Optimal. Leaf size=82 \[ -\frac{i (a+i a \tan (c+d x))^8}{8 a^5 d}+\frac{4 i (a+i a \tan (c+d x))^7}{7 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^6}{3 a^3 d} \]
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Rubi [A] time = 0.0563161, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 43} \[ -\frac{i (a+i a \tan (c+d x))^8}{8 a^5 d}+\frac{4 i (a+i a \tan (c+d x))^7}{7 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^6}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^5 \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^5-4 a (a+x)^6+(a+x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{2 i (a+i a \tan (c+d x))^6}{3 a^3 d}+\frac{4 i (a+i a \tan (c+d x))^7}{7 a^4 d}-\frac{i (a+i a \tan (c+d x))^8}{8 a^5 d}\\ \end{align*}
Mathematica [A] time = 1.20586, size = 106, normalized size = 1.29 \[ \frac{a^3 \sec (c) \sec ^8(c+d x) (28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+28 \sin (3 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x)+28 i \cos (c+2 d x)+28 i \cos (3 c+2 d x)-35 \sin (c)+35 i \cos (c))}{168 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 174, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ( -i{a}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{12\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{24\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) -3\,{a}^{3} \left ( 1/7\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{{\frac{i}{2}}{a}^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}-{a}^{3} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13342, size = 146, normalized size = 1.78 \begin{align*} \frac{-105 i \, a^{3} \tan \left (d x + c\right )^{8} - 360 \, a^{3} \tan \left (d x + c\right )^{7} + 140 i \, a^{3} \tan \left (d x + c\right )^{6} - 840 \, a^{3} \tan \left (d x + c\right )^{5} + 1050 i \, a^{3} \tan \left (d x + c\right )^{4} - 280 \, a^{3} \tan \left (d x + c\right )^{3} + 1260 i \, a^{3} \tan \left (d x + c\right )^{2} + 840 \, a^{3} \tan \left (d x + c\right )}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.03985, size = 547, normalized size = 6.67 \begin{align*} \frac{1792 i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} + 2240 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 1792 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 896 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 256 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, a^{3}}{21 \,{\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, 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^{3} \left (\int - 3 \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int 3 i \tan{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int - i \tan ^{3}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \sec ^{6}{\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 = 1.2548, size = 146, normalized size = 1.78 \begin{align*} -\frac{21 i \, a^{3} \tan \left (d x + c\right )^{8} + 72 \, a^{3} \tan \left (d x + c\right )^{7} - 28 i \, a^{3} \tan \left (d x + c\right )^{6} + 168 \, a^{3} \tan \left (d x + c\right )^{5} - 210 i \, a^{3} \tan \left (d x + c\right )^{4} + 56 \, a^{3} \tan \left (d x + c\right )^{3} - 252 i \, a^{3} \tan \left (d x + c\right )^{2} - 168 \, a^{3} \tan \left (d x + c\right )}{168 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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