Optimal. Leaf size=109 \[ \frac{i (a+i a \tan (c+d x))^9}{9 a^7 d}-\frac{3 i (a+i a \tan (c+d x))^8}{4 a^6 d}+\frac{12 i (a+i a \tan (c+d x))^7}{7 a^5 d}-\frac{4 i (a+i a \tan (c+d x))^6}{3 a^4 d} \]
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Rubi [A] time = 0.0662851, antiderivative size = 109, 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} \[ \frac{i (a+i a \tan (c+d x))^9}{9 a^7 d}-\frac{3 i (a+i a \tan (c+d x))^8}{4 a^6 d}+\frac{12 i (a+i a \tan (c+d x))^7}{7 a^5 d}-\frac{4 i (a+i a \tan (c+d x))^6}{3 a^4 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^5 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 (a+x)^5-12 a^2 (a+x)^6+6 a (a+x)^7-(a+x)^8\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{4 i (a+i a \tan (c+d x))^6}{3 a^4 d}+\frac{12 i (a+i a \tan (c+d x))^7}{7 a^5 d}-\frac{3 i (a+i a \tan (c+d x))^8}{4 a^6 d}+\frac{i (a+i a \tan (c+d x))^9}{9 a^7 d}\\ \end{align*}
Mathematica [A] time = 1.24373, size = 99, normalized size = 0.91 \[ \frac{a^2 \sec (c) \sec ^9(c+d x) (-63 \sin (2 c+d x)+84 \sin (2 c+3 d x)+36 \sin (4 c+5 d x)+9 \sin (6 c+7 d x)+\sin (8 c+9 d x)+63 i \cos (2 c+d x)+63 \sin (d x)+63 i \cos (d x))}{504 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 141, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ( -{a}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{21\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{16\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{315\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{{\frac{i}{4}}{a}^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}-{a}^{2} \left ( -{\frac{16}{35}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{35}} \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 = 1.10516, size = 146, normalized size = 1.34 \begin{align*} -\frac{140 \, a^{2} \tan \left (d x + c\right )^{9} - 315 i \, a^{2} \tan \left (d x + c\right )^{8} + 360 \, a^{2} \tan \left (d x + c\right )^{7} - 1260 i \, a^{2} \tan \left (d x + c\right )^{6} - 1890 i \, a^{2} \tan \left (d x + c\right )^{4} - 840 \, a^{2} \tan \left (d x + c\right )^{3} - 1260 i \, a^{2} \tan \left (d x + c\right )^{2} - 1260 \, a^{2} \tan \left (d x + c\right )}{1260 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.19107, size = 590, normalized size = 5.41 \begin{align*} \frac{8064 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} + 8064 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 5376 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 2304 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 576 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 64 i \, a^{2}}{63 \,{\left (d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, 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^{2} \left (\int - \tan ^{2}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int 2 i \tan{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \sec ^{8}{\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 = 1.20306, size = 146, normalized size = 1.34 \begin{align*} -\frac{28 \, a^{2} \tan \left (d x + c\right )^{9} - 63 i \, a^{2} \tan \left (d x + c\right )^{8} + 72 \, a^{2} \tan \left (d x + c\right )^{7} - 252 i \, a^{2} \tan \left (d x + c\right )^{6} - 378 i \, a^{2} \tan \left (d x + c\right )^{4} - 168 \, a^{2} \tan \left (d x + c\right )^{3} - 252 i \, a^{2} \tan \left (d x + c\right )^{2} - 252 \, a^{2} \tan \left (d x + c\right )}{252 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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