Integrand size = 15, antiderivative size = 200 \[ \int (a+i a \tan (c+d x))^8 \, dx=128 a^8 x-\frac {128 i a^8 \log (\cos (c+d x))}{d}-\frac {64 a^8 \tan (c+d x)}{d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {16 i a^2 \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d} \]
128*a^8*x-128*I*a^8*ln(cos(d*x+c))/d-64*a^8*tan(d*x+c)/d+4/5*I*a^3*(a+I*a* tan(d*x+c))^5/d+1/3*I*a^2*(a+I*a*tan(d*x+c))^6/d+1/7*I*a*(a+I*a*tan(d*x+c) )^7/d+16/3*I*a^2*(a^2+I*a^2*tan(d*x+c))^3/d+2*I*(a^2+I*a^2*tan(d*x+c))^4/d +16*I*(a^4+I*a^4*tan(d*x+c))^2/d
Time = 0.51 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.50 \[ \int (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \left (13440 i \log (i+\tan (c+d x))-13335 \tan (c+d x)-6300 i \tan ^2(c+d x)+3465 \tan ^3(c+d x)+1680 i \tan ^4(c+d x)-609 \tan ^5(c+d x)-140 i \tan ^6(c+d x)+15 \tan ^7(c+d x)\right )}{105 d} \]
(a^8*((13440*I)*Log[I + Tan[c + d*x]] - 13335*Tan[c + d*x] - (6300*I)*Tan[ c + d*x]^2 + 3465*Tan[c + d*x]^3 + (1680*I)*Tan[c + d*x]^4 - 609*Tan[c + d *x]^5 - (140*I)*Tan[c + d*x]^6 + 15*Tan[c + d*x]^7))/(105*d)
Time = 0.97 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.067, Rules used = {3042, 3959, 3042, 3959, 3042, 3959, 3042, 3959, 3042, 3959, 3042, 3959, 3042, 3958, 3042, 3956}
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.
| \(\displaystyle \int (a+i a \tan (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (c+d x))^8dx\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle 2 a \int (i \tan (c+d x) a+a)^7dx+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \int (i \tan (c+d x) a+a)^7dx+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle 2 a \left (2 a \int (i \tan (c+d x) a+a)^6dx+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \left (2 a \int (i \tan (c+d x) a+a)^6dx+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle 2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^5dx+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^5dx+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^4dx+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^4dx+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^3dx+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^3dx+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^2dx+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^2dx+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3958 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (2 i a^2 \int \tan (c+d x)dx-\frac {a^2 \tan (c+d x)}{d}+2 a^2 x\right )+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (2 i a^2 \int \tan (c+d x)dx-\frac {a^2 \tan (c+d x)}{d}+2 a^2 x\right )+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle 2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (2 a \left (-\frac {a^2 \tan (c+d x)}{d}-\frac {2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x\right )+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\right )+\frac {i a (a+i a \tan (c+d x))^5}{5 d}\right )+\frac {i a (a+i a \tan (c+d x))^6}{6 d}\right )+\frac {i a (a+i a \tan (c+d x))^7}{7 d}\) |
((I/7)*a*(a + I*a*Tan[c + d*x])^7)/d + 2*a*(((I/6)*a*(a + I*a*Tan[c + d*x] )^6)/d + 2*a*(((I/5)*a*(a + I*a*Tan[c + d*x])^5)/d + 2*a*(((I/4)*a*(a + I* a*Tan[c + d*x])^4)/d + 2*a*(((I/3)*a*(a + I*a*Tan[c + d*x])^3)/d + 2*a*((( I/2)*a*(a + I*a*Tan[c + d*x])^2)/d + 2*a*(2*a^2*x - ((2*I)*a^2*Log[Cos[c + d*x]])/d - (a^2*Tan[c + d*x])/d))))))
3.1.81.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2) *x, x] + (Simp[b^2*(Tan[c + d*x]/d), x] + Simp[2*a*b Int[Tan[c + d*x], x] , x]) /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[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] && GtQ[n , 1]
Time = 0.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.52
| method | result | size |
| derivativedivides | \(\frac {a^{8} \left (-127 \tan \left (d x +c \right )+\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {4 i \left (\tan ^{6}\left (d x +c \right )\right )}{3}-\frac {29 \left (\tan ^{5}\left (d x +c \right )\right )}{5}+16 i \left (\tan ^{4}\left (d x +c \right )\right )+33 \left (\tan ^{3}\left (d x +c \right )\right )-60 i \left (\tan ^{2}\left (d x +c \right )\right )+64 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+128 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(103\) |
