\(\int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) [184]

Optimal result

Integrand size = 24, antiderivative size = 301 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {160 \sin (c+d x)}{4199 a^8 d}-\frac {320 \sin ^3(c+d x)}{12597 a^8 d}+\frac {32 \sin ^5(c+d x)}{4199 a^8 d}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}+\frac {11 i \cos ^3(c+d x)}{323 a d (a+i a \tan (c+d x))^7}+\frac {22 i \cos ^3(c+d x)}{969 a^2 d (a+i a \tan (c+d x))^6}+\frac {66 i \cos ^3(c+d x)}{4199 a^3 d (a+i a \tan (c+d x))^5}+\frac {48 i \cos ^3(c+d x)}{4199 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {112 i \cos ^3(c+d x)}{12597 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {64 i \cos ^5(c+d x)}{4199 d \left (a^8+i a^8 \tan (c+d x)\right )} \] Output:

160/4199*sin(d*x+c)/a^8/d-320/12597*sin(d*x+c)^3/a^8/d+32/4199*sin(d*x+c)^ 
5/a^8/d+1/19*I*cos(d*x+c)^3/d/(a+I*a*tan(d*x+c))^8+11/323*I*cos(d*x+c)^3/a 
/d/(a+I*a*tan(d*x+c))^7+22/969*I*cos(d*x+c)^3/a^2/d/(a+I*a*tan(d*x+c))^6+6 
6/4199*I*cos(d*x+c)^3/a^3/d/(a+I*a*tan(d*x+c))^5+48/4199*I*cos(d*x+c)^3/d/ 
(a^2+I*a^2*tan(d*x+c))^4+112/12597*I*cos(d*x+c)^3/a^2/d/(a^2+I*a^2*tan(d*x 
+c))^3+64/4199*I*cos(d*x+c)^5/d/(a^8+I*a^8*tan(d*x+c))
 

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {i \sec ^8(c+d x) (-739024 \cos (c+d x)-604656 \cos (3 (c+d x))-426360 \cos (5 (c+d x))-369512 \cos (7 (c+d x))+65208 \cos (9 (c+d x))+1768 \cos (11 (c+d x))-92378 i \sin (c+d x)-226746 i \sin (3 (c+d x))-266475 i \sin (5 (c+d x))-323323 i \sin (7 (c+d x))+73359 i \sin (9 (c+d x))+2431 i \sin (11 (c+d x)))}{12899328 a^8 d (-i+\tan (c+d x))^8} \] Input:

Integrate[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^8,x]
 

Output:

((-1/12899328*I)*Sec[c + d*x]^8*(-739024*Cos[c + d*x] - 604656*Cos[3*(c + 
d*x)] - 426360*Cos[5*(c + d*x)] - 369512*Cos[7*(c + d*x)] + 65208*Cos[9*(c 
 + d*x)] + 1768*Cos[11*(c + d*x)] - (92378*I)*Sin[c + d*x] - (226746*I)*Si 
n[3*(c + d*x)] - (266475*I)*Sin[5*(c + d*x)] - (323323*I)*Sin[7*(c + d*x)] 
 + (73359*I)*Sin[9*(c + d*x)] + (2431*I)*Sin[11*(c + d*x)]))/(a^8*d*(-I + 
Tan[c + d*x])^8)
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {3042, 3983, 3042, 3983, 3042, 3983, 3042, 3983, 3042, 3983, 3042, 3983, 3042, 3981, 3042, 3113, 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.

\(\displaystyle \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sec (c+d x)^3 (a+i a \tan (c+d x))^8}dx\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {11 \int \frac {\cos ^3(c+d x)}{(i \tan (c+d x) a+a)^7}dx}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {11 \int \frac {1}{\sec (c+d x)^3 (i \tan (c+d x) a+a)^7}dx}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {11 \left (\frac {10 \int \frac {\cos ^3(c+d x)}{(i \tan (c+d x) a+a)^6}dx}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {11 \left (\frac {10 \int \frac {1}{\sec (c+d x)^3 (i \tan (c+d x) a+a)^6}dx}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \int \frac {\cos ^3(c+d x)}{(i \tan (c+d x) a+a)^5}dx}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \int \frac {1}{\sec (c+d x)^3 (i \tan (c+d x) a+a)^5}dx}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \int \frac {\cos ^3(c+d x)}{(i \tan (c+d x) a+a)^4}dx}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \int \frac {1}{\sec (c+d x)^3 (i \tan (c+d x) a+a)^4}dx}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \int \frac {\cos ^3(c+d x)}{(i \tan (c+d x) a+a)^3}dx}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \int \frac {1}{\sec (c+d x)^3 (i \tan (c+d x) a+a)^3}dx}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3983

