Integrand size = 24, antiderivative size = 68 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7} \]
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Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3583, 3569} \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8} \]
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Rule 3569
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {\int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{9 a} \\ & = \frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.59 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {\sec ^7(c+d x) (-8 i+\tan (c+d x))}{63 a^8 d (-i+\tan (c+d x))^8} \]
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Time = 0.90 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.56
method | result | size |
risch | \(\frac {i {\mathrm e}^{-7 i \left (d x +c \right )}}{14 a^{8} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{18 a^{8} d}\) | \(38\) |
derivativedivides | \(\frac {-\frac {1856}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {152 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {172}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {992 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {272}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {256}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}}{a^{8} d}\) | \(156\) |
default | \(\frac {-\frac {1856}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {152 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {172}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {992 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {272}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {256}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}}{a^{8} d}\) | \(156\) |
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.44 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (9 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{126 \, a^{8} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (54) = 108\).
Time = 8.97 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.57 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} - \frac {\tan {\left (c + d x \right )} \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} + \frac {8 i \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{7}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {7 i \, \cos \left (9 \, d x + 9 \, c\right ) + 9 i \, \cos \left (7 \, d x + 7 \, c\right ) + 7 \, \sin \left (9 \, d x + 9 \, c\right ) + 9 \, \sin \left (7 \, d x + 7 \, c\right )}{126 \, a^{8} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (56) = 112\).
Time = 1.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.84 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2 \, {\left (63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 63 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 483 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 189 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{9}} \]
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Time = 4.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.54 \[ \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2\,\left (\frac {{\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,9{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{-c\,9{}\mathrm {i}-d\,x\,9{}\mathrm {i}}\,7{}\mathrm {i}}{4}\right )}{63\,a^8\,d} \]
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