Optimal. Leaf size=167 \[ \frac{63 i a^5 \sec (c+d x)}{8 d}+\frac{63 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac{21 i a \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{21 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{8 d}+\frac{i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d} \]
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Rubi [A] time = 0.126343, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3498, 3486, 3770} \[ \frac{63 i a^5 \sec (c+d x)}{8 d}+\frac{63 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{9 i a^2 \sec (c+d x) (a+i a \tan (c+d x))^3}{20 d}+\frac{21 i a \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac{21 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{8 d}+\frac{i a \sec (c+d x) (a+i a \tan (c+d x))^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 3498
Rule 3486
Rule 3770
Rubi steps
\begin{align*} \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx &=\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 \tanh ^{-1}(\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*}
Mathematica [A] time = 1.01146, size = 115, normalized size = 0.69 \[ \frac{a^5 (\cos (5 d x)+i \sin (5 d x)) \left (5040 \tanh ^{-1}\left (\cos (c) \tan \left (\frac{d x}{2}\right )+\sin (c)\right )+i \sec ^5(c+d x) (450 i \sin (2 (c+d x))+325 i \sin (4 (c+d x))+1920 \cos (2 (c+d x))+640 \cos (4 (c+d x))+1344)\right )}{320 d (\cos (d x)+i \sin (d x))^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 329, normalized size = 2. \begin{align*}{\frac{{\frac{36\,i}{5}}{a}^{5}\cos \left ( dx+c \right ) }{d}}+{\frac{{\frac{10\,i}{3}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}-{\frac{{\frac{10\,i}{3}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{\frac{i}{5}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{\frac{18\,i}{5}}{a}^{5}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{5\,i{a}^{5}}{d\cos \left ( dx+c \right ) }}+{\frac{5\,{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5\,{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5\,{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d}}-{\frac{55\,{a}^{5}\sin \left ( dx+c \right ) }{8\,d}}+{\frac{63\,{a}^{5}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{\frac{i}{15}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{\frac{i}{5}}{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d\cos \left ( dx+c \right ) }}+{\frac{{\frac{i}{5}}{a}^{5}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}-5\,{\frac{{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11655, size = 290, normalized size = 1.74 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.13587, size = 891, normalized size = 5.34 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{5} \left (\int - 10 \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 5 \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 5 i \tan{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int - 10 i \tan ^{3}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int i \tan ^{5}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \sec{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46856, size = 258, normalized size = 1.54 \begin{align*} \frac{315 \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 315 \, a^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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