Integrand size = 24, antiderivative size = 183 \[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {63 \text {arctanh}(\sin (c+d x))}{2 a^8 d}-\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3581, 3853, 3855} \[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {63 \text {arctanh}(\sin (c+d x))}{2 a^8 d}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \tan (c+d x) \sec (c+d x)}{2 a^8 d}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7} \]
[In]
[Out]
Rule 3581
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {9 \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{5 a^2} \\ & = \frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {21 \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{5 a^4} \\ & = \frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}-\frac {21 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^6} \\ & = \frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \int \sec ^3(c+d x) \, dx}{a^8} \\ & = -\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \int \sec (c+d x) \, dx}{2 a^8} \\ & = -\frac {63 \text {arctanh}(\sin (c+d x))}{2 a^8 d}-\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1244\) vs. \(2(183)=366\).
Time = 7.08 (sec) , antiderivative size = 1244, normalized size of antiderivative = 6.80 \[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {63 \cos (8 c) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}-\frac {63 \cos (8 c) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}+\frac {\cos (5 d x) \sec ^8(c+d x) \left (\frac {8}{5} i \cos (3 c)-\frac {8}{5} \sin (3 c)\right ) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\cos (3 d x) \sec ^8(c+d x) (-8 i \cos (5 c)+8 \sin (5 c)) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\cos (d x) \sec ^8(c+d x) (48 i \cos (7 c)-48 \sin (7 c)) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\sec (c) \sec ^8(c+d x) (8 i \cos (8 c)-8 \sin (8 c)) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {63 i \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}-\frac {63 i \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) (48 \cos (7 c)+48 i \sin (7 c)) (\cos (d x)+i \sin (d x))^8 \sin (d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) (-8 \cos (5 c)-8 i \sin (5 c)) (\cos (d x)+i \sin (d x))^8 \sin (3 d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (\frac {8}{5} \cos (3 c)+\frac {8}{5} i \sin (3 c)\right ) (\cos (d x)+i \sin (d x))^8 \sin (5 d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (\frac {1}{4} \cos (8 c)+\frac {1}{4} i \sin (8 c)\right ) (\cos (d x)+i \sin (d x))^8}{d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2 (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (-\frac {1}{4} \cos (8 c)-\frac {1}{4} i \sin (8 c)\right ) (\cos (d x)+i \sin (d x))^8}{d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2 (a+i a \tan (c+d x))^8}+\frac {8 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}+\frac {8 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (-\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8} \]
[In]
[Out]
Time = 0.96 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {48 i {\mathrm e}^{-i \left (d x +c \right )}}{a^{8} d}-\frac {8 i {\mathrm e}^{-3 i \left (d x +c \right )}}{a^{8} d}+\frac {8 i {\mathrm e}^{-5 i \left (d x +c \right )}}{5 a^{8} d}+\frac {i \left (15 \,{\mathrm e}^{3 i \left (d x +c \right )}+17 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \,a^{8} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {63 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{8} d}+\frac {63 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{8} d}\) | \(143\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {1}{4}+4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\frac {2 \left (\frac {1}{4}-4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}-\frac {32 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {256}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {64}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {64}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\) | \(184\) |
default | \(\frac {\frac {2 \left (\frac {1}{4}+4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\frac {2 \left (\frac {1}{4}-4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}-\frac {32 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {256}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {64}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {64}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\) | \(184\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {315 \, {\left (e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, e^{\left (7 i \, d x + 7 i \, c\right )} + e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 315 \, {\left (e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, e^{\left (7 i \, d x + 7 i \, c\right )} + e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 630 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 1050 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 336 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 48 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 16 i}{10 \, {\left (a^{8} d e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, a^{8} d e^{\left (7 i \, d x + 7 i \, c\right )} + a^{8} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )}} \]
[In]
[Out]
\[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\int \frac {\sec ^{11}{\left (c + d x \right )}}{\tan ^{8}{\left (c + d x \right )} - 8 i \tan ^{7}{\left (c + d x \right )} - 28 \tan ^{6}{\left (c + d x \right )} + 56 i \tan ^{5}{\left (c + d x \right )} + 70 \tan ^{4}{\left (c + d x \right )} - 56 i \tan ^{3}{\left (c + d x \right )} - 28 \tan ^{2}{\left (c + d x \right )} + 8 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{8}} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (157) = 314\).
