Integrand size = 24, antiderivative size = 213 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac {20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac {20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {8 i \sec ^3(c+d x)}{9009 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3} \]
1/13*I*sec(d*x+c)^3/d/(a+I*a*tan(d*x+c))^8+5/143*I*sec(d*x+c)^3/a/d/(a+I*a *tan(d*x+c))^7+20/1287*I*sec(d*x+c)^3/a^2/d/(a+I*a*tan(d*x+c))^6+20/3003*I *sec(d*x+c)^3/a^3/d/(a+I*a*tan(d*x+c))^5+8/3003*I*sec(d*x+c)^3/d/(a^2+I*a^ 2*tan(d*x+c))^4+8/9009*I*sec(d*x+c)^3/a^2/d/(a^2+I*a^2*tan(d*x+c))^3
Time = 0.75 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.45 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^8(c+d x) (11440 \cos (c+d x)+6552 \cos (3 (c+d x))+1848 \cos (5 (c+d x))+1430 i \sin (c+d x)+2457 i \sin (3 (c+d x))+1155 i \sin (5 (c+d x)))}{144144 a^8 d (-i+\tan (c+d x))^8} \]
((I/144144)*Sec[c + d*x]^8*(11440*Cos[c + d*x] + 6552*Cos[3*(c + d*x)] + 1 848*Cos[5*(c + d*x)] + (1430*I)*Sin[c + d*x] + (2457*I)*Sin[3*(c + d*x)] + (1155*I)*Sin[5*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)
Time = 1.02 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3983, 3042, 3983, 3042, 3983, 3042, 3983, 3042, 3983, 3042, 3969}
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.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^3}{(a+i a \tan (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {5 \int \frac {\sec ^3(c+d x)}{(i \tan (c+d x) a+a)^7}dx}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \int \frac {\sec (c+d x)^3}{(i \tan (c+d x) a+a)^7}dx}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {5 \left (\frac {4 \int \frac {\sec ^3(c+d x)}{(i \tan (c+d x) a+a)^6}dx}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \int \frac {\sec (c+d x)^3}{(i \tan (c+d x) a+a)^6}dx}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\int \frac {\sec ^3(c+d x)}{(i \tan (c+d x) a+a)^5}dx}{3 a}+\frac {i \sec ^3(c+d x)}{9 d (a+i a \tan (c+d x))^6}\right )}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\int \frac {\sec (c+d x)^3}{(i \tan (c+d x) a+a)^5}dx}{3 a}+\frac {i \sec ^3(c+d x)}{9 d (a+i a \tan (c+d x))^6}\right )}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \int \frac {\sec ^3(c+d x)}{(i \tan (c+d x) a+a)^4}dx}{7 a}+\frac {i \sec ^3(c+d x)}{7 d (a+i a \tan (c+d x))^5}}{3 a}+\frac {i \sec ^3(c+d x)}{9 d (a+i a \tan (c+d x))^6}\right )}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \int \frac {\sec (c+d x)^3}{(i \tan (c+d x) a+a)^4}dx}{7 a}+\frac {i \sec ^3(c+d x)}{7 d (a+i a \tan (c+d x))^5}}{3 a}+\frac {i \sec ^3(c+d x)}{9 d (a+i a \tan (c+d x))^6}\right )}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \left (\frac {\int \frac {\sec ^3(c+d x)}{(i \tan (c+d x) a+a)^3}dx}{5 a}+\frac {i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}\right )}{7 a}+\frac {i \sec ^3(c+d x)}{7 d (a+i a \tan (c+d x))^5}}{3 a}+\frac {i \sec ^3(c+d x)}{9 d (a+i a \tan (c+d x))^6}\right )}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {4 \left (\frac {\frac {2 \left (\frac {\int \frac {\sec (c+d x)^3}{(i \tan (c+d x) a+a)^3}dx}{5 a}+\frac {i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}\right )}{7 a}+\frac {i \sec ^3(c+d x)}{7 d (a+i a \tan (c+d x))^5}}{3 a}+\frac {i \sec ^3(c+d x)}{9 d (a+i a \tan (c+d x))^6}\right )}{11 a}+\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}\right )}{13 a}+\frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3969 |
| \(\displaystyle \frac {i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac {5 \left (\frac {i \sec ^3(c+d x)}{11 d (a+i a \tan (c+d x))^7}+\frac {4 \left (\frac {i \sec ^3(c+d x)}{9 d (a+i a \tan (c+d x))^6}+\frac {\frac {i \sec ^3(c+d x)}{7 d (a+i a \tan (c+d x))^5}+\frac {2 \left (\frac {i \sec ^3(c+d x)}{15 a d (a+i a \tan (c+d x))^3}+\frac {i \sec ^3(c+d x)}{5 d (a+i a \tan (c+d x))^4}\right )}{7 a}}{3 a}\right )}{11 a}\right )}{13 a}\) |
((I/13)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^8) + (5*(((I/11)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^7) + (4*(((I/9)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^6) + (((I/7)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^ 5) + (2*(((I/5)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^4) + ((I/15)*Sec [c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*x])^3)))/(7*a))/(3*a)))/(11*a)))/(13* a)
3.2.81.3.1 Defintions of rubi rules used
