\(\int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [77]

Optimal result

Integrand size = 24, antiderivative size = 109 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2 i (a+i a \tan (c+d x))^{12}}{3 a^4 d}+\frac {12 i (a+i a \tan (c+d x))^{13}}{13 a^5 d}-\frac {3 i (a+i a \tan (c+d x))^{14}}{7 a^6 d}+\frac {i (a+i a \tan (c+d x))^{15}}{15 a^7 d} \] Output:

-2/3*I*(a+I*a*tan(d*x+c))^12/a^4/d+12/13*I*(a+I*a*tan(d*x+c))^13/a^5/d-3/7 
*I*(a+I*a*tan(d*x+c))^14/a^6/d+1/15*I*(a+I*a*tan(d*x+c))^15/a^7/d
 

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 (-i+\tan (c+d x))^{12} \left (-144 i-363 \tan (c+d x)+312 i \tan ^2(c+d x)+91 \tan ^3(c+d x)\right )}{1365 d} \] Input:

Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^8,x]
 

Output:

(a^8*(-I + Tan[c + d*x])^12*(-144*I - 363*Tan[c + d*x] + (312*I)*Tan[c + d 
*x]^2 + 91*Tan[c + d*x]^3))/(1365*d)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3968, 49, 2009}

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.

\(\displaystyle \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (c+d x)^8 (a+i a \tan (c+d x))^8dx\)

\(\Big \downarrow \) 3968

\(\displaystyle -\frac {i \int (a-i a \tan (c+d x))^3 (i \tan (c+d x) a+a)^{11}d(i a \tan (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {i \int \left (-(i \tan (c+d x) a+a)^{14}+6 a (i \tan (c+d x) a+a)^{13}-12 a^2 (i \tan (c+d x) a+a)^{12}+8 a^3 (i \tan (c+d x) a+a)^{11}\right )d(i a \tan (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i \left (\frac {2}{3} a^3 (a+i a \tan (c+d x))^{12}-\frac {12}{13} a^2 (a+i a \tan (c+d x))^{13}-\frac {1}{15} (a+i a \tan (c+d x))^{15}+\frac {3}{7} a (a+i a \tan (c+d x))^{14}\right )}{a^7 d}\)

Input:

Int[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^8,x]
 

Output:

((-I)*((2*a^3*(a + I*a*Tan[c + d*x])^12)/3 - (12*a^2*(a + I*a*Tan[c + d*x] 
)^13)/13 + (3*a*(a + I*a*Tan[c + d*x])^14)/7 - (a + I*a*Tan[c + d*x])^15/1 
5))/(a^7*d)
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3968

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ 
), x_Symbol] :> Simp[1/(a^(m - 2)*b*f)   Subst[Int[(a - x)^(m/2 - 1)*(a + x 
)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && 
 EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (93 ) = 186\).

Time = 1.24 (sec) , antiderivative size = 611, normalized size of antiderivative = 5.61

