos ^2(c+d x) \left (-840 i \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )\right ) \cos ^3(c+d x) (\cos (5 c)-i \sin (5 c))+(\cos (4 c-d x)-i \sin (4 c-d x)) (511 \cos (c+d x)+153 \cos (3 (c+d x))-i (49 \sin (c+d x)+57 \sin (3 (c+d x))))\right ) (-i+\tan (c+d x))^5}{24 d (\cos (d x)+i \sin (d x))^5} \]
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Time = 7.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {16 i a^{5} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{5} \left (87 \,{\mathrm e}^{5 i \left (d x +c \right )}+136 \,{\mathrm e}^{3 i \left (d x +c \right )}+57 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {35 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {35 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(118\) |
derivativedivides | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-10 a^{5} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-5 i a^{5} \cos \left (d x +c \right )+a^{5} \sin \left (d x +c \right )}{d}\) | \(226\) |
default | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-10 a^{5} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-5 i a^{5} \cos \left (d x +c \right )+a^{5} \sin \left (d x +c \right )}{d}\) | \(226\) |
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Time = 0.24 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.66 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {-96 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 462 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 560 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 210 i \, a^{5} e^{\left (i \, d x + i \, c\right )} - 105 \, {\left (a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 105 \, {\left (a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{6 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.52 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {35 a^{5} \left (\frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{2} - \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{2}\right )}{d} + \frac {- 87 i a^{5} e^{5 i c} e^{5 i d x} - 136 i a^{5} e^{3 i c} e^{3 i d x} - 57 i a^{5} e^{i c} e^{i d x}}{3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} + 3 d} + \begin {cases} - \frac {16 i a^{5} e^{i c} e^{i d x}}{d} & \text {for}\: d \neq 0 \\16 a^{5} x e^{i c} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.33 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {15 \, a^{5} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} + 120 i \, a^{5} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 4 i \, a^{5} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} + 60 \, a^{5} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 60 i \, a^{5} \cos \left (d x + c\right ) - 12 \, a^{5} \sin \left (d x + c\right )}{12 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (108) = 216\).
Time = 0.81 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.92 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\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 )}} \]
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Time = 8.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.71 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {35\,a^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {37\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,27{}\mathrm {i}-118\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,48{}\mathrm {i}+139\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,55{}\mathrm {i}}{3}-\frac {166\,a^5}{3}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,1{}\mathrm {i}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,3{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
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