Integrand size = 26, antiderivative size = 117 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {16 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}+\frac {24 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{19/2}}{19 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{21/2}}{21 a^7 d} \]
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Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 45} \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 i (a+i a \tan (c+d x))^{21/2}}{21 a^7 d}-\frac {12 i (a+i a \tan (c+d x))^{19/2}}{19 a^6 d}+\frac {24 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}-\frac {16 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x)^3 (a+x)^{13/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d} \\ & = -\frac {i \text {Subst}\left (\int \left (8 a^3 (a+x)^{13/2}-12 a^2 (a+x)^{15/2}+6 a (a+x)^{17/2}-(a+x)^{19/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d} \\ & = -\frac {16 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}+\frac {24 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{19/2}}{19 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{21/2}}{21 a^7 d} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 a^3 (-i+\tan (c+d x))^7 \sqrt {a+i a \tan (c+d x)} \left (-3243+7365 i \tan (c+d x)+5865 \tan ^2(c+d x)-1615 i \tan ^3(c+d x)\right )}{33915 d} \]
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Time = 0.61 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.70
\[\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {21}{2}}}{21}-\frac {6 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {19}{2}}}{19}+\frac {12 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {17}{2}}}{17}-\frac {8 a^{3} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {15}{2}}}{15}\right )}{d \,a^{7}}\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (85) = 170\).
Time = 0.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.73 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {2048 \, \sqrt {2} {\left (16 i \, a^{3} e^{\left (21 i \, d x + 21 i \, c\right )} + 168 i \, a^{3} e^{\left (19 i \, d x + 19 i \, c\right )} + 798 i \, a^{3} e^{\left (17 i \, d x + 17 i \, c\right )} + 2261 i \, a^{3} e^{\left (15 i \, d x + 15 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{33915 \, {\left (d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Timed out. \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]
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none
Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 i \, {\left (1615 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {21}{2}} - 10710 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {19}{2}} a + 23940 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {17}{2}} a^{2} - 18088 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a^{3}\right )}}{33915 \, a^{7} d} \]
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\[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{8} \,d x } \]
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Time = 16.01 (sec) , antiderivative size = 690, normalized size of antiderivative = 5.90 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Too large to display} \]
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