Time = 0.34 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.90 \[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {630 \, {\left (\cos \left (9 \, d x + 9 \, c\right ) + 2 \, \cos \left (7 \, d x + 7 \, c\right ) + \cos \left (5 \, d x + 5 \, c\right ) + i \, \sin \left (9 \, d x + 9 \, c\right ) + 2 i \, \sin \left (7 \, d x + 7 \, c\right ) + i \, \sin \left (5 \, d x + 5 \, c\right )\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) + 630 \, {\left (\cos \left (9 \, d x + 9 \, c\right ) + 2 \, \cos \left (7 \, d x + 7 \, c\right ) + \cos \left (5 \, d x + 5 \, c\right ) + i \, \sin \left (9 \, d x + 9 \, c\right ) + 2 i \, \sin \left (7 \, d x + 7 \, c\right ) + i \, \sin \left (5 \, d x + 5 \, c\right )\right )} \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) + 315 \, {\left (i \, \cos \left (9 \, d x + 9 \, c\right ) + 2 i \, \cos \left (7 \, d x + 7 \, c\right ) + i \, \cos \left (5 \, d x + 5 \, c\right ) - \sin \left (9 \, d x + 9 \, c\right ) - 2 \, \sin \left (7 \, d x + 7 \, c\right ) - \sin \left (5 \, d x + 5 \, c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + 315 \, {\left (-i \, \cos \left (9 \, d x + 9 \, c\right ) - 2 i \, \cos \left (7 \, d x + 7 \, c\right ) - i \, \cos \left (5 \, d x + 5 \, c\right ) + \sin \left (9 \, d x + 9 \, c\right ) + 2 \, \sin \left (7 \, d x + 7 \, c\right ) + \sin \left (5 \, d x + 5 \, c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 1260 \, \cos \left (8 \, d x + 8 \, c\right ) + 2100 \, \cos \left (6 \, d x + 6 \, c\right ) + 672 \, \cos \left (4 \, d x + 4 \, c\right ) - 96 \, \cos \left (2 \, d x + 2 \, c\right ) + 1260 i \, \sin \left (8 \, d x + 8 \, c\right ) + 2100 i \, \sin \left (6 \, d x + 6 \, c\right ) + 672 i \, \sin \left (4 \, d x + 4 \, c\right ) - 96 i \, \sin \left (2 \, d x + 2 \, c\right ) + 32}{-20 \, {\left (i \, a^{8} \cos \left (9 \, d x + 9 \, c\right ) + 2 i \, a^{8} \cos \left (7 \, d x + 7 \, c\right ) + i \, a^{8} \cos \left (5 \, d x + 5 \, c\right ) - a^{8} \sin \left (9 \, d x + 9 \, c\right ) - 2 \, a^{8} \sin \left (7 \, d x + 7 \, c\right ) - a^{8} \sin \left (5 \, d x + 5 \, c\right )\right )} d} \]
[In]
[Out]
none
Time = 1.81 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {\frac {315 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} - \frac {315 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} - \frac {10 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{8}} - \frac {64 \, {\left (10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 85 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 55 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{5}}}{10 \, d} \]
[In]
[Out]
Time = 8.67 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.55 \[ \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {63\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^8\,d}+\frac {\frac {1223\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{a^8}-\frac {1109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^8}+\frac {309\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{a^8}-\frac {431\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,4407{}\mathrm {i}}{5\,a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,7351{}\mathrm {i}}{5\,a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,761{}\mathrm {i}}{a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,65{}\mathrm {i}}{a^8}+\frac {496{}\mathrm {i}}{5\,a^8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,12{}\mathrm {i}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,26{}\mathrm {i}+26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,20{}\mathrm {i}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,5{}\mathrm {i}+1\right )} \]
[In]
[Out]