Integrand size = 22, antiderivative size = 269 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac {14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac {14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac {8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {8 i \sec (c+d x)}{2145 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac {16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )} \] Output:
1/15*I*sec(d*x+c)/d/(a+I*a*tan(d*x+c))^8+7/195*I*sec(d*x+c)/a/d/(a+I*a*tan (d*x+c))^7+14/715*I*sec(d*x+c)/a^2/d/(a+I*a*tan(d*x+c))^6+14/1287*I*sec(d* x+c)/a^3/d/(a+I*a*tan(d*x+c))^5+8/1287*I*sec(d*x+c)/d/(a^2+I*a^2*tan(d*x+c ))^4+8/2145*I*sec(d*x+c)/a^2/d/(a^2+I*a^2*tan(d*x+c))^3+16/6435*I*sec(d*x+ c)/d/(a^4+I*a^4*tan(d*x+c))^2+16/6435*I*sec(d*x+c)/d/(a^8+I*a^8*tan(d*x+c) )
Time = 0.70 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.43 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^8(c+d x) (28600 \cos (c+d x)+19656 \cos (3 (c+d x))+9240 \cos (5 (c+d x))+3432 \cos (7 (c+d x))+3575 i \sin (c+d x)+7371 i \sin (3 (c+d x))+5775 i \sin (5 (c+d x))+3003 i \sin (7 (c+d x)))}{411840 a^8 d (-i+\tan (c+d x))^8} \] Input:
Integrate[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^8,x]
Output:
((I/411840)*Sec[c + d*x]^8*(28600*Cos[c + d*x] + 19656*Cos[3*(c + d*x)] + 9240*Cos[5*(c + d*x)] + 3432*Cos[7*(c + d*x)] + (3575*I)*Sin[c + d*x] + (7 371*I)*Sin[3*(c + d*x)] + (5775*I)*Sin[5*(c + d*x)] + (3003*I)*Sin[7*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)
Time = 1.25 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 3983, 3042, 3983, 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.
| \(\displaystyle \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {7 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^7}dx}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^7}dx}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {7 \left (\frac {6 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^6}dx}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {6 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^6}dx}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^5}dx}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^5}dx}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^4}dx}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^4}dx}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^3}dx}{7 a}+\frac {i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}\right )}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^3}dx}{7 a}+\frac {i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}\right )}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \left (\frac {2 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^2}dx}{5 a}+\frac {i \sec (c+d x)}{5 d (a+i a \tan (c+d x))^3}\right )}{7 a}+\frac {i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}\right )}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \left (\frac {2 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^2}dx}{5 a}+\frac {i \sec (c+d x)}{5 d (a+i a \tan (c+d x))^3}\right )}{7 a}+\frac {i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}\right )}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\sec (c+d x)}{i \tan (c+d x) a+a}dx}{3 a}+\frac {i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2}\right )}{5 a}+\frac {i \sec (c+d x)}{5 d (a+i a \tan (c+d x))^3}\right )}{7 a}+\frac {i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}\right )}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\sec (c+d x)}{i \tan (c+d x) a+a}dx}{3 a}+\frac {i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2}\right )}{5 a}+\frac {i \sec (c+d x)}{5 d (a+i a \tan (c+d x))^3}\right )}{7 a}+\frac {i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}\right )}{9 a}+\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}\right )}{11 a}+\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}\right )}{13 a}+\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}\right )}{15 a}+\frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3969 |
| \(\displaystyle \frac {i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac {7 \left (\frac {i \sec (c+d x)}{13 d (a+i a \tan (c+d x))^7}+\frac {6 \left (\frac {i \sec (c+d x)}{11 d (a+i a \tan (c+d x))^6}+\frac {5 \left (\frac {i \sec (c+d x)}{9 d (a+i a \tan (c+d x))^5}+\frac {4 \left (\frac {i \sec (c+d x)}{7 d (a+i a \tan (c+d x))^4}+\frac {3 \left (\frac {i \sec (c+d x)}{5 d (a+i a \tan (c+d x))^3}+\frac {2 \left (\frac {i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))}+\frac {i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\) |
Input:
Int[Sec[c + d*x]/(a + I*a*Tan[c + d*x])^8,x]
Output:
((I/15)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (7*(((I/13)*Sec[c + d *x])/(d*(a + I*a*Tan[c + d*x])^7) + (6*(((I/11)*Sec[c + d*x])/(d*(a + I*a* Tan[c + d*x])^6) + (5*(((I/9)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^5) + (4*(((I/7)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^4) + (3*(((I/5)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^3) + (2*(((I/3)*Sec[c + d*x])/(d*(a + I* a*Tan[c + d*x])^2) + ((I/3)*Sec[c + d*x])/(a*d*(a + I*a*Tan[c + d*x]))))/( 5*a)))/(7*a)))/(9*a)))/(11*a)))/(13*a)))/(15*a)