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ (a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ [Simplify[m + n], 0]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[a*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ (b*f*(m + 2*n))), x] + Simp[Simplify[m + n]/(a*(m + 2*n)) Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x ] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2* n]
Time = 0.78 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{96 a^{8} d}+\frac {i {\mathrm e}^{-5 i \left (d x +c \right )}}{32 a^{8} d}+\frac {5 i {\mathrm e}^{-7 i \left (d x +c \right )}}{112 a^{8} d}+\frac {5 i {\mathrm e}^{-9 i \left (d x +c \right )}}{144 a^{8} d}+\frac {5 i {\mathrm e}^{-11 i \left (d x +c \right )}}{352 a^{8} d}+\frac {i {\mathrm e}^{-13 i \left (d x +c \right )}}{416 a^{8} d}\) | \(110\) |
| derivativedivides | \(\frac {-\frac {188}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {256}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}+\frac {480}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {2672 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {9056}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {11680}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {864 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {200 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {1472 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {4544}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\) | \(222\) |
| default | \(\frac {-\frac {188}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {256}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}+\frac {480}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {2672 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {9056}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {11680}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {864 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {200 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {1472 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {4544}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\) | \(222\) |
1/96*I/a^8/d*exp(-3*I*(d*x+c))+1/32*I/a^8/d*exp(-5*I*(d*x+c))+5/112*I/a^8/ d*exp(-7*I*(d*x+c))+5/144*I/a^8/d*exp(-9*I*(d*x+c))+5/352*I/a^8/d*exp(-11* I*(d*x+c))+1/416*I/a^8/d*exp(-13*I*(d*x+c))
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.35 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (3003 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 9009 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 12870 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10010 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4095 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 693 i\right )} e^{\left (-13 i \, d x - 13 i \, c\right )}}{288288 \, a^{8} d} \]
1/288288*(3003*I*e^(10*I*d*x + 10*I*c) + 9009*I*e^(8*I*d*x + 8*I*c) + 1287 0*I*e^(6*I*d*x + 6*I*c) + 10010*I*e^(4*I*d*x + 4*I*c) + 4095*I*e^(2*I*d*x + 2*I*c) + 693*I)*e^(-13*I*d*x - 13*I*c)/(a^8*d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (189) = 378\).
Time = 8.74 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.36 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Too large to display} \]
Piecewise((-8*tan(c + d*x)**5*sec(c + d*x)**3/(9009*a**8*d*tan(c + d*x)**8 - 72072*I*a**8*d*tan(c + d*x)**7 - 252252*a**8*d*tan(c + d*x)**6 + 504504 *I*a**8*d*tan(c + d*x)**5 + 630630*a**8*d*tan(c + d*x)**4 - 504504*I*a**8* d*tan(c + d*x)**3 - 252252*a**8*d*tan(c + d*x)**2 + 72072*I*a**8*d*tan(c + d*x) + 9009*a**8*d) + 64*I*tan(c + d*x)**4*sec(c + d*x)**3/(9009*a**8*d*t an(c + d*x)**8 - 72072*I*a**8*d*tan(c + d*x)**7 - 252252*a**8*d*tan(c + d* x)**6 + 504504*I*a**8*d*tan(c + d*x)**5 + 630630*a**8*d*tan(c + d*x)**4 - 504504*I*a**8*d*tan(c + d*x)**3 - 252252*a**8*d*tan(c + d*x)**2 + 72072*I* a**8*d*tan(c + d*x) + 9009*a**8*d) + 236*tan(c + d*x)**3*sec(c + d*x)**3/( 9009*a**8*d*tan(c + d*x)**8 - 72072*I*a**8*d*tan(c + d*x)**7 - 252252*a**8 *d*tan(c + d*x)**6 + 504504*I*a**8*d*tan(c + d*x)**5 + 630630*a**8*d*tan(c + d*x)**4 - 504504*I*a**8*d*tan(c + d*x)**3 - 252252*a**8*d*tan(c + d*x)* *2 + 72072*I*a**8*d*tan(c + d*x) + 9009*a**8*d) - 544*I*tan(c + d*x)**2*se c(c + d*x)**3/(9009*a**8*d*tan(c + d*x)**8 - 72072*I*a**8*d*tan(c + d*x)** 7 - 252252*a**8*d*tan(c + d*x)**6 + 504504*I*a**8*d*tan(c + d*x)**5 + 6306 30*a**8*d*tan(c + d*x)**4 - 504504*I*a**8*d*tan(c + d*x)**3 - 252252*a**8* d*tan(c + d*x)**2 + 72072*I*a**8*d*tan(c + d*x) + 9009*a**8*d) - 911*tan(c + d*x)*sec(c + d*x)**3/(9009*a**8*d*tan(c + d*x)**8 - 72072*I*a**8*d*tan( c + d*x)**7 - 252252*a**8*d*tan(c + d*x)**6 + 504504*I*a**8*d*tan(c + d*x) **5 + 630630*a**8*d*tan(c + d*x)**4 - 504504*I*a**8*d*tan(c + d*x)**3 -...
Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.66 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {693 i \, \cos \left (13 \, d x + 13 \, c\right ) + 4095 i \, \cos \left (11 \, d x + 11 \, c\right ) + 10010 i \, \cos \left (9 \, d x + 9 \, c\right ) + 12870 i \, \cos \left (7 \, d x + 7 \, c\right ) + 9009 i \, \cos \left (5 \, d x + 5 \, c\right ) + 3003 i \, \cos \left (3 \, d x + 3 \, c\right ) + 693 \, \sin \left (13 \, d x + 13 \, c\right ) + 4095 \, \sin \left (11 \, d x + 11 \, c\right ) + 10010 \, \sin \left (9 \, d x + 9 \, c\right ) + 12870 \, \sin \left (7 \, d x + 7 \, c\right ) + 9009 \, \sin \left (5 \, d x + 5 \, c\right ) + 3003 \, \sin \left (3 \, d x + 3 \, c\right )}{288288 \, a^{8} d} \]
1/288288*(693*I*cos(13*d*x + 13*c) + 4095*I*cos(11*d*x + 11*c) + 10010*I*c os(9*d*x + 9*c) + 12870*I*cos(7*d*x + 7*c) + 9009*I*cos(5*d*x + 5*c) + 300 3*I*cos(3*d*x + 3*c) + 693*sin(13*d*x + 13*c) + 4095*sin(11*d*x + 11*c) + 10010*sin(9*d*x + 9*c) + 12870*sin(7*d*x + 7*c) + 9009*sin(5*d*x + 5*c) + 3003*sin(3*d*x + 3*c))/(a^8*d)
Time = 1.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.83 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2 \, {\left (9009 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 45045 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 183183 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 435435 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 810810 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1051050 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1076790 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 785070 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 451165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 171457 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 51675 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7111 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1240\right )}}{9009 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{13}} \]
2/9009*(9009*tan(1/2*d*x + 1/2*c)^12 - 45045*I*tan(1/2*d*x + 1/2*c)^11 - 1 83183*tan(1/2*d*x + 1/2*c)^10 + 435435*I*tan(1/2*d*x + 1/2*c)^9 + 810810*t an(1/2*d*x + 1/2*c)^8 - 1051050*I*tan(1/2*d*x + 1/2*c)^7 - 1076790*tan(1/2 *d*x + 1/2*c)^6 + 785070*I*tan(1/2*d*x + 1/2*c)^5 + 451165*tan(1/2*d*x + 1 /2*c)^4 - 171457*I*tan(1/2*d*x + 1/2*c)^3 - 51675*tan(1/2*d*x + 1/2*c)^2 + 7111*I*tan(1/2*d*x + 1/2*c) + 1240)/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^13)
Time = 5.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\frac {{\cos \left (3\,c+3\,d\,x\right )}^3\,5{}\mathrm {i}}{36}+\frac {5\,\sin \left (3\,c+3\,d\,x\right )\,{\cos \left (3\,c+3\,d\,x\right )}^2}{36}-\frac {\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{32}+\frac {\cos \left (5\,c+5\,d\,x\right )\,1{}\mathrm {i}}{32}+\frac {\cos \left (7\,c+7\,d\,x\right )\,5{}\mathrm {i}}{112}+\frac {\cos \left (11\,c+11\,d\,x\right )\,5{}\mathrm {i}}{352}+\frac {\cos \left (13\,c+13\,d\,x\right )\,1{}\mathrm {i}}{416}-\frac {7\,\sin \left (3\,c+3\,d\,x\right )}{288}+\frac {\sin \left (5\,c+5\,d\,x\right )}{32}+\frac {5\,\sin \left (7\,c+7\,d\,x\right )}{112}+\frac {5\,\sin \left (11\,c+11\,d\,x\right )}{352}+\frac {\sin \left (13\,c+13\,d\,x\right )}{416}}{a^8\,d} \]
((cos(5*c + 5*d*x)*1i)/32 - (cos(3*c + 3*d*x)*3i)/32 + (cos(7*c + 7*d*x)*5 i)/112 + (cos(11*c + 11*d*x)*5i)/352 + (cos(13*c + 13*d*x)*1i)/416 - (7*si n(3*c + 3*d*x))/288 + sin(5*c + 5*d*x)/32 + (5*sin(7*c + 7*d*x))/112 + (5* sin(11*c + 11*d*x))/352 + sin(13*c + 13*d*x)/416 + (cos(3*c + 3*d*x)^3*5i) /36 + (5*cos(3*c + 3*d*x)^2*sin(3*c + 3*d*x))/36)/(a^8*d)