\[\frac {a^{8} \left (\frac {\sin \left (d x +c \right )^{9}}{15 \cos \left (d x +c \right )^{15}}+\frac {2 \sin \left (d x +c \right )^{9}}{65 \cos \left (d x +c \right )^{13}}+\frac {8 \sin \left (d x +c \right )^{9}}{715 \cos \left (d x +c \right )^{11}}+\frac {16 \sin \left (d x +c \right )^{9}}{6435 \cos \left (d x +c \right )^{9}}\right )-8 i a^{8} \left (\frac {\sin \left (d x +c \right )^{8}}{14 \cos \left (d x +c \right )^{14}}+\frac {\sin \left (d x +c \right )^{8}}{28 \cos \left (d x +c \right )^{12}}+\frac {\sin \left (d x +c \right )^{8}}{70 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{8}}{280 \cos \left (d x +c \right )^{8}}\right )-28 a^{8} \left (\frac {\sin \left (d x +c \right )^{7}}{13 \cos \left (d x +c \right )^{13}}+\frac {6 \sin \left (d x +c \right )^{7}}{143 \cos \left (d x +c \right )^{11}}+\frac {8 \sin \left (d x +c \right )^{7}}{429 \cos \left (d x +c \right )^{9}}+\frac {16 \sin \left (d x +c \right )^{7}}{3003 \cos \left (d x +c \right )^{7}}\right )+56 i a^{8} \left (\frac {\sin \left (d x +c \right )^{6}}{12 \cos \left (d x +c \right )^{12}}+\frac {\sin \left (d x +c \right )^{6}}{20 \cos \left (d x +c \right )^{10}}+\frac {\sin \left (d x +c \right )^{6}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{6}}{120 \cos \left (d x +c \right )^{6}}\right )+70 a^{8} \left (\frac {\sin \left (d x +c \right )^{5}}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \sin \left (d x +c \right )^{5}}{33 \cos \left (d x +c \right )^{9}}+\frac {8 \sin \left (d x +c \right )^{5}}{231 \cos \left (d x +c \right )^{7}}+\frac {16 \sin \left (d x +c \right )^{5}}{1155 \cos \left (d x +c \right )^{5}}\right )+\frac {i a^{8}}{\cos \left (d x +c \right )^{8}}-28 a^{8} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )-56 i a^{8} \left (\frac {\sin \left (d x +c \right )^{4}}{10 \cos \left (d x +c \right )^{10}}+\frac {3 \sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{4}}{20 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{4}}{40 \cos \left (d x +c \right )^{4}}\right )-a^{8} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\]

Input:

int(sec(d*x+c)^8*(a+I*a*tan(d*x+c))^8,x)
 

Output:

1/d*(a^8*(1/15*sin(d*x+c)^9/cos(d*x+c)^15+2/65*sin(d*x+c)^9/cos(d*x+c)^13+ 
8/715*sin(d*x+c)^9/cos(d*x+c)^11+16/6435*sin(d*x+c)^9/cos(d*x+c)^9)-8*I*a^ 
8*(1/14*sin(d*x+c)^8/cos(d*x+c)^14+1/28*sin(d*x+c)^8/cos(d*x+c)^12+1/70*si 
n(d*x+c)^8/cos(d*x+c)^10+1/280*sin(d*x+c)^8/cos(d*x+c)^8)-28*a^8*(1/13*sin 
(d*x+c)^7/cos(d*x+c)^13+6/143*sin(d*x+c)^7/cos(d*x+c)^11+8/429*sin(d*x+c)^ 
7/cos(d*x+c)^9+16/3003*sin(d*x+c)^7/cos(d*x+c)^7)+56*I*a^8*(1/12*sin(d*x+c 
)^6/cos(d*x+c)^12+1/20*sin(d*x+c)^6/cos(d*x+c)^10+1/40*sin(d*x+c)^6/cos(d* 
x+c)^8+1/120*sin(d*x+c)^6/cos(d*x+c)^6)+70*a^8*(1/11*sin(d*x+c)^5/cos(d*x+ 
c)^11+2/33*sin(d*x+c)^5/cos(d*x+c)^9+8/231*sin(d*x+c)^5/cos(d*x+c)^7+16/11 
55*sin(d*x+c)^5/cos(d*x+c)^5)+I*a^8/cos(d*x+c)^8-28*a^8*(1/9*sin(d*x+c)^3/ 
cos(d*x+c)^9+2/21*sin(d*x+c)^3/cos(d*x+c)^7+8/105*sin(d*x+c)^3/cos(d*x+c)^ 
5+16/315*sin(d*x+c)^3/cos(d*x+c)^3)-56*I*a^8*(1/10*sin(d*x+c)^4/cos(d*x+c) 
^10+3/40*sin(d*x+c)^4/cos(d*x+c)^8+1/20*sin(d*x+c)^4/cos(d*x+c)^6+1/40*sin 
(d*x+c)^4/cos(d*x+c)^4)-a^8*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/3 
5*sec(d*x+c)^2)*tan(d*x+c))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (85) = 170\).

