Integrand size = 15, antiderivative size = 229 \[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\frac {x}{256 a^8}+\frac {i}{16 d (a+i a \tan (c+d x))^8}+\frac {i}{28 a d (a+i a \tan (c+d x))^7}+\frac {i}{48 a^2 d (a+i a \tan (c+d x))^6}+\frac {i}{80 a^3 d (a+i a \tan (c+d x))^5}+\frac {i}{128 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {i}{192 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {i}{256 d \left (a^8+i a^8 \tan (c+d x)\right )} \] Output:
1/256*x/a^8+1/16*I/d/(a+I*a*tan(d*x+c))^8+1/28*I/a/d/(a+I*a*tan(d*x+c))^7+ 1/48*I/a^2/d/(a+I*a*tan(d*x+c))^6+1/80*I/a^3/d/(a+I*a*tan(d*x+c))^5+1/128* I/d/(a^2+I*a^2*tan(d*x+c))^4+1/192*I/a^2/d/(a^2+I*a^2*tan(d*x+c))^3+1/256* I/d/(a^4+I*a^4*tan(d*x+c))^2+1/256*I/d/(a^8+I*a^8*tan(d*x+c))
Time = 0.44 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^8(c+d x) (7350+12544 \cos (2 (c+d x))+7840 \cos (4 (c+d x))+3840 \cos (6 (c+d x))+1194 \cos (8 (c+d x))+3136 i \sin (2 (c+d x))+3920 i \sin (4 (c+d x))+2880 i \sin (6 (c+d x))+1089 i \sin (8 (c+d x))+840 \arctan (\tan (c+d x)) (-i \cos (8 (c+d x))+\sin (8 (c+d x))))}{215040 a^8 d (-i+\tan (c+d x))^8} \] Input:
Integrate[(a + I*a*Tan[c + d*x])^(-8),x]
Output:
((I/215040)*Sec[c + d*x]^8*(7350 + 12544*Cos[2*(c + d*x)] + 7840*Cos[4*(c + d*x)] + 3840*Cos[6*(c + d*x)] + 1194*Cos[8*(c + d*x)] + (3136*I)*Sin[2*( c + d*x)] + (3920*I)*Sin[4*(c + d*x)] + (2880*I)*Sin[6*(c + d*x)] + (1089* I)*Sin[8*(c + d*x)] + 840*ArcTan[Tan[c + d*x]]*((-I)*Cos[8*(c + d*x)] + Si n[8*(c + d*x)])))/(a^8*d*(-I + Tan[c + d*x])^8)
Time = 1.00 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.12, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.133, Rules used = {3042, 3960, 3042, 3960, 3042, 3960, 3042, 3960, 3042, 3960, 3042, 3960, 3042, 3960, 3042, 3960, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+i a \tan (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\int \frac {1}{(i \tan (c+d x) a+a)^7}dx}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{(i \tan (c+d x) a+a)^7}dx}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^6}dx}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^6}dx}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^5}dx}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^5}dx}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^4}dx}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^4}dx}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^3}dx}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^3}dx}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^2}dx}{2 a}+\frac {i}{6 d (a+i a \tan (c+d x))^3}}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \frac {1}{(i \tan (c+d x) a+a)^2}dx}{2 a}+\frac {i}{6 d (a+i a \tan (c+d x))^3}}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\int \frac {1}{i \tan (c+d x) a+a}dx}{2 a}+\frac {i}{4 d (a+i a \tan (c+d x))^2}}{2 a}+\frac {i}{6 d (a+i a \tan (c+d x))^3}}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\int \frac {1}{i \tan (c+d x) a+a}dx}{2 a}+\frac {i}{4 d (a+i a \tan (c+d x))^2}}{2 a}+\frac {i}{6 d (a+i a \tan (c+d x))^3}}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\int 1dx}{2 a}+\frac {i}{2 d (a+i a \tan (c+d x))}}{2 a}+\frac {i}{4 d (a+i a \tan (c+d x))^2}}{2 a}+\frac {i}{6 d (a+i a \tan (c+d x))^3}}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {x}{2 a}+\frac {i}{2 d (a+i a \tan (c+d x))}}{2 a}+\frac {i}{4 d (a+i a \tan (c+d x))^2}}{2 a}+\frac {i}{6 d (a+i a \tan (c+d x))^3}}{2 a}+\frac {i}{8 d (a+i a \tan (c+d x))^4}}{2 a}+\frac {i}{10 d (a+i a \tan (c+d x))^5}}{2 a}+\frac {i}{12 d (a+i a \tan (c+d x))^6}}{2 a}+\frac {i}{14 d (a+i a \tan (c+d x))^7}}{2 a}+\frac {i}{16 d (a+i a \tan (c+d x))^8}\) |
