Integrand size = 24, antiderivative size = 213 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac {20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {8 i \sec ^3(c+d x)}{9009 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3} \]
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Time = 0.36 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3583, 3569} \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec ^3(c+d x)}{9009 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8} \]
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Rule 3569
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{13 a} \\ & = \frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {20 \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{143 a^2} \\ & = \frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac {20 \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{429 a^3} \\ & = \frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac {20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac {40 \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{3003 a^4} \\ & = \frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac {20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {8 \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{3003 a^5} \\ & = \frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac {20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec ^3(c+d x)}{9009 a^5 d (a+i a \tan (c+d x))^3}+\frac {8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.45 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^8(c+d x) (11440 \cos (c+d x)+6552 \cos (3 (c+d x))+1848 \cos (5 (c+d x))+1430 i \sin (c+d x)+2457 i \sin (3 (c+d x))+1155 i \sin (5 (c+d x)))}{144144 a^8 d (-i+\tan (c+d x))^8} \]
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Time = 0.78 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{96 a^{8} d}+\frac {i {\mathrm e}^{-5 i \left (d x +c \right )}}{32 a^{8} d}+\frac {5 i {\mathrm e}^{-7 i \left (d x +c \right )}}{112 a^{8} d}+\frac {5 i {\mathrm e}^{-9 i \left (d x +c \right )}}{144 a^{8} d}+\frac {5 i {\mathrm e}^{-11 i \left (d x +c \right )}}{352 a^{8} d}+\frac {i {\mathrm e}^{-13 i \left (d x +c \right )}}{416 a^{8} d}\) | \(110\) |
derivativedivides | \(\frac {-\frac {188}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {256}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}+\frac {480}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {2672 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {9056}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {11680}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {864 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {200 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {1472 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {4544}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\) | \(222\) |
default | \(\frac {-\frac {188}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {256}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}+\frac {480}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {2672 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {9056}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {11680}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {864 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {200 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {1472 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {4544}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\) | \(222\) |
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Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.35 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (3003 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 9009 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 12870 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10010 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4095 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 693 i\right )} e^{\left (-13 i \, d x - 13 i \, c\right )}}{288288 \, a^{8} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (189) = 378\).
Time = 8.74 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.36 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.66 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {693 i \, \cos \left (13 \, d x + 13 \, c\right ) + 4095 i \, \cos \left (11 \, d x + 11 \, c\right ) + 10010 i \, \cos \left (9 \, d x + 9 \, c\right ) + 12870 i \, \cos \left (7 \, d x + 7 \, c\right ) + 9009 i \, \cos \left (5 \, d x + 5 \, c\right ) + 3003 i \, \cos \left (3 \, d x + 3 \, c\right ) + 693 \, \sin \left (13 \, d x + 13 \, c\right ) + 4095 \, \sin \left (11 \, d x + 11 \, c\right ) + 10010 \, \sin \left (9 \, d x + 9 \, c\right ) + 12870 \, \sin \left (7 \, d x + 7 \, c\right ) + 9009 \, \sin \left (5 \, d x + 5 \, c\right ) + 3003 \, \sin \left (3 \, d x + 3 \, c\right )}{288288 \, a^{8} d} \]
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Time = 1.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.83 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2 \, {\left (9009 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 45045 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 183183 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 435435 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 810810 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1051050 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1076790 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 785070 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 451165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 171457 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 51675 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7111 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1240\right )}}{9009 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{13}} \]
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Time = 5.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\frac {{\cos \left (3\,c+3\,d\,x\right )}^3\,5{}\mathrm {i}}{36}+\frac {5\,\sin \left (3\,c+3\,d\,x\right )\,{\cos \left (3\,c+3\,d\,x\right )}^2}{36}-\frac {\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{32}+\frac {\cos \left (5\,c+5\,d\,x\right )\,1{}\mathrm {i}}{32}+\frac {\cos \left (7\,c+7\,d\,x\right )\,5{}\mathrm {i}}{112}+\frac {\cos \left (11\,c+11\,d\,x\right )\,5{}\mathrm {i}}{352}+\frac {\cos \left (13\,c+13\,d\,x\right )\,1{}\mathrm {i}}{416}-\frac {7\,\sin \left (3\,c+3\,d\,x\right )}{288}+\frac {\sin \left (5\,c+5\,d\,x\right )}{32}+\frac {5\,\sin \left (7\,c+7\,d\,x\right )}{112}+\frac {5\,\sin \left (11\,c+11\,d\,x\right )}{352}+\frac {\sin \left (13\,c+13\,d\,x\right )}{416}}{a^8\,d} \]
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