Optimal. Leaf size=27 \[ -\frac{i (a+i a \tan (c+d x))^9}{9 a d} \]
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Rubi [A] time = 0.0377482, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 32} \[ -\frac{i (a+i a \tan (c+d x))^9}{9 a d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a+x)^8 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{i (a+i a \tan (c+d x))^9}{9 a d}\\ \end{align*}
Mathematica [B] time = 2.86396, size = 212, normalized size = 7.85 \[ \frac{a^8 \sec (c) \sec ^9(c+d x) (-126 \sin (2 c+d x)+84 \sin (2 c+3 d x)-84 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)-36 \sin (6 c+5 d x)+9 \sin (6 c+7 d x)-9 \sin (8 c+7 d x)+2 \sin (8 c+9 d x)+126 i \cos (2 c+d x)+84 i \cos (2 c+3 d x)+84 i \cos (4 c+3 d x)+36 i \cos (4 c+5 d x)+36 i \cos (6 c+5 d x)+9 i \cos (6 c+7 d x)+9 i \cos (8 c+7 d x)+126 \sin (d x)+126 i \cos (d x))}{18 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.089, size = 180, normalized size = 6.7 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}-{\frac{14\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-4\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{{\frac{28\,i}{3}}{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+14\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}-{\frac{28\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{4\,i{a}^{8}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{a}^{8}\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 = 1.16394, size = 28, normalized size = 1.04 \begin{align*} -\frac{i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.29112, size = 741, normalized size = 27.44 \begin{align*} \frac{4608 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 18432 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 43008 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 64512 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 64512 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 43008 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 18432 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4608 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 512 i \, a^{8}}{9 \,{\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^{8} \left (\int - 28 \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - 28 \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 8 i \tan{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int - 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.79316, size = 162, normalized size = 6. \begin{align*} \frac{a^{8} \tan \left (d x + c\right )^{9} - 9 i \, a^{8} \tan \left (d x + c\right )^{8} - 36 \, a^{8} \tan \left (d x + c\right )^{7} + 84 i \, a^{8} \tan \left (d x + c\right )^{6} + 126 \, a^{8} \tan \left (d x + c\right )^{5} - 126 i \, a^{8} \tan \left (d x + c\right )^{4} - 84 \, a^{8} \tan \left (d x + c\right )^{3} + 36 i \, a^{8} \tan \left (d x + c\right )^{2} + 9 \, a^{8} \tan \left (d x + c\right )}{9 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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