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \int \frac {\cos ^3(c+d x)}{(i \tan (c+d x) a+a)^2}dx}{3 a}+\frac {i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}\right )}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \int \frac {1}{\sec (c+d x)^3 (i \tan (c+d x) a+a)^2}dx}{3 a}+\frac {i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}\right )}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3981

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \int \cos ^5(c+d x)dx}{7 a^2}+\frac {2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{3 a}+\frac {i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}\right )}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (\frac {5 \int \sin \left (c+d x+\frac {\pi }{2}\right )^5dx}{7 a^2}+\frac {2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{3 a}+\frac {i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}\right )}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (-\frac {5 \int \left (\sin ^4(c+d x)-2 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{7 a^2 d}+\frac {2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{3 a}+\frac {i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}\right )}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {11 \left (\frac {10 \left (\frac {3 \left (\frac {8 \left (\frac {7 \left (\frac {2 \left (-\frac {5 \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{7 a^2 d}+\frac {2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{3 a}+\frac {i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}\right )}{11 a}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}\right )}{13 a}+\frac {i \cos ^3(c+d x)}{13 d (a+i a \tan (c+d x))^5}\right )}{5 a}+\frac {i \cos ^3(c+d x)}{15 d (a+i a \tan (c+d x))^6}\right )}{17 a}+\frac {i \cos ^3(c+d x)}{17 d (a+i a \tan (c+d x))^7}\right )}{19 a}+\frac {i \cos ^3(c+d x)}{19 d (a+i a \tan (c+d x))^8}\)

Input:

Int[Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^8,x]
 

Output:

((I/19)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^8) + (11*(((I/17)*Cos[c 
+ d*x]^3)/(d*(a + I*a*Tan[c + d*x])^7) + (10*(((I/15)*Cos[c + d*x]^3)/(d*( 
a + I*a*Tan[c + d*x])^6) + (3*(((I/13)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + 
 d*x])^5) + (8*(((I/11)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^4) + (7* 
(((I/9)*Cos[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^3) + (2*((-5*(-Sin[c + d 
*x] + (2*Sin[c + d*x]^3)/3 - Sin[c + d*x]^5/5))/(7*a^2*d) + (((2*I)/7)*Cos 
[c + d*x]^5)/(d*(a^2 + I*a^2*Tan[c + d*x]))))/(3*a)))/(11*a)))/(13*a)))/(5 
*a)))/(17*a)))/(19*a)
 

Defintions of rubi rules used

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 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3981

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)])^(n_), x_Symbol] :> Simp[2*d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + 
 f*x])^(n + 1)/(b*f*(m + 2*n))), x] - Simp[d^2*((m - 2)/(b^2*(m + 2*n))) 
Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[ 
{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] 
 && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (IntegersQ[n, m + 
1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]
 

rule 3983

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]
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.70

method	result	size
risch	\(\frac {33 i {\mathrm e}^{-5 i \left (d x +c \right )}}{1024 a^{8} d}+\frac {33 i {\mathrm e}^{-7 i \left (d x +c \right )}}{1024 a^{8} d}+\frac {77 i {\mathrm e}^{-9 i \left (d x +c \right )}}{3072 a^{8} d}+\frac {15 i {\mathrm e}^{-11 i \left (d x +c \right )}}{1024 a^{8} d}+\frac {165 i {\mathrm e}^{-13 i \left (d x +c \right )}}{26624 a^{8} d}+\frac {11 i {\mathrm e}^{-15 i \left (d x +c \right )}}{6144 a^{8} d}+\frac {11 i {\mathrm e}^{-17 i \left (d x +c \right )}}{34816 a^{8} d}+\frac {i {\mathrm e}^{-19 i \left (d x +c \right )}}{38912 a^{8} d}+\frac {11 i \cos \left (d x +c \right )}{512 a^{8} d}+\frac {33 \sin \left (d x +c \right )}{1024 a^{8} d}+\frac {41 i \cos \left (3 d x +3 c \right )}{1536 a^{8} d}+\frac {83 \sin \left (3 d x +3 c \right )}{3072 a^{8} d}\)	\(210\)
derivativedivides	\(\frac {\frac {8856 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}-\frac {1}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {3}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}+\frac {7181 i}{512 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {i}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {32417 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{18}}-\frac {2177 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {50936 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {32525 i}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {204605 i}{96 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {1984 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{16}}-\frac {256}{19 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{19}}+\frac {10496}{17 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{17}}-\frac {14192}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}+\frac {175016}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {18011}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {12430}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {72425}{16 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {26871}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {54229}{768 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {509}{256 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{8} d}\)	\(372\)
default	\(\frac {\frac {8856 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}-\frac {1}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {3}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}+\frac {7181 i}{512 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {i}{512 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {32417 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{18}}-\frac {2177 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {50936 i}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}-\frac {32525 i}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {204605 i}{96 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {1984 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{16}}-\frac {256}{19 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{19}}+\frac {10496}{17 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{17}}-\frac {14192}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}+\frac {175016}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {18011}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {12430}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {72425}{16 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {26871}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {54229}{768 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {509}{256 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{8} d}\)	\(372\)

