Integrand size = 24, antiderivative size = 126 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {80 x}{a^8}+\frac {80 i \log (\cos (c+d x))}{a^8 d}-\frac {31 \tan (c+d x)}{a^8 d}+\frac {4 i \tan ^2(c+d x)}{a^8 d}+\frac {\tan ^3(c+d x)}{3 a^8 d}+\frac {16 i}{d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac {80 i}{d \left (a^8+i a^8 \tan (c+d x)\right )} \]
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Time = 0.11 (sec) , antiderivative size = 126, 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} \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\tan ^3(c+d x)}{3 a^8 d}+\frac {4 i \tan ^2(c+d x)}{a^8 d}-\frac {31 \tan (c+d x)}{a^8 d}-\frac {80 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {80 i \log (\cos (c+d x))}{a^8 d}+\frac {80 x}{a^8}+\frac {16 i}{d \left (a^4+i a^4 \tan (c+d x)\right )^2} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {(a-x)^5}{(a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{a^{11} d} \\ & = -\frac {i \text {Subst}\left (\int \left (-31 a^2+8 a x-x^2+\frac {32 a^5}{(a+x)^3}-\frac {80 a^4}{(a+x)^2}+\frac {80 a^3}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d} \\ & = \frac {80 x}{a^8}+\frac {80 i \log (\cos (c+d x))}{a^8 d}-\frac {31 \tan (c+d x)}{a^8 d}+\frac {4 i \tan ^2(c+d x)}{a^8 d}+\frac {\tan ^3(c+d x)}{3 a^8 d}+\frac {16 i}{d \left (a^4+i a^4 \tan (c+d x)\right )^2}-\frac {80 i}{d \left (a^8+i a^8 \tan (c+d x)\right )} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {i \left (-93 i \tan (c+d x)-12 \tan ^2(c+d x)+i \tan ^3(c+d x)+48 \left (5 \log (i-\tan (c+d x))+\frac {-4-5 i \tan (c+d x)}{(-i+\tan (c+d x))^2}\right )\right )}{3 a^8 d} \]
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Time = 0.52 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {32 i {\mathrm e}^{-2 i \left (d x +c \right )}}{a^{8} d}+\frac {4 i {\mathrm e}^{-4 i \left (d x +c \right )}}{a^{8} d}+\frac {160 x}{a^{8}}+\frac {160 c}{a^{8} d}-\frac {4 i \left (36 \,{\mathrm e}^{4 i \left (d x +c \right )}+81 \,{\mathrm e}^{2 i \left (d x +c \right )}+47\right )}{3 d \,a^{8} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {80 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{8} d}\) | \(120\) |
derivativedivides | \(-\frac {31 \tan \left (d x +c \right )}{a^{8} d}+\frac {\tan ^{3}\left (d x +c \right )}{3 a^{8} d}+\frac {4 i \left (\tan ^{2}\left (d x +c \right )\right )}{a^{8} d}+\frac {80 \arctan \left (\tan \left (d x +c \right )\right )}{a^{8} d}-\frac {40 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{a^{8} d}-\frac {80}{a^{8} d \left (\tan \left (d x +c \right )-i\right )}-\frac {16 i}{a^{8} d \left (\tan \left (d x +c \right )-i\right )^{2}}\) | \(123\) |
default | \(-\frac {31 \tan \left (d x +c \right )}{a^{8} d}+\frac {\tan ^{3}\left (d x +c \right )}{3 a^{8} d}+\frac {4 i \left (\tan ^{2}\left (d x +c \right )\right )}{a^{8} d}+\frac {80 \arctan \left (\tan \left (d x +c \right )\right )}{a^{8} d}-\frac {40 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{a^{8} d}-\frac {80}{a^{8} d \left (\tan \left (d x +c \right )-i\right )}-\frac {16 i}{a^{8} d \left (\tan \left (d x +c \right )-i\right )^{2}}\) | \(123\) |
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Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.58 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {4 \, {\left (120 \, d x e^{\left (10 i \, d x + 10 i \, c\right )} + 60 \, {\left (6 \, d x - i\right )} e^{\left (8 i \, d x + 8 i \, c\right )} + 30 \, {\left (12 \, d x - 5 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, {\left (12 \, d x - 11 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 60 \, {\left (-i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 3 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 3 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 15 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )}}{3 \, {\left (a^{8} d e^{\left (10 i \, d x + 10 i \, c\right )} + 3 \, a^{8} d e^{\left (8 i \, d x + 8 i \, c\right )} + 3 \, a^{8} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{8} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
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\[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\int \frac {\sec ^{12}{\left (c + d x \right )}}{\tan ^{8}{\left (c + d x \right )} - 8 i \tan ^{7}{\left (c + d x \right )} - 28 \tan ^{6}{\left (c + d x \right )} + 56 i \tan ^{5}{\left (c + d x \right )} + 70 \tan ^{4}{\left (c + d x \right )} - 56 i \tan ^{3}{\left (c + d x \right )} - 28 \tan ^{2}{\left (c + d x \right )} + 8 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{8}} \]
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Time = 0.22 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {\frac {48 \, {\left (5 \, \tan \left (d x + c\right )^{6} - 29 i \, \tan \left (d x + c\right )^{5} - 70 \, \tan \left (d x + c\right )^{4} + 90 i \, \tan \left (d x + c\right )^{3} + 65 \, \tan \left (d x + c\right )^{2} - 25 i \, \tan \left (d x + c\right ) - 4\right )}}{a^{8} \tan \left (d x + c\right )^{7} - 7 i \, a^{8} \tan \left (d x + c\right )^{6} - 21 \, a^{8} \tan \left (d x + c\right )^{5} + 35 i \, a^{8} \tan \left (d x + c\right )^{4} + 35 \, a^{8} \tan \left (d x + c\right )^{3} - 21 i \, a^{8} \tan \left (d x + c\right )^{2} - 7 \, a^{8} \tan \left (d x + c\right ) + i \, a^{8}} - \frac {\tan \left (d x + c\right )^{3} + 12 i \, \tan \left (d x + c\right )^{2} - 93 \, \tan \left (d x + c\right )}{a^{8}} + \frac {240 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{8}}}{3 \, d} \]
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Time = 1.98 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.78 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {2 \, {\left (-\frac {120 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} + \frac {240 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{8}} - \frac {120 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} + \frac {220 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 93 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 684 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 190 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 684 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 93 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 220 i}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{8}} + \frac {4 \, {\left (-125 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 536 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 846 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 536 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 125 i\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{4}}\right )}}{3 \, d} \]
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Time = 3.81 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,a^8\,d}-\frac {31\,\mathrm {tan}\left (c+d\,x\right )}{a^8\,d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,4{}\mathrm {i}}{a^8\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,80{}\mathrm {i}}{a^8\,d}-\frac {\frac {64}{a^8}+\frac {\mathrm {tan}\left (c+d\,x\right )\,80{}\mathrm {i}}{a^8}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )} \]
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