| default | \(\frac {a^{8} \left (-127 \tan \left (d x +c \right )+\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {4 i \left (\tan ^{6}\left (d x +c \right )\right )}{3}-\frac {29 \left (\tan ^{5}\left (d x +c \right )\right )}{5}+16 i \left (\tan ^{4}\left (d x +c \right )\right )+33 \left (\tan ^{3}\left (d x +c \right )\right )-60 i \left (\tan ^{2}\left (d x +c \right )\right )+64 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+128 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(103\) |
| risch | \(-\frac {256 a^{8} c}{d}-\frac {32 i a^{8} \left (2940 \,{\mathrm e}^{12 i \left (d x +c \right )}+13230 \,{\mathrm e}^{10 i \left (d x +c \right )}+26950 \,{\mathrm e}^{8 i \left (d x +c \right )}+30625 \,{\mathrm e}^{6 i \left (d x +c \right )}+20139 \,{\mathrm e}^{4 i \left (d x +c \right )}+7203 \,{\mathrm e}^{2 i \left (d x +c \right )}+1089\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {128 i a^{8} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(122\) |
| parallelrisch | \(\frac {-140 i a^{8} \left (\tan ^{6}\left (d x +c \right )\right )+15 \left (\tan ^{7}\left (d x +c \right )\right ) a^{8}+1680 i a^{8} \left (\tan ^{4}\left (d x +c \right )\right )-609 \left (\tan ^{5}\left (d x +c \right )\right ) a^{8}-6300 i a^{8} \left (\tan ^{2}\left (d x +c \right )\right )+3465 \left (\tan ^{3}\left (d x +c \right )\right ) a^{8}+6720 i a^{8} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+13440 a^{8} x d -13335 a^{8} \tan \left (d x +c \right )}{105 d}\) | \(123\) |
| norman | \(128 a^{8} x -\frac {127 a^{8} \tan \left (d x +c \right )}{d}+\frac {33 a^{8} \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {29 a^{8} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{8} \left (\tan ^{7}\left (d x +c \right )\right )}{7 d}-\frac {60 i a^{8} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {16 i a^{8} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {4 i a^{8} \left (\tan ^{6}\left (d x +c \right )\right )}{3 d}+\frac {64 i a^{8} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(141\) |
| parts | \(a^{8} x +\frac {a^{8} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}-\frac {56 i a^{8} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}-\frac {8 i a^{8} \left (\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {4 i a^{8} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {56 i a^{8} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}-\frac {28 a^{8} \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {70 a^{8} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}-\frac {28 a^{8} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(311\) |
1/d*a^8*(-127*tan(d*x+c)+1/7*tan(d*x+c)^7-4/3*I*tan(d*x+c)^6-29/5*tan(d*x+ c)^5+16*I*tan(d*x+c)^4+33*tan(d*x+c)^3-60*I*tan(d*x+c)^2+64*I*ln(1+tan(d*x +c)^2)+128*arctan(tan(d*x+c)))
Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.48 \[ \int (a+i a \tan (c+d x))^8 \, dx=-\frac {32 \, {\left (2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 13230 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 26950 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 30625 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 20139 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7203 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 1089 i \, a^{8} + 420 \, {\left (i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 7 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 21 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 35 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 35 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 21 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{105 \, {\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-32/105*(2940*I*a^8*e^(12*I*d*x + 12*I*c) + 13230*I*a^8*e^(10*I*d*x + 10*I *c) + 26950*I*a^8*e^(8*I*d*x + 8*I*c) + 30625*I*a^8*e^(6*I*d*x + 6*I*c) + 20139*I*a^8*e^(4*I*d*x + 4*I*c) + 7203*I*a^8*e^(2*I*d*x + 2*I*c) + 1089*I* a^8 + 420*(I*a^8*e^(14*I*d*x + 14*I*c) + 7*I*a^8*e^(12*I*d*x + 12*I*c) + 2 1*I*a^8*e^(10*I*d*x + 10*I*c) + 35*I*a^8*e^(8*I*d*x + 8*I*c) + 35*I*a^8*e^ (6*I*d*x + 6*I*c) + 21*I*a^8*e^(4*I*d*x + 4*I*c) + 7*I*a^8*e^(2*I*d*x + 2* I*c) + I*a^8)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(14*I*d*x + 14*I*c) + 7*d *e^(12*I*d*x + 12*I*c) + 21*d*e^(10*I*d*x + 10*I*c) + 35*d*e^(8*I*d*x + 8* I*c) + 35*d*e^(6*I*d*x + 6*I*c) + 21*d*e^(4*I*d*x + 4*I*c) + 7*d*e^(2*I*d* x + 2*I*c) + d)