Input:

int(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)
 

Output:

33/1024*I/a^8/d*exp(-5*I*(d*x+c))+33/1024*I/a^8/d*exp(-7*I*(d*x+c))+77/307 
2*I/a^8/d*exp(-9*I*(d*x+c))+15/1024*I/a^8/d*exp(-11*I*(d*x+c))+165/26624*I 
/a^8/d*exp(-13*I*(d*x+c))+11/6144*I/a^8/d*exp(-15*I*(d*x+c))+11/34816*I/a^ 
8/d*exp(-17*I*(d*x+c))+1/38912*I/a^8/d*exp(-19*I*(d*x+c))+11/512*I/a^8/d*c 
os(d*x+c)+33/1024*sin(d*x+c)/a^8/d+41/1536*I/a^8/d*cos(3*d*x+3*c)+83/3072/ 
a^8/d*sin(3*d*x+3*c)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.47 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (-4199 i \, e^{\left (22 i \, d x + 22 i \, c\right )} - 138567 i \, e^{\left (20 i \, d x + 20 i \, c\right )} + 692835 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 692835 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 831402 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 831402 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 646646 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 377910 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 159885 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 46189 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 8151 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 663 i\right )} e^{\left (-19 i \, d x - 19 i \, c\right )}}{25798656 \, a^{8} d} \] Input:

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
 

Output:

1/25798656*(-4199*I*e^(22*I*d*x + 22*I*c) - 138567*I*e^(20*I*d*x + 20*I*c) 
 + 692835*I*e^(18*I*d*x + 18*I*c) + 692835*I*e^(16*I*d*x + 16*I*c) + 83140 
2*I*e^(14*I*d*x + 14*I*c) + 831402*I*e^(12*I*d*x + 12*I*c) + 646646*I*e^(1 
0*I*d*x + 10*I*c) + 377910*I*e^(8*I*d*x + 8*I*c) + 159885*I*e^(6*I*d*x + 6 
*I*c) + 46189*I*e^(4*I*d*x + 4*I*c) + 8151*I*e^(2*I*d*x + 2*I*c) + 663*I)* 
e^(-19*I*d*x - 19*I*c)/(a^8*d)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} \frac {\left (- 6279106898588469469113471576881812733952 i a^{88} d^{11} e^{103 i c} e^{3 i d x} - 207210527653419492480744562037099820220416 i a^{88} d^{11} e^{101 i c} e^{i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{99 i c} e^{- i d x} + 1036052638267097462403722810185499101102080 i a^{88} d^{11} e^{97 i c} e^{- 3 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{95 i c} e^{- 5 i d x} + 1243263165920516954884467372222598921322496 i a^{88} d^{11} e^{93 i c} e^{- 7 i d x} + 966982462382624298243474622839799161028608 i a^{88} d^{11} e^{91 i c} e^{- 9 i d x} + 565119620872962252220212441919363146055680 i a^{88} d^{11} e^{89 i c} e^{- 11 i d x} + 239089070369330183631628340812038254100480 i a^{88} d^{11} e^{87 i c} e^{- 13 i d x} + 69070175884473164160248187345699940073472 i a^{88} d^{11} e^{85 i c} e^{- 15 i d x} + 12188854567848205440043797766888224718848 i a^{88} d^{11} e^{83 i c} e^{- 17 i d x} + 991437931356074126702127091086602010624 i a^{88} d^{11} e^{81 i c} e^{- 19 i d x}\right ) e^{- 100 i c}}{38578832784927556418233169368361857437401088 a^{96} d^{12}} & \text {for}\: a^{96} d^{12} e^{100 i c} \neq 0 \\\frac {x \left (e^{22 i c} + 11 e^{20 i c} + 55 e^{18 i c} + 165 e^{16 i c} + 330 e^{14 i c} + 462 e^{12 i c} + 462 e^{10 i c} + 330 e^{8 i c} + 165 e^{6 i c} + 55 e^{4 i c} + 11 e^{2 i c} + 1\right ) e^{- 19 i c}}{2048 a^{8}} & \text {otherwise} \end {cases} \] Input:

integrate(cos(d*x+c)**3/(a+I*a*tan(d*x+c))**8,x)
 

