Integrand size = 24, antiderivative size = 82 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {2 i (a-i a \tan (c+d x))^6}{3 a^9 d}-\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac {i (a-i a \tan (c+d x))^8}{8 a^{11} d} \]
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Time = 0.08 (sec) , antiderivative size = 82, 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} \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {i (a-i a \tan (c+d x))^8}{8 a^{11} d}-\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac {2 i (a-i a \tan (c+d x))^6}{3 a^9 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x)^5 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^{11} d} \\ & = -\frac {i \text {Subst}\left (\int \left (4 a^2 (a-x)^5-4 a (a-x)^6+(a-x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{11} d} \\ & = \frac {2 i (a-i a \tan (c+d x))^6}{3 a^9 d}-\frac {4 i (a-i a \tan (c+d x))^7}{7 a^{10} d}+\frac {i (a-i a \tan (c+d x))^8}{8 a^{11} d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.56 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {(i+\tan (c+d x))^6 \left (-37 i+54 \tan (c+d x)+21 i \tan ^2(c+d x)\right )}{168 a^3 d} \]
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Time = 0.46 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {32 i \left (28 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{21 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\) | \(47\) |
derivativedivides | \(\frac {i \left (\frac {\left (\tan ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {3 i \left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {5 \left (\tan ^{4}\left (d x +c \right )\right )}{4}+i \left (\tan ^{5}\left (d x +c \right )\right )-\frac {3 \left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {i \left (\tan ^{3}\left (d x +c \right )\right )}{3}-i \tan \left (d x +c \right )\right )}{a^{3} d}\) | \(93\) |
default | \(\frac {i \left (\frac {\left (\tan ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {3 i \left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {5 \left (\tan ^{4}\left (d x +c \right )\right )}{4}+i \left (\tan ^{5}\left (d x +c \right )\right )-\frac {3 \left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {i \left (\tan ^{3}\left (d x +c \right )\right )}{3}-i \tan \left (d x +c \right )\right )}{a^{3} d}\) | \(93\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (64) = 128\).
Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.87 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {32 \, {\left (-28 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 8 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}}{21 \, {\left (a^{3} d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a^{3} d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a^{3} d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d\right )}} \]
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\[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \int \frac {\sec ^{12}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \]
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Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \]
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Time = 0.77 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {-21 i \, \tan \left (d x + c\right )^{8} + 72 \, \tan \left (d x + c\right )^{7} + 28 i \, \tan \left (d x + c\right )^{6} + 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} + 56 \, \tan \left (d x + c\right )^{3} + 252 i \, \tan \left (d x + c\right )^{2} - 168 \, \tan \left (d x + c\right )}{168 \, a^{3} d} \]
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Time = 4.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.26 \[ \int \frac {\sec ^{12}(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {{\cos \left (c+d\,x\right )}^8\,91{}\mathrm {i}+128\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^7+64\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^5+48\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^3-{\cos \left (c+d\,x\right )}^2\,112{}\mathrm {i}-72\,\sin \left (c+d\,x\right )\,\cos \left (c+d\,x\right )+21{}\mathrm {i}}{168\,a^3\,d\,{\cos \left (c+d\,x\right )}^8} \]
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