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 = 0.92 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{128 a^{8} d}+\frac {7 i {\mathrm e}^{-3 i \left (d x +c \right )}}{384 a^{8} d}+\frac {21 i {\mathrm e}^{-5 i \left (d x +c \right )}}{640 a^{8} d}+\frac {5 i {\mathrm e}^{-7 i \left (d x +c \right )}}{128 a^{8} d}+\frac {35 i {\mathrm e}^{-9 i \left (d x +c \right )}}{1152 a^{8} d}+\frac {21 i {\mathrm e}^{-11 i \left (d x +c \right )}}{1408 a^{8} d}+\frac {7 i {\mathrm e}^{-13 i \left (d x +c \right )}}{1664 a^{8} d}+\frac {i {\mathrm e}^{-15 i \left (d x +c \right )}}{1920 a^{8} d}\) | \(146\) |
| derivativedivides | \(\frac {-\frac {196}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {29792}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {2128}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {3584 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}+\frac {15008 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {6272}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {2944 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {23744}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2968}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {256}{15 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}-\frac {224 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3752 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}}{a^{8} d}\) | \(255\) |
| default | \(\frac {-\frac {196}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {29792}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {2128}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {3584 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}+\frac {15008 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {6272}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {2944 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {23744}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2968}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {256}{15 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}-\frac {224 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3752 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}}{a^{8} d}\) | \(255\) |
Input:
int(sec(d*x+c)/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/128*I/a^8/d*exp(-I*(d*x+c))+7/384*I/a^8/d*exp(-3*I*(d*x+c))+21/640*I/a^8 /d*exp(-5*I*(d*x+c))+5/128*I/a^8/d*exp(-7*I*(d*x+c))+35/1152*I/a^8/d*exp(- 9*I*(d*x+c))+21/1408*I/a^8/d*exp(-11*I*(d*x+c))+7/1664*I/a^8/d*exp(-13*I*( d*x+c))+1/1920*I/a^8/d*exp(-15*I*(d*x+c))
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.36 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (6435 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 15015 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 27027 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 32175 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 25025 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 12285 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3465 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 429 i\right )} e^{\left (-15 i \, d x - 15 i \, c\right )}}{823680 \, a^{8} d} \] Input:
integrate(sec(d*x+c)/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
Output:
1/823680*(6435*I*e^(14*I*d*x + 14*I*c) + 15015*I*e^(12*I*d*x + 12*I*c) + 2 7027*I*e^(10*I*d*x + 10*I*c) + 32175*I*e^(8*I*d*x + 8*I*c) + 25025*I*e^(6* I*d*x + 6*I*c) + 12285*I*e^(4*I*d*x + 4*I*c) + 3465*I*e^(2*I*d*x + 2*I*c) + 429*I)*e^(-15*I*d*x - 15*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. 1221 vs. \(2 (238) = 476\).
Time = 10.26 (sec) , antiderivative size = 1221, normalized size of antiderivative = 4.54 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)/(a+I*a*tan(d*x+c))**8,x)
Output:
Piecewise((16*tan(c + d*x)**7*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I* a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*t an(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8*d*tan(c + d* x) + 6435*a**8*d) - 128*I*tan(c + d*x)**6*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**4 - 36036 0*I*a**8*d*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8* d*tan(c + d*x) + 6435*a**8*d) - 456*tan(c + d*x)**5*sec(c + d*x)/(6435*a** 8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)* *4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 514 80*I*a**8*d*tan(c + d*x) + 6435*a**8*d) + 960*I*tan(c + d*x)**4*sec(c + d* x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180* a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*t an(c + d*x)**4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d *x)**2 + 51480*I*a**8*d*tan(c + d*x) + 6435*a**8*d) + 1350*tan(c + d*x)**3 *sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)** 7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 4504 50*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a*...
Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {429 i \, \cos \left (15 \, d x + 15 \, c\right ) + 3465 i \, \cos \left (13 \, d x + 13 \, c\right ) + 12285 i \, \cos \left (11 \, d x + 11 \, c\right ) + 25025 i \, \cos \left (9 \, d x + 9 \, c\right ) + 32175 i \, \cos \left (7 \, d x + 7 \, c\right ) + 27027 i \, \cos \left (5 \, d x + 5 \, c\right ) + 15015 i \, \cos \left (3 \, d x + 3 \, c\right ) + 6435 i \, \cos \left (d x + c\right ) + 429 \, \sin \left (15 \, d x + 15 \, c\right ) + 3465 \, \sin \left (13 \, d x + 13 \, c\right ) + 12285 \, \sin \left (11 \, d x + 11 \, c\right ) + 25025 \, \sin \left (9 \, d x + 9 \, c\right ) + 32175 \, \sin \left (7 \, d x + 7 \, c\right ) + 27027 \, \sin \left (5 \, d x + 5 \, c\right ) + 15015 \, \sin \left (3 \, d x + 3 \, c\right ) + 6435 \, \sin \left (d x + c\right )}{823680 \, a^{8} d} \] Input:
integrate(sec(d*x+c)/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
Output:
1/823680*(429*I*cos(15*d*x + 15*c) + 3465*I*cos(13*d*x + 13*c) + 12285*I*c os(11*d*x + 11*c) + 25025*I*cos(9*d*x + 9*c) + 32175*I*cos(7*d*x + 7*c) + 27027*I*cos(5*d*x + 5*c) + 15015*I*cos(3*d*x + 3*c) + 6435*I*cos(d*x + c) + 429*sin(15*d*x + 15*c) + 3465*sin(13*d*x + 13*c) + 12285*sin(11*d*x + 11 *c) + 25025*sin(9*d*x + 9*c) + 32175*sin(7*d*x + 7*c) + 27027*sin(5*d*x + 5*c) + 15015*sin(3*d*x + 3*c) + 6435*sin(d*x + c))/(a^8*d)
Time = 0.55 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.75 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2 \, {\left (6435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 45045 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 210210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 630630 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1414413 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 2357355 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3063060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 3063060 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2407405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1444443 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 668850 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 222950 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7845 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 952\right )}}{6435 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{15}} \] Input:
integrate(sec(d*x+c)/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
Output:
2/6435*(6435*tan(1/2*d*x + 1/2*c)^14 - 45045*I*tan(1/2*d*x + 1/2*c)^13 - 2 10210*tan(1/2*d*x + 1/2*c)^12 + 630630*I*tan(1/2*d*x + 1/2*c)^11 + 1414413 *tan(1/2*d*x + 1/2*c)^10 - 2357355*I*tan(1/2*d*x + 1/2*c)^9 - 3063060*tan( 1/2*d*x + 1/2*c)^8 + 3063060*I*tan(1/2*d*x + 1/2*c)^7 + 2407405*tan(1/2*d* x + 1/2*c)^6 - 1444443*I*tan(1/2*d*x + 1/2*c)^5 - 668850*tan(1/2*d*x + 1/2 *c)^4 + 222950*I*tan(1/2*d*x + 1/2*c)^3 + 54915*tan(1/2*d*x + 1/2*c)^2 - 7 845*I*tan(1/2*d*x + 1/2*c) - 952)/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^15)