Output:

Piecewise(((-6279106898588469469113471576881812733952*I*a**88*d**11*exp(10 
3*I*c)*exp(3*I*d*x) - 207210527653419492480744562037099820220416*I*a**88*d 
**11*exp(101*I*c)*exp(I*d*x) + 1036052638267097462403722810185499101102080 
*I*a**88*d**11*exp(99*I*c)*exp(-I*d*x) + 103605263826709746240372281018549 
9101102080*I*a**88*d**11*exp(97*I*c)*exp(-3*I*d*x) + 124326316592051695488 
4467372222598921322496*I*a**88*d**11*exp(95*I*c)*exp(-5*I*d*x) + 124326316 
5920516954884467372222598921322496*I*a**88*d**11*exp(93*I*c)*exp(-7*I*d*x) 
 + 966982462382624298243474622839799161028608*I*a**88*d**11*exp(91*I*c)*ex 
p(-9*I*d*x) + 565119620872962252220212441919363146055680*I*a**88*d**11*exp 
(89*I*c)*exp(-11*I*d*x) + 239089070369330183631628340812038254100480*I*a** 
88*d**11*exp(87*I*c)*exp(-13*I*d*x) + 690701758844731641602481873456999400 
73472*I*a**88*d**11*exp(85*I*c)*exp(-15*I*d*x) + 1218885456784820544004379 
7766888224718848*I*a**88*d**11*exp(83*I*c)*exp(-17*I*d*x) + 99143793135607 
4126702127091086602010624*I*a**88*d**11*exp(81*I*c)*exp(-19*I*d*x))*exp(-1 
00*I*c)/(38578832784927556418233169368361857437401088*a**96*d**12), Ne(a** 
96*d**12*exp(100*I*c), 0)), (x*(exp(22*I*c) + 11*exp(20*I*c) + 55*exp(18*I 
*c) + 165*exp(16*I*c) + 330*exp(14*I*c) + 462*exp(12*I*c) + 462*exp(10*I*c 
) + 330*exp(8*I*c) + 165*exp(6*I*c) + 55*exp(4*I*c) + 11*exp(2*I*c) + 1)*e 
xp(-19*I*c)/(2048*a**8), True))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [A] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\frac {4199 \, {\left (18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 33 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 17\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}^{3}} + \frac {12823746 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{18} - 140368371 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} - 879644311 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 3693272440 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 11467502592 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 27403194676 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 51919375300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 79183835016 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 98304418212 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 99750226290 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 82860874122 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 56110430792 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30766700912 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 13462452660 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4616712644 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1197851960 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 226248618 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27911475 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2143959}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{19}}}{6449664 \, d} \] Input:

integrate(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
 

Output:

1/6449664*(4199*(18*tan(1/2*d*x + 1/2*c)^2 + 33*I*tan(1/2*d*x + 1/2*c) - 1 
7)/(a^8*(tan(1/2*d*x + 1/2*c) + I)^3) + (12823746*tan(1/2*d*x + 1/2*c)^18 
- 140368371*I*tan(1/2*d*x + 1/2*c)^17 - 879644311*tan(1/2*d*x + 1/2*c)^16 
+ 3693272440*I*tan(1/2*d*x + 1/2*c)^15 + 11467502592*tan(1/2*d*x + 1/2*c)^ 
14 - 27403194676*I*tan(1/2*d*x + 1/2*c)^13 - 51919375300*tan(1/2*d*x + 1/2 
*c)^12 + 79183835016*I*tan(1/2*d*x + 1/2*c)^11 + 98304418212*tan(1/2*d*x + 
 1/2*c)^10 - 99750226290*I*tan(1/2*d*x + 1/2*c)^9 - 82860874122*tan(1/2*d* 
x + 1/2*c)^8 + 56110430792*I*tan(1/2*d*x + 1/2*c)^7 + 30766700912*tan(1/2* 
d*x + 1/2*c)^6 - 13462452660*I*tan(1/2*d*x + 1/2*c)^5 - 4616712644*tan(1/2 
*d*x + 1/2*c)^4 + 1197851960*I*tan(1/2*d*x + 1/2*c)^3 + 226248618*tan(1/2* 
d*x + 1/2*c)^2 - 27911475*I*tan(1/2*d*x + 1/2*c) - 2143959)/(a^8*(tan(1/2* 
d*x + 1/2*c) - I)^19))/d
 