Time = 0.09 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.17 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {8192 \, {\left (-1365 i \, a^{8} e^{\left (22 i \, d x + 22 i \, c\right )} - 3003 i \, a^{8} e^{\left (20 i \, d x + 20 i \, c\right )} - 5005 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 6435 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 6435 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 5005 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 3003 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 1365 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 455 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 105 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 15 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{1365 \, {\left (d e^{\left (30 i \, d x + 30 i \, c\right )} + 15 \, d e^{\left (28 i \, d x + 28 i \, c\right )} + 105 \, d e^{\left (26 i \, d x + 26 i \, c\right )} + 455 \, d e^{\left (24 i \, d x + 24 i \, c\right )} + 1365 \, d e^{\left (22 i \, d x + 22 i \, c\right )} + 3003 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 5005 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 6435 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 6435 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 5005 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 3003 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 1365 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 455 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 105 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 15 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:

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

Output:

-8192/1365*(-1365*I*a^8*e^(22*I*d*x + 22*I*c) - 3003*I*a^8*e^(20*I*d*x + 2 
0*I*c) - 5005*I*a^8*e^(18*I*d*x + 18*I*c) - 6435*I*a^8*e^(16*I*d*x + 16*I* 
c) - 6435*I*a^8*e^(14*I*d*x + 14*I*c) - 5005*I*a^8*e^(12*I*d*x + 12*I*c) - 
 3003*I*a^8*e^(10*I*d*x + 10*I*c) - 1365*I*a^8*e^(8*I*d*x + 8*I*c) - 455*I 
*a^8*e^(6*I*d*x + 6*I*c) - 105*I*a^8*e^(4*I*d*x + 4*I*c) - 15*I*a^8*e^(2*I 
*d*x + 2*I*c) - I*a^8)/(d*e^(30*I*d*x + 30*I*c) + 15*d*e^(28*I*d*x + 28*I* 
c) + 105*d*e^(26*I*d*x + 26*I*c) + 455*d*e^(24*I*d*x + 24*I*c) + 1365*d*e^ 
(22*I*d*x + 22*I*c) + 3003*d*e^(20*I*d*x + 20*I*c) + 5005*d*e^(18*I*d*x + 
18*I*c) + 6435*d*e^(16*I*d*x + 16*I*c) + 6435*d*e^(14*I*d*x + 14*I*c) + 50 
05*d*e^(12*I*d*x + 12*I*c) + 3003*d*e^(10*I*d*x + 10*I*c) + 1365*d*e^(8*I* 
d*x + 8*I*c) + 455*d*e^(6*I*d*x + 6*I*c) + 105*d*e^(4*I*d*x + 4*I*c) + 15* 
d*e^(2*I*d*x + 2*I*c) + d)
 

Sympy [F]

\[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int \sec ^{8}{\left (c + d x \right )}\, dx\right ) \] Input:

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

Output:

a**8*(Integral(-28*tan(c + d*x)**2*sec(c + d*x)**8, x) + Integral(70*tan(c 
 + d*x)**4*sec(c + d*x)**8, x) + Integral(-28*tan(c + d*x)**6*sec(c + d*x) 
**8, x) + Integral(tan(c + d*x)**8*sec(c + d*x)**8, x) + Integral(8*I*tan( 
c + d*x)*sec(c + d*x)**8, x) + Integral(-56*I*tan(c + d*x)**3*sec(c + d*x) 
**8, x) + Integral(56*I*tan(c + d*x)**5*sec(c + d*x)**8, x) + Integral(-8* 
I*tan(c + d*x)**7*sec(c + d*x)**8, x) + Integral(sec(c + d*x)**8, x))
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (85) = 170\).