Time = 0.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.50 \[ \int (a+i a \tan (c+d x))^8 \, dx=- \frac {128 i a^{8} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 94080 i a^{8} e^{12 i c} e^{12 i d x} - 423360 i a^{8} e^{10 i c} e^{10 i d x} - 862400 i a^{8} e^{8 i c} e^{8 i d x} - 980000 i a^{8} e^{6 i c} e^{6 i d x} - 644448 i a^{8} e^{4 i c} e^{4 i d x} - 230496 i a^{8} e^{2 i c} e^{2 i d x} - 34848 i a^{8}}{105 d e^{14 i c} e^{14 i d x} + 735 d e^{12 i c} e^{12 i d x} + 2205 d e^{10 i c} e^{10 i d x} + 3675 d e^{8 i c} e^{8 i d x} + 3675 d e^{6 i c} e^{6 i d x} + 2205 d e^{4 i c} e^{4 i d x} + 735 d e^{2 i c} e^{2 i d x} + 105 d} \]
-128*I*a**8*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-94080*I*a**8*exp(12*I*c) *exp(12*I*d*x) - 423360*I*a**8*exp(10*I*c)*exp(10*I*d*x) - 862400*I*a**8*e xp(8*I*c)*exp(8*I*d*x) - 980000*I*a**8*exp(6*I*c)*exp(6*I*d*x) - 644448*I* a**8*exp(4*I*c)*exp(4*I*d*x) - 230496*I*a**8*exp(2*I*c)*exp(2*I*d*x) - 348 48*I*a**8)/(105*d*exp(14*I*c)*exp(14*I*d*x) + 735*d*exp(12*I*c)*exp(12*I*d *x) + 2205*d*exp(10*I*c)*exp(10*I*d*x) + 3675*d*exp(8*I*c)*exp(8*I*d*x) + 3675*d*exp(6*I*c)*exp(6*I*d*x) + 2205*d*exp(4*I*c)*exp(4*I*d*x) + 735*d*ex p(2*I*c)*exp(2*I*d*x) + 105*d)
Time = 0.48 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.60 \[ \int (a+i a \tan (c+d x))^8 \, dx=\frac {15 \, a^{8} \tan \left (d x + c\right )^{7} - 140 i \, a^{8} \tan \left (d x + c\right )^{6} - 609 \, a^{8} \tan \left (d x + c\right )^{5} + 1680 i \, a^{8} \tan \left (d x + c\right )^{4} + 3465 \, a^{8} \tan \left (d x + c\right )^{3} - 6300 i \, a^{8} \tan \left (d x + c\right )^{2} + 13440 \, {\left (d x + c\right )} a^{8} + 6720 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 13335 \, a^{8} \tan \left (d x + c\right )}{105 \, d} \]
1/105*(15*a^8*tan(d*x + c)^7 - 140*I*a^8*tan(d*x + c)^6 - 609*a^8*tan(d*x + c)^5 + 1680*I*a^8*tan(d*x + c)^4 + 3465*a^8*tan(d*x + c)^3 - 6300*I*a^8* tan(d*x + c)^2 + 13440*(d*x + c)*a^8 + 6720*I*a^8*log(tan(d*x + c)^2 + 1) - 13335*a^8*tan(d*x + c))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (166) = 332\).
Time = 0.54 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.89 \[ \int (a+i a \tan (c+d x))^8 \, dx=-\frac {32 \, {\left (420 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8820 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 14700 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 14700 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8820 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 13230 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 26950 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 30625 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 20139 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7203 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 420 i \, a^{8} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1089 i \, a^{8}\right )}}{105 \, {\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-32/105*(420*I*a^8*e^(14*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 29 40*I*a^8*e^(12*I*d*x + 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 8820*I*a^8*e ^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 14700*I*a^8*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 14700*I*a^8*e^(6*I*d*x + 6*I*c)*l og(e^(2*I*d*x + 2*I*c) + 1) + 8820*I*a^8*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d* x + 2*I*c) + 1) + 2940*I*a^8*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 2940*I*a^8*e^(12*I*d*x + 12*I*c) + 13230*I*a^8*e^(10*I*d*x + 10*I*c) + 26950*I*a^8*e^(8*I*d*x + 8*I*c) + 30625*I*a^8*e^(6*I*d*x + 6*I*c) + 201 39*I*a^8*e^(4*I*d*x + 4*I*c) + 7203*I*a^8*e^(2*I*d*x + 2*I*c) + 420*I*a^8* log(e^(2*I*d*x + 2*I*c) + 1) + 1089*I*a^8)/(d*e^(14*I*d*x + 14*I*c) + 7*d* e^(12*I*d*x + 12*I*c) + 21*d*e^(10*I*d*x + 10*I*c) + 35*d*e^(8*I*d*x + 8*I *c) + 35*d*e^(6*I*d*x + 6*I*c) + 21*d*e^(4*I*d*x + 4*I*c) + 7*d*e^(2*I*d*x + 2*I*c) + d)
Time = 4.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int (a+i a \tan (c+d x))^8 \, dx=\frac {33\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^3-127\,a^8\,\mathrm {tan}\left (c+d\,x\right )-\frac {29\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+a^8\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,128{}\mathrm {i}-a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2\,60{}\mathrm {i}+a^8\,{\mathrm {tan}\left (c+d\,x\right )}^4\,16{}\mathrm {i}-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^6\,4{}\mathrm {i}}{3}}{d} \]