Mupad [B] (verification not implemented)

Time = 6.93 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {46189\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {46189\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {20995\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {20995\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {221255\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{128}+\frac {221255\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{128}-\frac {66861\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}+\frac {2093\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{32}-\frac {221\,\cos \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )}{128}+\frac {221\,\cos \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )}{128}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,309861{}\mathrm {i}}{256}-\frac {\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,665911{}\mathrm {i}}{512}+\frac {\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,665911{}\mathrm {i}}{512}-\frac {\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,194821{}\mathrm {i}}{128}+\frac {\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,194821{}\mathrm {i}}{128}-\frac {\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1825043{}\mathrm {i}}{1024}+\frac {\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1825043{}\mathrm {i}}{1024}-\frac {\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,1074183{}\mathrm {i}}{512}+\frac {\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )\,37895{}\mathrm {i}}{512}-\frac {\sin \left (\frac {19\,c}{2}+\frac {19\,d\,x}{2}\right )\,2431{}\mathrm {i}}{1024}+\frac {\sin \left (\frac {21\,c}{2}+\frac {21\,d\,x}{2}\right )\,2431{}\mathrm {i}}{1024}\right )}{12597\,a^8\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^{19}\,{\left (\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^3} \] Input:

int(cos(c + d*x)^3/(a + a*tan(c + d*x)*1i)^8,x)
 

Output:

-(2*cos(c/2 + (d*x)/2)*((46189*cos((5*c)/2 + (5*d*x)/2))/64 - (46189*cos(( 
3*c)/2 + (3*d*x)/2))/64 - (20995*cos((7*c)/2 + (7*d*x)/2))/16 + (20995*cos 
((9*c)/2 + (9*d*x)/2))/16 - (221255*cos((11*c)/2 + (11*d*x)/2))/128 + (221 
255*cos((13*c)/2 + (13*d*x)/2))/128 - (66861*cos((15*c)/2 + (15*d*x)/2))/3 
2 + (2093*cos((17*c)/2 + (17*d*x)/2))/32 - (221*cos((19*c)/2 + (19*d*x)/2) 
)/128 + (221*cos((21*c)/2 + (21*d*x)/2))/128 + (sin(c/2 + (d*x)/2)*309861i 
)/256 - (sin((3*c)/2 + (3*d*x)/2)*665911i)/512 + (sin((5*c)/2 + (5*d*x)/2) 
*665911i)/512 - (sin((7*c)/2 + (7*d*x)/2)*194821i)/128 + (sin((9*c)/2 + (9 
*d*x)/2)*194821i)/128 - (sin((11*c)/2 + (11*d*x)/2)*1825043i)/1024 + (sin( 
(13*c)/2 + (13*d*x)/2)*1825043i)/1024 - (sin((15*c)/2 + (15*d*x)/2)*107418 
3i)/512 + (sin((17*c)/2 + (17*d*x)/2)*37895i)/512 - (sin((19*c)/2 + (19*d* 
x)/2)*2431i)/1024 + (sin((21*c)/2 + (21*d*x)/2)*2431i)/1024))/(12597*a^8*d 
*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2)*1i)^19*(cos(c/2 + (d*x)/2)*1i + 
sin(c/2 + (d*x)/2))^3)
 

Reduce [F]

\[ \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\int \frac {\cos \left (d x +c \right )^{3}}{\tan \left (d x +c \right )^{8}-8 \tan \left (d x +c \right )^{7} i -28 \tan \left (d x +c \right )^{6}+56 \tan \left (d x +c \right )^{5} i +70 \tan \left (d x +c \right )^{4}-56 \tan \left (d x +c \right )^{3} i -28 \tan \left (d x +c \right )^{2}+8 \tan \left (d x +c \right ) i +1}d x}{a^{8}} \] Input:

int(cos(d*x+c)^3/(a+I*a*tan(d*x+c))^8,x)
 

Output:

int(cos(c + d*x)**3/(tan(c + d*x)**8 - 8*tan(c + d*x)**7*i - 28*tan(c + d* 
x)**6 + 56*tan(c + d*x)**5*i + 70*tan(c + d*x)**4 - 56*tan(c + d*x)**3*i - 
 28*tan(c + d*x)**2 + 8*tan(c + d*x)*i + 1),x)/a**8