\(\int (a+i a \tan (c+d x))^8 \, dx\) [81]

Optimal result

Integrand size = 15, antiderivative size = 200 \[ \int (a+i a \tan (c+d x))^8 \, dx=128 a^8 x-\frac {128 i a^8 \log (\cos (c+d x))}{d}-\frac {64 a^8 \tan (c+d x)}{d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {16 i a^2 \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3559, 3558, 3556} \[ \int (a+i a \tan (c+d x))^8 \, dx=-\frac {64 a^8 \tan (c+d x)}{d}-\frac {128 i a^8 \log (\cos (c+d x))}{d}+128 a^8 x+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {16 i a^2 \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d} \]

[In]

Int[(a + I*a*Tan[c + d*x])^8,x]

[Out]

128*a^8*x - ((128*I)*a^8*Log[Cos[c + d*x]])/d - (64*a^8*Tan[c + d*x])/d + (((4*I)/5)*a^3*(a + I*a*Tan[c + d*x]
)^5)/d + ((I/3)*a^2*(a + I*a*Tan[c + d*x])^6)/d + ((I/7)*a*(a + I*a*Tan[c + d*x])^7)/d + (((16*I)/3)*a^2*(a^2
+ I*a^2*Tan[c + d*x])^3)/d + ((2*I)*(a^2 + I*a^2*Tan[c + d*x])^4)/d + ((16*I)*(a^4 + I*a^4*Tan[c + d*x])^2)/d

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3558

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2)*x, x] + (Dist[2*a*b, Int[Tan[c + d
*x], x], x] + Simp[b^2*(Tan[c + d*x]/d), x]) /; FreeQ[{a, b, c, d}, x]

Rule 3559

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Dist[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] && G
tQ[n, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {i a (a+i a \tan (c+d x))^7}{7 d}+(2 a) \int (a+i a \tan (c+d x))^7 \, dx \\ & = \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}+\left (4 a^2\right ) \int (a+i a \tan (c+d x))^6 \, dx \\ & = \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}+\left (8 a^3\right ) \int (a+i a \tan (c+d x))^5 \, dx \\ & = \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 {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\left (16 a^4\right ) \int (a+i a \tan (c+d x))^4 \, dx \\ & = \frac {16 i a^5 (a+i a \tan (c+d x))^3}{3 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 {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\left (32 a^5\right ) \int (a+i a \tan (c+d x))^3 \, dx \\ & = \frac {16 i a^5 (a+i a \tan (c+d x))^3}{3 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 {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}+\left (64 a^6\right ) \int (a+i a \tan (c+d x))^2 \, dx \\ & = 128 a^8 x-\frac {64 a^8 \tan (c+d x)}{d}+\frac {16 i a^5 (a+i a \tan (c+d x))^3}{3 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 {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}+\left (128 i a^8\right ) \int \tan (c+d x) \, 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 {16 i a^5 (a+i a \tan (c+d x))^3}{3 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 {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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.50 \[ \int (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \left (13440 i \log (i+\tan (c+d x))-13335 \tan (c+d x)-6300 i \tan ^2(c+d x)+3465 \tan ^3(c+d x)+1680 i \tan ^4(c+d x)-609 \tan ^5(c+d x)-140 i \tan ^6(c+d x)+15 \tan ^7(c+d x)\right )}{105 d} \]

[In]

Integrate[(a + I*a*Tan[c + d*x])^8,x]

[Out]

(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)

Maple [A] (verified)

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	\(a^{8} x +\frac {a^{8} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}-\frac {56 i a^{8} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}-\frac {8 i a^{8} \left (\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {4 i a^{8} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {56 i a^{8} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}-\frac {28 a^{8} \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {70 a^{8} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}-\frac {28 a^{8} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(311\)

[In]

int((a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

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)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.48 \[ \int (a+i a \tan (c+d x))^8 \, dx=-\frac {32 \, {\left (2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 13230 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 26950 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 30625 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 20139 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7203 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 1089 i \, a^{8} + 420 \, {\left (i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 7 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 21 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 35 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 35 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 21 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{105 \, {\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate((a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

-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) + 21*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)

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.50 \[ \int (a+i a \tan (c+d x))^8 \, dx=- \frac {128 i a^{8} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 94080 i a^{8} e^{12 i c} e^{12 i d x} - 423360 i a^{8} e^{10 i c} e^{10 i d x} - 862400 i a^{8} e^{8 i c} e^{8 i d x} - 980000 i a^{8} e^{6 i c} e^{6 i d x} - 644448 i a^{8} e^{4 i c} e^{4 i d x} - 230496 i a^{8} e^{2 i c} e^{2 i d x} - 34848 i a^{8}}{105 d e^{14 i c} e^{14 i d x} + 735 d e^{12 i c} e^{12 i d x} + 2205 d e^{10 i c} e^{10 i d x} + 3675 d e^{8 i c} e^{8 i d x} + 3675 d e^{6 i c} e^{6 i d x} + 2205 d e^{4 i c} e^{4 i d x} + 735 d e^{2 i c} e^{2 i d x} + 105 d} \]

[In]

integrate((a+I*a*tan(d*x+c))**8,x)

[Out]

-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(1
0*I*c)*exp(10*I*d*x) - 862400*I*a**8*exp(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) - 34848*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*exp(2*I*c)*exp(2*I*d*x) + 105*d)

Maxima [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.60 \[ \int (a+i a \tan (c+d x))^8 \, dx=\frac {15 \, a^{8} \tan \left (d x + c\right )^{7} - 140 i \, a^{8} \tan \left (d x + c\right )^{6} - 609 \, a^{8} \tan \left (d x + c\right )^{5} + 1680 i \, a^{8} \tan \left (d x + c\right )^{4} + 3465 \, a^{8} \tan \left (d x + c\right )^{3} - 6300 i \, a^{8} \tan \left (d x + c\right )^{2} + 13440 \, {\left (d x + c\right )} a^{8} + 6720 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 13335 \, a^{8} \tan \left (d x + c\right )}{105 \, d} \]

[In]

integrate((a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

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 \[ \int (a+i a \tan (c+d x))^8 \, dx=-\frac {32 \, {\left (420 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8820 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 14700 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 14700 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8820 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 13230 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 26950 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 30625 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 20139 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7203 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 420 i \, a^{8} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1089 i \, a^{8}\right )}}{105 \, {\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate((a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-32/105*(420*I*a^8*e^(14*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 2940*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)*log(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) + 20139*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)

Mupad [B] (verification not implemented)

Time = 4.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int (a+i a \tan (c+d x))^8 \, dx=\frac {33\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^3-127\,a^8\,\mathrm {tan}\left (c+d\,x\right )-\frac {29\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+a^8\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,128{}\mathrm {i}-a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2\,60{}\mathrm {i}+a^8\,{\mathrm {tan}\left (c+d\,x\right )}^4\,16{}\mathrm {i}-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^6\,4{}\mathrm {i}}{3}}{d} \]

[In]

int((a + a*tan(c + d*x)*1i)^8,x)

[Out]

(a^8*log(tan(c + d*x) + 1i)*128i - 127*a^8*tan(c + d*x) - a^8*tan(c + d*x)^2*60i + 33*a^8*tan(c + d*x)^3 + a^8
*tan(c + d*x)^4*16i - (29*a^8*tan(c + d*x)^5)/5 - (a^8*tan(c + d*x)^6*4i)/3 + (a^8*tan(c + d*x)^7)/7)/d