Input:
Int[(a + I*a*Tan[c + d*x])^(-8),x]
Output:
(I/16)/(d*(a + I*a*Tan[c + d*x])^8) + ((I/14)/(d*(a + I*a*Tan[c + d*x])^7) + ((I/12)/(d*(a + I*a*Tan[c + d*x])^6) + ((I/10)/(d*(a + I*a*Tan[c + d*x] )^5) + ((I/8)/(d*(a + I*a*Tan[c + d*x])^4) + ((I/6)/(d*(a + I*a*Tan[c + d* x])^3) + ((I/4)/(d*(a + I*a*Tan[c + d*x])^2) + (x/(2*a) + (I/2)/(d*(a + I* a*Tan[c + d*x])))/(2*a))/(2*a))/(2*a))/(2*a))/(2*a))/(2*a))/(2*a)
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[1/(2*a) Int[(a + b*Tan[c + d*x])^ (n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
Time = 0.70 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {x}{256 a^{8}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{64 a^{8} d}+\frac {7 i {\mathrm e}^{-4 i \left (d x +c \right )}}{256 a^{8} d}+\frac {7 i {\mathrm e}^{-6 i \left (d x +c \right )}}{192 a^{8} d}+\frac {35 i {\mathrm e}^{-8 i \left (d x +c \right )}}{1024 a^{8} d}+\frac {7 i {\mathrm e}^{-10 i \left (d x +c \right )}}{320 a^{8} d}+\frac {7 i {\mathrm e}^{-12 i \left (d x +c \right )}}{768 a^{8} d}+\frac {i {\mathrm e}^{-14 i \left (d x +c \right )}}{448 a^{8} d}+\frac {i {\mathrm e}^{-16 i \left (d x +c \right )}}{4096 a^{8} d}\) | \(152\) |
| derivativedivides | \(\frac {\arctan \left (\tan \left (d x +c \right )\right )}{256 a^{8} d}+\frac {i}{16 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{8}}+\frac {i}{128 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{4}}-\frac {i}{48 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{6}}-\frac {i}{256 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{2}}-\frac {1}{28 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{7}}+\frac {1}{80 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{5}}-\frac {1}{192 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{3}}+\frac {1}{256 a^{8} d \left (-i+\tan \left (d x +c \right )\right )}\) | \(173\) |
| default | \(\frac {\arctan \left (\tan \left (d x +c \right )\right )}{256 a^{8} d}+\frac {i}{16 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{8}}+\frac {i}{128 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{4}}-\frac {i}{48 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{6}}-\frac {i}{256 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{2}}-\frac {1}{28 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{7}}+\frac {1}{80 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{5}}-\frac {1}{192 d \,a^{8} \left (-i+\tan \left (d x +c \right )\right )^{3}}+\frac {1}{256 a^{8} d \left (-i+\tan \left (d x +c \right )\right )}\) | \(173\) |
| norman | \(\frac {\frac {255 \tan \left (d x +c \right )}{256 a d}+\frac {x}{256 a}+\frac {7 x \tan \left (d x +c \right )^{4}}{64 a}+\frac {x \tan \left (d x +c \right )^{2}}{32 a}-\frac {1117 \tan \left (d x +c \right )^{3}}{256 a d}+\frac {7 x \tan \left (d x +c \right )^{6}}{32 a}+\frac {961 \tan \left (d x +c \right )^{7}}{8960 a d}+\frac {5053 \tan \left (d x +c \right )^{9}}{26880 a d}+\frac {383 \tan \left (d x +c \right )^{11}}{3840 a d}+\frac {23 \tan \left (d x +c \right )^{13}}{768 a d}+\frac {\tan \left (d x +c \right )^{15}}{256 a d}+\frac {3371 \tan \left (d x +c \right )^{5}}{1280 a d}+\frac {7 x \tan \left (d x +c \right )^{12}}{64 a}+\frac {x \tan \left (d x +c \right )^{14}}{32 a}+\frac {x \tan \left (d x +c \right )^{16}}{256 a}+\frac {64 i \tan \left (d x +c \right )^{4}}{15 a d}-\frac {4 i \tan \left (d x +c \right )^{6}}{5 a d}-\frac {292 i \tan \left (d x +c \right )^{2}}{105 a d}+\frac {35 x \tan \left (d x +c \right )^{8}}{128 a}+\frac {7 x \tan \left (d x +c \right )^{10}}{32 a}+\frac {16 i}{105 a d}}{a^{7} \left (1+\tan \left (d x +c \right )^{2}\right )^{8}}\) | \(322\) |
Input:
int(1/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/256*x/a^8+1/64*I/a^8/d*exp(-2*I*(d*x+c))+7/256*I/a^8/d*exp(-4*I*(d*x+c)) +7/192*I/a^8/d*exp(-6*I*(d*x+c))+35/1024*I/a^8/d*exp(-8*I*(d*x+c))+7/320*I /a^8/d*exp(-10*I*(d*x+c))+7/768*I/a^8/d*exp(-12*I*(d*x+c))+1/448*I/a^8/d*e xp(-14*I*(d*x+c))+1/4096*I/a^8/d*exp(-16*I*(d*x+c))
Time = 0.07 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (1680 \, d x e^{\left (16 i \, d x + 16 i \, c\right )} + 6720 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 11760 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 15680 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 14700 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 9408 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 3920 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 960 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105 i\right )} e^{\left (-16 i \, d x - 16 i \, c\right )}}{430080 \, a^{8} d} \] Input:
integrate(1/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
Output:
1/430080*(1680*d*x*e^(16*I*d*x + 16*I*c) + 6720*I*e^(14*I*d*x + 14*I*c) + 11760*I*e^(12*I*d*x + 12*I*c) + 15680*I*e^(10*I*d*x + 10*I*c) + 14700*I*e^ (8*I*d*x + 8*I*c) + 9408*I*e^(6*I*d*x + 6*I*c) + 3920*I*e^(4*I*d*x + 4*I*c ) + 960*I*e^(2*I*d*x + 2*I*c) + 105*I)*e^(-16*I*d*x - 16*I*c)/(a^8*d)
Time = 0.41 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} \frac {\left (22698142121947299840 i a^{56} d^{7} e^{70 i c} e^{- 2 i d x} + 39721748713407774720 i a^{56} d^{7} e^{68 i c} e^{- 4 i d x} + 52962331617877032960 i a^{56} d^{7} e^{66 i c} e^{- 6 i d x} + 49652185891759718400 i a^{56} d^{7} e^{64 i c} e^{- 8 i d x} + 31777398970726219776 i a^{56} d^{7} e^{62 i c} e^{- 10 i d x} + 13240582904469258240 i a^{56} d^{7} e^{60 i c} e^{- 12 i d x} + 3242591731706757120 i a^{56} d^{7} e^{58 i c} e^{- 14 i d x} + 354658470655426560 i a^{56} d^{7} e^{56 i c} e^{- 16 i d x}\right ) e^{- 72 i c}}{1452681095804627189760 a^{64} d^{8}} & \text {for}\: a^{64} d^{8} e^{72 i c} \neq 0 \\x \left (\frac {\left (e^{16 i c} + 8 e^{14 i c} + 28 e^{12 i c} + 56 e^{10 i c} + 70 e^{8 i c} + 56 e^{6 i c} + 28 e^{4 i c} + 8 e^{2 i c} + 1\right ) e^{- 16 i c}}{256 a^{8}} - \frac {1}{256 a^{8}}\right ) & \text {otherwise} \end {cases} + \frac {x}{256 a^{8}} \] Input:
integrate(1/(a+I*a*tan(d*x+c))**8,x)
Output:
Piecewise(((22698142121947299840*I*a**56*d**7*exp(70*I*c)*exp(-2*I*d*x) + 39721748713407774720*I*a**56*d**7*exp(68*I*c)*exp(-4*I*d*x) + 529623316178 77032960*I*a**56*d**7*exp(66*I*c)*exp(-6*I*d*x) + 49652185891759718400*I*a **56*d**7*exp(64*I*c)*exp(-8*I*d*x) + 31777398970726219776*I*a**56*d**7*ex p(62*I*c)*exp(-10*I*d*x) + 13240582904469258240*I*a**56*d**7*exp(60*I*c)*e xp(-12*I*d*x) + 3242591731706757120*I*a**56*d**7*exp(58*I*c)*exp(-14*I*d*x ) + 354658470655426560*I*a**56*d**7*exp(56*I*c)*exp(-16*I*d*x))*exp(-72*I* c)/(1452681095804627189760*a**64*d**8), Ne(a**64*d**8*exp(72*I*c), 0)), (x *((exp(16*I*c) + 8*exp(14*I*c) + 28*exp(12*I*c) + 56*exp(10*I*c) + 70*exp( 8*I*c) + 56*exp(6*I*c) + 28*exp(4*I*c) + 8*exp(2*I*c) + 1)*exp(-16*I*c)/(2 56*a**8) - 1/(256*a**8)), True)) + x/(256*a**8)
