Optimal. Leaf size=163 \[ \frac{7 i a^4 \sec ^3(c+d x)}{8 d}+\frac{21 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{21 i \sec ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{40 d}+\frac{21 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]
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Rubi [A] time = 0.161987, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3498, 3486, 3768, 3770} \[ \frac{7 i a^4 \sec ^3(c+d x)}{8 d}+\frac{21 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{21 i \sec ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{40 d}+\frac{21 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 3498
Rule 3486
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx &=\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{1}{2} (3 a) \int \sec ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{3 i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{1}{10} \left (21 a^2\right ) \int \sec ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{3 i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{21 i \sec ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{40 d}+\frac{1}{8} \left (21 a^3\right ) \int \sec ^3(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=\frac{7 i a^4 \sec ^3(c+d x)}{8 d}+\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{3 i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{21 i \sec ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{40 d}+\frac{1}{8} \left (21 a^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{7 i a^4 \sec ^3(c+d x)}{8 d}+\frac{21 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{3 i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{21 i \sec ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{40 d}+\frac{1}{16} \left (21 a^4\right ) \int \sec (c+d x) \, dx\\ &=\frac{21 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{7 i a^4 \sec ^3(c+d x)}{8 d}+\frac{21 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{i a \sec ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{3 i \sec ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{21 i \sec ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{40 d}\\ \end{align*}
Mathematica [A] time = 1.78356, size = 171, normalized size = 1.05 \[ -\frac{a^4 (\cos (4 c)-i \sin (4 c)) (\tan (c+d x)-i)^4 \sec ^2(c+d x) \left (-4608 i \cos (c+d x)+5 (90 \sin (c+d x)+155 \sin (3 (c+d x))-63 \sin (5 (c+d x))-512 i \cos (3 (c+d x)))+5040 \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{3840 d (\cos (d x)+i \sin (d x))^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 324, normalized size = 2. \begin{align*}{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{24\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{48\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{48\,d}}-{\frac{13\,{a}^{4}\sin \left ( dx+c \right ) }{16\,d}}+{\frac{21\,{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{{\frac{8\,i}{15}}{a}^{4}\cos \left ( dx+c \right ) }{d}}+{\frac{{\frac{4\,i}{15}}{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}-{\frac{{\frac{4\,i}{15}}{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{\frac{4\,i}{15}}{a}^{4}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{{\frac{4\,i}{5}}{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{\frac{4\,i}{3}}{a}^{4}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10252, size = 332, normalized size = 2.04 \begin{align*} -\frac{5 \, a^{4}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{5} + 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \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 )} + 180 \, a^{4}{\left (\frac{2 \,{\left (\sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \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 )} + 120 \, a^{4}{\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 )} - \frac{640 i \, a^{4}}{\cos \left (d x + c\right )^{3}} - \frac{128 i \,{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} a^{4}}{\cos \left (d x + c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.21524, size = 1064, normalized size = 6.53 \begin{align*} \frac{-630 i \, a^{4} e^{\left (11 i \, d x + 11 i \, c\right )} + 6670 i \, a^{4} e^{\left (9 i \, d x + 9 i \, c\right )} + 10116 i \, a^{4} e^{\left (7 i \, d x + 7 i \, c\right )} + 8316 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} + 3570 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} + 630 i \, a^{4} e^{\left (i \, d x + i \, c\right )} + 315 \,{\left (a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 315 \,{\left (a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{240 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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^{4} \left (\int - 6 \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 i \tan{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int - 4 i \tan ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\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 = 1.35276, size = 323, normalized size = 1.98 \begin{align*} \frac{315 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 315 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (75 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 960 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 1175 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 4800 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1890 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 4480 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1890 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1920 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1175 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1728 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 75 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 448 i \, a^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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