Integrand size = 22, antiderivative size = 167 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {63 a^5 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {63 i a^5 \sec (c+d x)}{8 d}+\frac {9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {21 i a \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {21 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{8 d} \]
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Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3579, 3567, 3855} \[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {63 a^5 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {63 i a^5 \sec (c+d x)}{8 d}+\frac {21 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{8 d}+\frac {9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac {21 i a \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d} \]
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Rule 3567
Rule 3579
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {1}{5} (9 a) \int \sec (c+d x) (a+i a \tan (c+d x))^4 \, dx \\ & = \frac {9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {1}{20} \left (63 a^2\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx \\ & = \frac {21 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{20 d}+\frac {9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {1}{4} \left (21 a^3\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx \\ & = \frac {21 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{20 d}+\frac {9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {21 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{8 d}+\frac {1}{8} \left (63 a^4\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx \\ & = \frac {63 i a^5 \sec (c+d x)}{8 d}+\frac {21 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{20 d}+\frac {9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {21 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{8 d}+\frac {1}{8} \left (63 a^5\right ) \int \sec (c+d x) \, dx \\ & = \frac {63 a^5 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {63 i a^5 \sec (c+d x)}{8 d}+\frac {21 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{20 d}+\frac {9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d}+\frac {21 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{8 d} \\ \end{align*}
Time = 2.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.69 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 (\cos (5 d x)+i \sin (5 d x)) \left (5040 \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )\right )+i \sec ^5(c+d x) (1344+1920 \cos (2 (c+d x))+640 \cos (4 (c+d x))+450 i \sin (2 (c+d x))+325 i \sin (4 (c+d x)))\right )}{320 d (\cos (d x)+i \sin (d x))^5} \]
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Time = 8.78 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {i a^{5} \left (965 \,{\mathrm e}^{9 i \left (d x +c \right )}+2370 \,{\mathrm e}^{7 i \left (d x +c \right )}+2688 \,{\mathrm e}^{5 i \left (d x +c \right )}+1470 \,{\mathrm e}^{3 i \left (d x +c \right )}+315 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{20 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {63 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {63 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(122\) |
derivativedivides | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{5 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {5 i a^{5}}{\cos \left (d x +c \right )}+a^{5} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(312\) |
default | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{5 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {3 \sin \left (d x +c \right )}{8}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {5 i a^{5}}{\cos \left (d x +c \right )}+a^{5} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(312\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (137) = 274\).
Time = 0.25 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.86 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {1930 i \, a^{5} e^{\left (9 i \, d x + 9 i \, c\right )} + 4740 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} + 5376 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} + 2940 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} + 630 i \, a^{5} e^{\left (i \, d x + i \, c\right )} + 315 \, {\left (a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 315 \, {\left (a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{40 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=i a^{5} \left (\int \left (- i \sec {\left (c + d x \right )}\right )\, dx + \int 5 \tan {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \left (- 10 \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\right )\, dx + \int \tan ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 10 i \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \left (- 5 i \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\right )\, dx\right ) \]
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Time = 0.33 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.29 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {75 \, a^{5} {\left (\frac {2 \, {\left (5 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 600 \, a^{5} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, a^{5} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + \frac {1200 i \, a^{5}}{\cos \left (d x + c\right )} + \frac {800 i \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{5}}{\cos \left (d x + c\right )^{3}} + \frac {16 i \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} + 3\right )} a^{5}}{\cos \left (d x + c\right )^{5}}}{240 \, d} \]
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Time = 0.72 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.13 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {315 \, a^{5} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 315 \, a^{5} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - \frac {2 \, {\left (275 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 200 i \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 750 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1600 i \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3280 i \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 750 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 i \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 275 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 488 i \, a^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{40 \, d} \]
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Time = 7.80 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.37 \[ \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {63\,a^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {55\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,10{}\mathrm {i}-\frac {75\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,80{}\mathrm {i}+a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,164{}\mathrm {i}+\frac {75\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,112{}\mathrm {i}-\frac {55\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {a^5\,122{}\mathrm {i}}{5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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