Time = 0.05 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.71 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {91 \, a^{8} \tan \left (d x + c\right )^{15} - 780 i \, a^{8} \tan \left (d x + c\right )^{14} - 2625 \, a^{8} \tan \left (d x + c\right )^{13} + 3640 i \, a^{8} \tan \left (d x + c\right )^{12} - 1365 \, a^{8} \tan \left (d x + c\right )^{11} + 12012 i \, a^{8} \tan \left (d x + c\right )^{10} + 15015 \, a^{8} \tan \left (d x + c\right )^{9} + 19305 \, a^{8} \tan \left (d x + c\right )^{7} - 20020 i \, a^{8} \tan \left (d x + c\right )^{6} - 3003 \, a^{8} \tan \left (d x + c\right )^{5} - 10920 i \, a^{8} \tan \left (d x + c\right )^{4} - 11375 \, a^{8} \tan \left (d x + c\right )^{3} + 5460 i \, a^{8} \tan \left (d x + c\right )^{2} + 1365 \, a^{8} \tan \left (d x + c\right )}{1365 \, d} \] Input:

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

Output:

1/1365*(91*a^8*tan(d*x + c)^15 - 780*I*a^8*tan(d*x + c)^14 - 2625*a^8*tan( 
d*x + c)^13 + 3640*I*a^8*tan(d*x + c)^12 - 1365*a^8*tan(d*x + c)^11 + 1201 
2*I*a^8*tan(d*x + c)^10 + 15015*a^8*tan(d*x + c)^9 + 19305*a^8*tan(d*x + c 
)^7 - 20020*I*a^8*tan(d*x + c)^6 - 3003*a^8*tan(d*x + c)^5 - 10920*I*a^8*t 
an(d*x + c)^4 - 11375*a^8*tan(d*x + c)^3 + 5460*I*a^8*tan(d*x + c)^2 + 136 
5*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. 186 vs. \(2 (85) = 170\).

Time = 0.34 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.71 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {91 \, a^{8} \tan \left (d x + c\right )^{15} - 780 i \, a^{8} \tan \left (d x + c\right )^{14} - 2625 \, a^{8} \tan \left (d x + c\right )^{13} + 3640 i \, a^{8} \tan \left (d x + c\right )^{12} - 1365 \, a^{8} \tan \left (d x + c\right )^{11} + 12012 i \, a^{8} \tan \left (d x + c\right )^{10} + 15015 \, a^{8} \tan \left (d x + c\right )^{9} + 19305 \, a^{8} \tan \left (d x + c\right )^{7} - 20020 i \, a^{8} \tan \left (d x + c\right )^{6} - 3003 \, a^{8} \tan \left (d x + c\right )^{5} - 10920 i \, a^{8} \tan \left (d x + c\right )^{4} - 11375 \, a^{8} \tan \left (d x + c\right )^{3} + 5460 i \, a^{8} \tan \left (d x + c\right )^{2} + 1365 \, a^{8} \tan \left (d x + c\right )}{1365 \, d} \] Input:

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

Output:

1/1365*(91*a^8*tan(d*x + c)^15 - 780*I*a^8*tan(d*x + c)^14 - 2625*a^8*tan( 
d*x + c)^13 + 3640*I*a^8*tan(d*x + c)^12 - 1365*a^8*tan(d*x + c)^11 + 1201 
2*I*a^8*tan(d*x + c)^10 + 15015*a^8*tan(d*x + c)^9 + 19305*a^8*tan(d*x + c 
)^7 - 20020*I*a^8*tan(d*x + c)^6 - 3003*a^8*tan(d*x + c)^5 - 10920*I*a^8*t 
an(d*x + c)^4 - 11375*a^8*tan(d*x + c)^3 + 5460*I*a^8*tan(d*x + c)^2 + 136 
5*a^8*tan(d*x + c))/d
 

Mupad [B] (verification not implemented)

