Optimal. Leaf size=55 \[ \frac{i (a+i a \tan (c+d x))^{11}}{11 a^3 d}-\frac{i (a+i a \tan (c+d x))^{10}}{5 a^2 d} \]
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Rubi [A] time = 0.0454868, antiderivative size = 55, 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} \[ \frac{i (a+i a \tan (c+d x))^{11}}{11 a^3 d}-\frac{i (a+i a \tan (c+d x))^{10}}{5 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x) (a+x)^9 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i (a+i a \tan (c+d x))^{10}}{5 a^2 d}+\frac{i (a+i a \tan (c+d x))^{11}}{11 a^3 d}\\ \end{align*}
Mathematica [B] time = 4.20502, size = 223, normalized size = 4.05 \[ \frac{a^8 \sec (c) \sec ^{11}(c+d x) (-462 \sin (2 c+d x)+330 \sin (2 c+3 d x)-330 \sin (4 c+3 d x)+165 \sin (4 c+5 d x)-165 \sin (6 c+5 d x)+55 \sin (6 c+7 d x)-55 \sin (8 c+7 d x)+22 \sin (8 c+9 d x)+2 \sin (10 c+11 d x)+462 i \cos (2 c+d x)+330 i \cos (2 c+3 d x)+330 i \cos (4 c+3 d x)+165 i \cos (4 c+5 d x)+165 i \cos (6 c+5 d x)+55 i \cos (6 c+7 d x)+55 i \cos (8 c+7 d x)+462 \sin (d x)+462 i \cos (d x))}{220 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.095, size = 339, normalized size = 6.2 \begin{align*}{\frac{1}{d} \left ({a}^{8} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{11\, \left ( \cos \left ( dx+c \right ) \right ) ^{11}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{99\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}} \right ) -56\,i{a}^{8} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{12\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) -28\,{a}^{8} \left ( 1/9\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{63\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}} \right ) +56\,i{a}^{8} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{24\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \right ) +70\,{a}^{8} \left ( 1/7\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \right ) -8\,i{a}^{8} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{10\, \left ( \cos \left ( dx+c \right ) \right ) ^{10}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{40\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}} \right ) -28\,{a}^{8} \left ( 1/5\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+2/15\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{2\,i{a}^{8}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{a}^{8} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.34338, size = 181, normalized size = 3.29 \begin{align*} \frac{45 \, a^{8} \tan \left (d x + c\right )^{11} - 396 i \, a^{8} \tan \left (d x + c\right )^{10} - 1485 \, a^{8} \tan \left (d x + c\right )^{9} + 2970 i \, a^{8} \tan \left (d x + c\right )^{8} + 2970 \, a^{8} \tan \left (d x + c\right )^{7} + 4158 \, a^{8} \tan \left (d x + c\right )^{5} - 5940 i \, a^{8} \tan \left (d x + c\right )^{4} - 4455 \, a^{8} \tan \left (d x + c\right )^{3} + 1980 i \, a^{8} \tan \left (d x + c\right )^{2} + 495 \, a^{8} \tan \left (d x + c\right )}{495 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54089, size = 890, normalized size = 16.18 \begin{align*} \frac{56320 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} + 168960 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 337920 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 473088 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 473088 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 337920 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 168960 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 56320 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 11264 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 1024 i \, a^{8}}{55 \,{\left (d e^{\left (22 i \, d x + 22 i \, c\right )} + 11 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 55 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 165 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 330 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 462 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 462 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 330 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 165 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 55 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 11 \, 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.81662, size = 181, normalized size = 3.29 \begin{align*} \frac{5 \, a^{8} \tan \left (d x + c\right )^{11} - 44 i \, a^{8} \tan \left (d x + c\right )^{10} - 165 \, a^{8} \tan \left (d x + c\right )^{9} + 330 i \, a^{8} \tan \left (d x + c\right )^{8} + 330 \, a^{8} \tan \left (d x + c\right )^{7} + 462 \, a^{8} \tan \left (d x + c\right )^{5} - 660 i \, a^{8} \tan \left (d x + c\right )^{4} - 495 \, a^{8} \tan \left (d x + c\right )^{3} + 220 i \, a^{8} \tan \left (d x + c\right )^{2} + 55 \, a^{8} \tan \left (d x + c\right )}{55 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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