Integrand size = 24, antiderivative size = 81 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {4 i}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac {i}{d \left (a^2+i a^2 \tan (c+d x)\right )^4} \]
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Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i}{3 a^5 d (a+i a \tan (c+d x))^3}+\frac {4 i}{5 a^3 d (a+i a \tan (c+d x))^5}-\frac {i}{d \left (a^2+i a^2 \tan (c+d x)\right )^4} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^6} \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = -\frac {i \text {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^6}-\frac {4 a}{(a+x)^5}+\frac {1}{(a+x)^4}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = \frac {4 i}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac {i}{d \left (a^2+i a^2 \tan (c+d x)\right )^4} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2-5 i \tan (c+d x)-5 \tan ^2(c+d x)}{15 a^8 d (-i+\tan (c+d x))^5} \]
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Time = 0.53 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(-\frac {\frac {i}{\left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {4}{5 \left (\tan \left (d x +c \right )-i\right )^{5}}+\frac {1}{3 \left (\tan \left (d x +c \right )-i\right )^{3}}}{a^{8} d}\) | \(50\) |
default | \(-\frac {\frac {i}{\left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {4}{5 \left (\tan \left (d x +c \right )-i\right )^{5}}+\frac {1}{3 \left (\tan \left (d x +c \right )-i\right )^{3}}}{a^{8} d}\) | \(50\) |
risch | \(\frac {i {\mathrm e}^{-6 i \left (d x +c \right )}}{24 a^{8} d}+\frac {i {\mathrm e}^{-8 i \left (d x +c \right )}}{16 a^{8} d}+\frac {i {\mathrm e}^{-10 i \left (d x +c \right )}}{40 a^{8} d}\) | \(56\) |
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none
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.51 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (10 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i\right )} e^{\left (-10 i \, d x - 10 i \, c\right )}}{240 \, a^{8} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (65) = 130\).
Time = 8.73 (sec) , antiderivative size = 466, normalized size of antiderivative = 5.75 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} - \frac {i \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}}{240 a^{8} d \tan ^{8}{\left (c + d x \right )} - 1920 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{6}{\left (c + d x \right )} + 13440 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 16800 a^{8} d \tan ^{4}{\left (c + d x \right )} - 13440 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{2}{\left (c + d x \right )} + 1920 i a^{8} d \tan {\left (c + d x \right )} + 240 a^{8} d} - \frac {8 \tan {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}}{240 a^{8} d \tan ^{8}{\left (c + d x \right )} - 1920 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{6}{\left (c + d x \right )} + 13440 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 16800 a^{8} d \tan ^{4}{\left (c + d x \right )} - 13440 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{2}{\left (c + d x \right )} + 1920 i a^{8} d \tan {\left (c + d x \right )} + 240 a^{8} d} + \frac {31 i \sec ^{6}{\left (c + d x \right )}}{240 a^{8} d \tan ^{8}{\left (c + d x \right )} - 1920 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{6}{\left (c + d x \right )} + 13440 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 16800 a^{8} d \tan ^{4}{\left (c + d x \right )} - 13440 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 6720 a^{8} d \tan ^{2}{\left (c + d x \right )} + 1920 i a^{8} d \tan {\left (c + d x \right )} + 240 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{6}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (65) = 130\).
Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.74 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {5 \, \tan \left (d x + c\right )^{4} - 5 i \, \tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )^{2} - i \, \tan \left (d x + c\right ) + 2}{15 \, {\left (a^{8} \tan \left (d x + c\right )^{7} - 7 i \, a^{8} \tan \left (d x + c\right )^{6} - 21 \, a^{8} \tan \left (d x + c\right )^{5} + 35 i \, a^{8} \tan \left (d x + c\right )^{4} + 35 \, a^{8} \tan \left (d x + c\right )^{3} - 21 i \, a^{8} \tan \left (d x + c\right )^{2} - 7 \, a^{8} \tan \left (d x + c\right ) + i \, a^{8}\right )} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (65) = 130\).
Time = 1.50 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.69 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 30 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 170 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 282 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 170 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{15 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{10}} \]
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Time = 4.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {-{\mathrm {tan}\left (c+d\,x\right )}^2\,5{}\mathrm {i}+5\,\mathrm {tan}\left (c+d\,x\right )+2{}\mathrm {i}}{15\,a^8\,d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,10{}\mathrm {i}-10\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,5{}\mathrm {i}+1\right )} \]
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