Exception generated. \[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \, \log \left (\tan \left (d x + c\right ) + i\right )}{512 \, a^{8} d} - \frac {i \, \log \left (\tan \left (d x + c\right ) - i\right )}{512 \, a^{8} d} + \frac {105 \, \tan \left (d x + c\right )^{7} - 840 i \, \tan \left (d x + c\right )^{6} - 2975 \, \tan \left (d x + c\right )^{5} + 6160 i \, \tan \left (d x + c\right )^{4} + 8351 \, \tan \left (d x + c\right )^{3} - 8008 i \, \tan \left (d x + c\right )^{2} - 5993 \, \tan \left (d x + c\right ) + 4096 i}{26880 \, a^{8} d {\left (\tan \left (d x + c\right ) - i\right )}^{8}} \] Input:
integrate(1/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
Output:
1/512*I*log(tan(d*x + c) + I)/(a^8*d) - 1/512*I*log(tan(d*x + c) - I)/(a^8 *d) + 1/26880*(105*tan(d*x + c)^7 - 840*I*tan(d*x + c)^6 - 2975*tan(d*x + c)^5 + 6160*I*tan(d*x + c)^4 + 8351*tan(d*x + c)^3 - 8008*I*tan(d*x + c)^2 - 5993*tan(d*x + c) + 4096*I)/(a^8*d*(tan(d*x + c) - I)^8)
Time = 2.17 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\frac {x}{256\,a^8}-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,5993{}\mathrm {i}}{26880\,a^8}+\frac {16}{105\,a^8}-\frac {143\,{\mathrm {tan}\left (c+d\,x\right )}^2}{480\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,1193{}\mathrm {i}}{3840\,a^8}+\frac {11\,{\mathrm {tan}\left (c+d\,x\right )}^4}{48\,a^8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,85{}\mathrm {i}}{768\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{32\,a^8}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,1{}\mathrm {i}}{256\,a^8}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^8\,1{}\mathrm {i}+8\,{\mathrm {tan}\left (c+d\,x\right )}^7-{\mathrm {tan}\left (c+d\,x\right )}^6\,28{}\mathrm {i}-56\,{\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,70{}\mathrm {i}+56\,{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,28{}\mathrm {i}-8\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \] Input:
int(1/(a + a*tan(c + d*x)*1i)^8,x)
Output:
x/(256*a^8) - ((tan(c + d*x)*5993i)/(26880*a^8) + 16/(105*a^8) - (143*tan( c + d*x)^2)/(480*a^8) - (tan(c + d*x)^3*1193i)/(3840*a^8) + (11*tan(c + d* x)^4)/(48*a^8) + (tan(c + d*x)^5*85i)/(768*a^8) - tan(c + d*x)^6/(32*a^8) - (tan(c + d*x)^7*1i)/(256*a^8))/(d*(56*tan(c + d*x)^3 - tan(c + d*x)^2*28 i - 8*tan(c + d*x) + tan(c + d*x)^4*70i - 56*tan(c + d*x)^5 - tan(c + d*x) ^6*28i + 8*tan(c + d*x)^7 + tan(c + d*x)^8*1i + 1i))
\[ \int \frac {1}{(a+i a \tan (c+d x))^8} \, dx=\frac {\int \frac {1}{\tan \left (d x +c \right )^{8}-8 \tan \left (d x +c \right )^{7} i -28 \tan \left (d x +c \right )^{6}+56 \tan \left (d x +c \right )^{5} i +70 \tan \left (d x +c \right )^{4}-56 \tan \left (d x +c \right )^{3} i -28 \tan \left (d x +c \right )^{2}+8 \tan \left (d x +c \right ) i +1}d x}{a^{8}} \] Input:
int(1/(a+I*a*tan(d*x+c))^8,x)
Output:
int(1/(tan(c + d*x)**8 - 8*tan(c + d*x)**7*i - 28*tan(c + d*x)**6 + 56*tan (c + d*x)**5*i + 70*tan(c + d*x)**4 - 56*tan(c + d*x)**3*i - 28*tan(c + d* x)**2 + 8*tan(c + d*x)*i + 1),x)/a**8