Time = 1.95 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8\,\left (\frac {\sin \left (9\,c+9\,d\,x\right )}{12}+\frac {\sin \left (11\,c+11\,d\,x\right )}{52}+\frac {\sin \left (13\,c+13\,d\,x\right )}{364}+\frac {\sin \left (15\,c+15\,d\,x\right )}{5460}+\frac {\cos \left (c+d\,x\right )\,297{}\mathrm {i}}{7168}+\frac {\cos \left (3\,c+3\,d\,x\right )\,33{}\mathrm {i}}{1024}+\frac {\cos \left (5\,c+5\,d\,x\right )\,99{}\mathrm {i}}{5120}+\frac {\cos \left (7\,c+7\,d\,x\right )\,9{}\mathrm {i}}{1024}-\frac {\cos \left (9\,c+9\,d\,x\right )\,247{}\mathrm {i}}{3072}-\frac {\cos \left (11\,c+11\,d\,x\right )\,19{}\mathrm {i}}{1024}-\frac {\cos \left (13\,c+13\,d\,x\right )\,19{}\mathrm {i}}{7168}-\frac {\cos \left (15\,c+15\,d\,x\right )\,19{}\mathrm {i}}{107520}\right )}{d\,{\cos \left (c+d\,x\right )}^{15}} \] Input:

int((a + a*tan(c + d*x)*1i)^8/cos(c + d*x)^8,x)
 

Output:

(a^8*((cos(c + d*x)*297i)/7168 + (cos(3*c + 3*d*x)*33i)/1024 + (cos(5*c + 
5*d*x)*99i)/5120 + (cos(7*c + 7*d*x)*9i)/1024 - (cos(9*c + 9*d*x)*247i)/30 
72 - (cos(11*c + 11*d*x)*19i)/1024 - (cos(13*c + 13*d*x)*19i)/7168 - (cos( 
15*c + 15*d*x)*19i)/107520 + sin(9*c + 9*d*x)/12 + sin(11*c + 11*d*x)/52 + 
 sin(13*c + 13*d*x)/364 + sin(15*c + 15*d*x)/5460))/(d*cos(c + d*x)^15)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.60 \[ \int \sec ^8(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {\sin \left (d x +c \right ) a^{8} \left (-3952 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{13} i +27664 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{11} i -82992 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9} i +138320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} i -116480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} i +43680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} i -5460 \cos \left (d x +c \right ) \sin \left (d x +c \right ) i +4096 \sin \left (d x +c \right )^{14}-30720 \sin \left (d x +c \right )^{12}+99840 \sin \left (d x +c \right )^{10}-183040 \sin \left (d x +c \right )^{8}+184080 \sin \left (d x +c \right )^{6}-93912 \sin \left (d x +c \right )^{4}+20930 \sin \left (d x +c \right )^{2}-1365\right )}{1365 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{14}-7 \sin \left (d x +c \right )^{12}+21 \sin \left (d x +c \right )^{10}-35 \sin \left (d x +c \right )^{8}+35 \sin \left (d x +c \right )^{6}-21 \sin \left (d x +c \right )^{4}+7 \sin \left (d x +c \right )^{2}-1\right )} \] Input:

int(sec(d*x+c)^8*(a+I*a*tan(d*x+c))^8,x)
 

Output:

(sin(c + d*x)*a**8*( - 3952*cos(c + d*x)*sin(c + d*x)**13*i + 27664*cos(c 
+ d*x)*sin(c + d*x)**11*i - 82992*cos(c + d*x)*sin(c + d*x)**9*i + 138320* 
cos(c + d*x)*sin(c + d*x)**7*i - 116480*cos(c + d*x)*sin(c + d*x)**5*i + 4 
3680*cos(c + d*x)*sin(c + d*x)**3*i - 5460*cos(c + d*x)*sin(c + d*x)*i + 4 
096*sin(c + d*x)**14 - 30720*sin(c + d*x)**12 + 99840*sin(c + d*x)**10 - 1 
83040*sin(c + d*x)**8 + 184080*sin(c + d*x)**6 - 93912*sin(c + d*x)**4 + 2 
0930*sin(c + d*x)**2 - 1365))/(1365*cos(c + d*x)*d*(sin(c + d*x)**14 - 7*s 
in(c + d*x)**12 + 21*sin(c + d*x)**10 - 35*sin(c + d*x)**8 + 35*sin(c + d* 
x)**6 - 21*sin(c + d*x)**4 + 7*sin(c + d*x)**2 - 1))