Time = 2.47 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.83 \[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )\,\left (-\frac {{\sin \left (c+d\,x\right )}^2\,44779{}\mathrm {i}}{32}+\frac {32175\,\sin \left (c+d\,x\right )}{128}-\frac {{\sin \left (2\,c+2\,d\,x\right )}^2\,26075{}\mathrm {i}}{16}-\frac {3575\,\sin \left (2\,c+2\,d\,x\right )}{8}+\frac {{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,114583{}\mathrm {i}}{64}-\frac {{\sin \left (3\,c+3\,d\,x\right )}^2\,57925{}\mathrm {i}}{32}+\frac {84227\,\sin \left (3\,c+3\,d\,x\right )}{128}+\frac {{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2\,116585{}\mathrm {i}}{64}+\frac {{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2\,119315{}\mathrm {i}}{64}+\frac {{\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}^2\,122285{}\mathrm {i}}{64}-754\,\sin \left (4\,c+4\,d\,x\right )+\frac {111527\,\sin \left (5\,c+5\,d\,x\right )}{128}-\frac {7187\,\sin \left (6\,c+6\,d\,x\right )}{8}+\frac {121427\,\sin \left (7\,c+7\,d\,x\right )}{128}-952{}\mathrm {i}\right )}{6435\,a^8\,d\,\left (-2\,{\sin \left (\frac {15\,c}{4}+\frac {15\,d\,x}{4}\right )}^2+\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )} \] Input:
int(1/(cos(c + d*x)*(a + a*tan(c + d*x)*1i)^8),x)
Output:
(2*(2*sin(c/4 + (d*x)/4)^2 - 1)*((32175*sin(c + d*x))/128 - (3575*sin(2*c + 2*d*x))/8 + (84227*sin(3*c + 3*d*x))/128 - 754*sin(4*c + 4*d*x) + (11152 7*sin(5*c + 5*d*x))/128 - (7187*sin(6*c + 6*d*x))/8 + (121427*sin(7*c + 7* d*x))/128 - (sin(2*c + 2*d*x)^2*26075i)/16 + (sin(c/2 + (d*x)/2)^2*114583i )/64 - (sin(3*c + 3*d*x)^2*57925i)/32 + (sin((3*c)/2 + (3*d*x)/2)^2*116585 i)/64 + (sin((5*c)/2 + (5*d*x)/2)^2*119315i)/64 + (sin((7*c)/2 + (7*d*x)/2 )^2*122285i)/64 - (sin(c + d*x)^2*44779i)/32 - 952i))/(6435*a^8*d*(sin((15 *c)/2 + (15*d*x)/2)*1i - 2*sin((15*c)/4 + (15*d*x)/4)^2 + 1))
\[ \int \frac {\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {too large to display} \] Input:
int(sec(d*x+c)/(a+I*a*tan(d*x+c))^8,x)
Output:
( - 4*int(cos(c + d*x)/(128*cos(c + d*x)*sin(c + d*x)**7*i - 192*cos(c + d *x)*sin(c + d*x)**5*i + 80*cos(c + d*x)*sin(c + d*x)**3*i - 8*cos(c + d*x) *sin(c + d*x)*i - 128*sin(c + d*x)**8 + 256*sin(c + d*x)**6 - 160*sin(c + d*x)**4 + 32*sin(c + d*x)**2 - 1),x)*d - 512*int(sin(c + d*x)**8/(128*cos( c + d*x)*sin(c + d*x)**7*i - 192*cos(c + d*x)*sin(c + d*x)**5*i + 80*cos(c + d*x)*sin(c + d*x)**3*i - 8*cos(c + d*x)*sin(c + d*x)*i - 128*sin(c + d* x)**8 + 256*sin(c + d*x)**6 - 160*sin(c + d*x)**4 + 32*sin(c + d*x)**2 - 1 ),x)*d - 1024*int(sin(c + d*x)**6/(128*cos(c + d*x)*sin(c + d*x)**7 - 192* cos(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)**3 - 8*cos(c + d*x)*sin(c + d*x) + 128*sin(c + d*x)**8*i - 256*sin(c + d*x)**6*i + 160*s in(c + d*x)**4*i - 32*sin(c + d*x)**2*i + i),x)*d*i + 640*int(sin(c + d*x) **4/(128*cos(c + d*x)*sin(c + d*x)**7 - 192*cos(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)**3 - 8*cos(c + d*x)*sin(c + d*x) + 128*sin(c + d*x)**8*i - 256*sin(c + d*x)**6*i + 160*sin(c + d*x)**4*i - 32*sin(c + d*x)**2*i + i),x)*d*i - 128*int(sin(c + d*x)**2/(128*cos(c + d*x)*sin(c + d*x)**7 - 192*cos(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)* *3 - 8*cos(c + d*x)*sin(c + d*x) + 128*sin(c + d*x)**8*i - 256*sin(c + d*x )**6*i + 160*sin(c + d*x)**4*i - 32*sin(c + d*x)**2*i + i),x)*d*i + 512*in t((cos(c + d*x)*sin(c + d*x)**7)/(128*cos(c + d*x)*sin(c + d*x)**7 - 192*c os(c + d*x)*sin(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)**3 - 8*cos(c...