Integrand size = 24, antiderivative size = 138 \[ \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5} \]
1/11*I*sec(d*x+c)^5/d/(a+I*a*tan(d*x+c))^8+1/33*I*sec(d*x+c)^5/a/d/(a+I*a* tan(d*x+c))^7+2/231*I*sec(d*x+c)^5/a^2/d/(a+I*a*tan(d*x+c))^6+2/1155*I*sec (d*x+c)^5/a^3/d/(a+I*a*tan(d*x+c))^5
Time = 0.86 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.53 \[ \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i \sec ^8(c+d x) (440 \cos (c+d x)+168 \cos (3 (c+d x))+55 i \sin (c+d x)+63 i \sin (3 (c+d x)))}{4620 a^8 d (-i+\tan (c+d x))^8} \]
((I/4620)*Sec[c + d*x]^8*(440*Cos[c + d*x] + 168*Cos[3*(c + d*x)] + (55*I) *Sin[c + d*x] + (63*I)*Sin[3*(c + d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)
Time = 0.63 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
| \(\displaystyle \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^5}{(a+i a \tan (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {3 \int \frac {\sec ^5(c+d x)}{(i \tan (c+d x) a+a)^7}dx}{11 a}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {\sec (c+d x)^5}{(i \tan (c+d x) a+a)^7}dx}{11 a}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\sec ^5(c+d x)}{(i \tan (c+d x) a+a)^6}dx}{9 a}+\frac {i \sec ^5(c+d x)}{9 d (a+i a \tan (c+d x))^7}\right )}{11 a}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\sec (c+d x)^5}{(i \tan (c+d x) a+a)^6}dx}{9 a}+\frac {i \sec ^5(c+d x)}{9 d (a+i a \tan (c+d x))^7}\right )}{11 a}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {\sec ^5(c+d x)}{(i \tan (c+d x) a+a)^5}dx}{7 a}+\frac {i \sec ^5(c+d x)}{7 d (a+i a \tan (c+d x))^6}\right )}{9 a}+\frac {i \sec ^5(c+d x)}{9 d (a+i a \tan (c+d x))^7}\right )}{11 a}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {\sec (c+d x)^5}{(i \tan (c+d x) a+a)^5}dx}{7 a}+\frac {i \sec ^5(c+d x)}{7 d (a+i a \tan (c+d x))^6}\right )}{9 a}+\frac {i \sec ^5(c+d x)}{9 d (a+i a \tan (c+d x))^7}\right )}{11 a}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}\) |
| \(\Big \downarrow \) 3969 |
| \(\displaystyle \frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {3 \left (\frac {i \sec ^5(c+d x)}{9 d (a+i a \tan (c+d x))^7}+\frac {2 \left (\frac {i \sec ^5(c+d x)}{35 a d (a+i a \tan (c+d x))^5}+\frac {i \sec ^5(c+d x)}{7 d (a+i a \tan (c+d x))^6}\right )}{9 a}\right )}{11 a}\) |
((I/11)*Sec[c + d*x]^5)/(d*(a + I*a*Tan[c + d*x])^8) + (3*(((I/9)*Sec[c + d*x]^5)/(d*(a + I*a*Tan[c + d*x])^7) + (2*(((I/7)*Sec[c + d*x]^5)/(d*(a + I*a*Tan[c + d*x])^6) + ((I/35)*Sec[c + d*x]^5)/(a*d*(a + I*a*Tan[c + d*x]) ^5)))/(9*a)))/(11*a)
3.2.80.3.1 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 = 0.97 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(\frac {i {\mathrm e}^{-5 i \left (d x +c \right )}}{40 a^{8} d}+\frac {3 i {\mathrm e}^{-7 i \left (d x +c \right )}}{56 a^{8} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{24 a^{8} d}+\frac {i {\mathrm e}^{-11 i \left (d x +c \right )}}{88 a^{8} d}\) | \(74\) |
| derivativedivides | \(\frac {\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {576 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {4752}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {1024}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {176 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1864}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {256}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {584 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {60}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}}{a^{8} d}\) | \(189\) |
| default | \(\frac {\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {576 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}-\frac {4752}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {1024}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {176 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1864}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {256}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {584 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {60}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}}{a^{8} d}\) | \(189\) |
1/40*I/a^8/d*exp(-5*I*(d*x+c))+3/56*I/a^8/d*exp(-7*I*(d*x+c))+1/24*I/a^8/d *exp(-9*I*(d*x+c))+1/88*I/a^8/d*exp(-11*I*(d*x+c))
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.38 \[ \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (231 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 495 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 385 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{9240 \, a^{8} d} \]
1/9240*(231*I*e^(6*I*d*x + 6*I*c) + 495*I*e^(4*I*d*x + 4*I*c) + 385*I*e^(2 *I*d*x + 2*I*c) + 105*I)*e^(-11*I*d*x - 11*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. 620 vs. \(2 (119) = 238\).
