Integrand size = 22, antiderivative size = 269 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {8 i \sec (c+d x)}{2145 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )} \]
[Out]
Time = 0.38 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3583, 3569} \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec (c+d x)}{2145 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8} \]
[In]
[Out]
Rule 3569
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{15 a} \\ & = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{65 a^2} \\ & = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{143 a^3} \\ & = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {56 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{1287 a^4} \\ & = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {8 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{429 a^5} \\ & = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac {8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {16 \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{2145 a^6} \\ & = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac {8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {16 \int \frac {\sec (c+d x)}{a+i a \tan (c+d x)} \, dx}{6435 a^7} \\ & = \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac {8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.43 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^8(c+d x) (28600 \cos (c+d x)+19656 \cos (3 (c+d x))+9240 \cos (5 (c+d x))+3432 \cos (7 (c+d x))+3575 i \sin (c+d x)+7371 i \sin (3 (c+d x))+5775 i \sin (5 (c+d x))+3003 i \sin (7 (c+d x)))}{411840 a^8 d (-i+\tan (c+d x))^8} \]
[In]
[Out]
Time = 0.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{128 a^{8} d}+\frac {7 i {\mathrm e}^{-3 i \left (d x +c \right )}}{384 a^{8} d}+\frac {21 i {\mathrm e}^{-5 i \left (d x +c \right )}}{640 a^{8} d}+\frac {5 i {\mathrm e}^{-7 i \left (d x +c \right )}}{128 a^{8} d}+\frac {35 i {\mathrm e}^{-9 i \left (d x +c \right )}}{1152 a^{8} d}+\frac {21 i {\mathrm e}^{-11 i \left (d x +c \right )}}{1408 a^{8} d}+\frac {7 i {\mathrm e}^{-13 i \left (d x +c \right )}}{1664 a^{8} d}+\frac {i {\mathrm e}^{-15 i \left (d x +c \right )}}{1920 a^{8} d}\) | \(146\) |
derivativedivides | \(\frac {\frac {29792}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {15008 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {196}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {224 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3584 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {2944 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {6272}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {256}{15 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}-\frac {2128}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {23744}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}+\frac {3752 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2968}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}}{a^{8} d}\) | \(255\) |
default | \(\frac {\frac {29792}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {15008 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {196}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {224 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3584 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {2944 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {6272}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {256}{15 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}-\frac {2128}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {23744}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}+\frac {3752 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2968}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}}{a^{8} d}\) | \(255\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.36 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (6435 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 15015 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 27027 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 32175 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 25025 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 12285 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3465 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 429 i\right )} e^{\left (-15 i \, d x - 15 i \, c\right )}}{823680 \, a^{8} d} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1221 vs. \(2 (238) = 476\).
Time = 8.86 (sec) , antiderivative size = 1221, normalized size of antiderivative = 4.54 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {429 i \, \cos \left (15 \, d x + 15 \, c\right ) + 3465 i \, \cos \left (13 \, d x + 13 \, c\right ) + 12285 i \, \cos \left (11 \, d x + 11 \, c\right ) + 25025 i \, \cos \left (9 \, d x + 9 \, c\right ) + 32175 i \, \cos \left (7 \, d x + 7 \, c\right ) + 27027 i \, \cos \left (5 \, d x + 5 \, c\right ) + 15015 i \, \cos \left (3 \, d x + 3 \, c\right ) + 6435 i \, \cos \left (d x + c\right ) + 429 \, \sin \left (15 \, d x + 15 \, c\right ) + 3465 \, \sin \left (13 \, d x + 13 \, c\right ) + 12285 \, \sin \left (11 \, d x + 11 \, c\right ) + 25025 \, \sin \left (9 \, d x + 9 \, c\right ) + 32175 \, \sin \left (7 \, d x + 7 \, c\right ) + 27027 \, \sin \left (5 \, d x + 5 \, c\right ) + 15015 \, \sin \left (3 \, d x + 3 \, c\right ) + 6435 \, \sin \left (d x + c\right )}{823680 \, a^{8} d} \]
[In]
[Out]
none
Time = 1.17 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.75 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2 \, {\left (6435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 45045 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 210210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 630630 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1414413 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 2357355 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3063060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 3063060 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2407405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1444443 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 668850 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 222950 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7845 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 952\right )}}{6435 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{15}} \]
[In]
[Out]
Time = 6.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.83 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )\,\left (-\frac {{\sin \left (c+d\,x\right )}^2\,44779{}\mathrm {i}}{32}+\frac {32175\,\sin \left (c+d\,x\right )}{128}-\frac {{\sin \left (2\,c+2\,d\,x\right )}^2\,26075{}\mathrm {i}}{16}-\frac {3575\,\sin \left (2\,c+2\,d\,x\right )}{8}+\frac {{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,114583{}\mathrm {i}}{64}-\frac {{\sin \left (3\,c+3\,d\,x\right )}^2\,57925{}\mathrm {i}}{32}+\frac {84227\,\sin \left (3\,c+3\,d\,x\right )}{128}+\frac {{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2\,116585{}\mathrm {i}}{64}+\frac {{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2\,119315{}\mathrm {i}}{64}+\frac {{\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}^2\,122285{}\mathrm {i}}{64}-754\,\sin \left (4\,c+4\,d\,x\right )+\frac {111527\,\sin \left (5\,c+5\,d\,x\right )}{128}-\frac {7187\,\sin \left (6\,c+6\,d\,x\right )}{8}+\frac {121427\,\sin \left (7\,c+7\,d\,x\right )}{128}-952{}\mathrm {i}\right )}{6435\,a^8\,d\,\left (-2\,{\sin \left (\frac {15\,c}{4}+\frac {15\,d\,x}{4}\right )}^2+\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )} \]
[In]
[Out]