Time = 8.79 (sec) , antiderivative size = 620, normalized size of antiderivative = 4.49 \[ \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} \frac {2 \tan ^{3}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} - \frac {16 i \tan ^{2}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} - \frac {61 \tan {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} + \frac {152 i \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{5}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \]
Piecewise((2*tan(c + d*x)**3*sec(c + d*x)**5/(1155*a**8*d*tan(c + d*x)**8 - 9240*I*a**8*d*tan(c + d*x)**7 - 32340*a**8*d*tan(c + d*x)**6 + 64680*I*a **8*d*tan(c + d*x)**5 + 80850*a**8*d*tan(c + d*x)**4 - 64680*I*a**8*d*tan( c + d*x)**3 - 32340*a**8*d*tan(c + d*x)**2 + 9240*I*a**8*d*tan(c + d*x) + 1155*a**8*d) - 16*I*tan(c + d*x)**2*sec(c + d*x)**5/(1155*a**8*d*tan(c + d *x)**8 - 9240*I*a**8*d*tan(c + d*x)**7 - 32340*a**8*d*tan(c + d*x)**6 + 64 680*I*a**8*d*tan(c + d*x)**5 + 80850*a**8*d*tan(c + d*x)**4 - 64680*I*a**8 *d*tan(c + d*x)**3 - 32340*a**8*d*tan(c + d*x)**2 + 9240*I*a**8*d*tan(c + d*x) + 1155*a**8*d) - 61*tan(c + d*x)*sec(c + d*x)**5/(1155*a**8*d*tan(c + d*x)**8 - 9240*I*a**8*d*tan(c + d*x)**7 - 32340*a**8*d*tan(c + d*x)**6 + 64680*I*a**8*d*tan(c + d*x)**5 + 80850*a**8*d*tan(c + d*x)**4 - 64680*I*a* *8*d*tan(c + d*x)**3 - 32340*a**8*d*tan(c + d*x)**2 + 9240*I*a**8*d*tan(c + d*x) + 1155*a**8*d) + 152*I*sec(c + d*x)**5/(1155*a**8*d*tan(c + d*x)**8 - 9240*I*a**8*d*tan(c + d*x)**7 - 32340*a**8*d*tan(c + d*x)**6 + 64680*I* a**8*d*tan(c + d*x)**5 + 80850*a**8*d*tan(c + d*x)**4 - 64680*I*a**8*d*tan (c + d*x)**3 - 32340*a**8*d*tan(c + d*x)**2 + 9240*I*a**8*d*tan(c + d*x) + 1155*a**8*d), Ne(d, 0)), (x*sec(c)**5/(I*a*tan(c) + a)**8, True))
Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {105 i \, \cos \left (11 \, d x + 11 \, c\right ) + 385 i \, \cos \left (9 \, d x + 9 \, c\right ) + 495 i \, \cos \left (7 \, d x + 7 \, c\right ) + 231 i \, \cos \left (5 \, d x + 5 \, c\right ) + 105 \, \sin \left (11 \, d x + 11 \, c\right ) + 385 \, \sin \left (9 \, d x + 9 \, c\right ) + 495 \, \sin \left (7 \, d x + 7 \, c\right ) + 231 \, \sin \left (5 \, d x + 5 \, c\right )}{9240 \, a^{8} d} \]
1/9240*(105*I*cos(11*d*x + 11*c) + 385*I*cos(9*d*x + 9*c) + 495*I*cos(7*d* x + 7*c) + 231*I*cos(5*d*x + 5*c) + 105*sin(11*d*x + 11*c) + 385*sin(9*d*x + 9*c) + 495*sin(7*d*x + 7*c) + 231*sin(5*d*x + 5*c))/(a^8*d)
Time = 1.42 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {2 \, {\left (1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 3465 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 23100 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 37422 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 32802 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 11220 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4895 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 517 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 152\right )}}{1155 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{11}} \]
2/1155*(1155*tan(1/2*d*x + 1/2*c)^10 - 3465*I*tan(1/2*d*x + 1/2*c)^9 - 138 60*tan(1/2*d*x + 1/2*c)^8 + 23100*I*tan(1/2*d*x + 1/2*c)^7 + 37422*tan(1/2 *d*x + 1/2*c)^6 - 32802*I*tan(1/2*d*x + 1/2*c)^5 - 27060*tan(1/2*d*x + 1/2 *c)^4 + 11220*I*tan(1/2*d*x + 1/2*c)^3 + 4895*tan(1/2*d*x + 1/2*c)^2 - 517 *I*tan(1/2*d*x + 1/2*c) - 152)/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^11)
Time = 4.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.46 \[ \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\frac {{\mathrm {e}}^{-c\,5{}\mathrm {i}-d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,3{}\mathrm {i}}{56}+\frac {{\mathrm {e}}^{-c\,9{}\mathrm {i}-d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^{-c\,11{}\mathrm {i}-d\,x\,11{}\mathrm {i}}\,1{}\mathrm {i}}{88}}{a^8\,d} \]