Integrand size = 22, antiderivative size = 271 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {192 \sin (c+d x)}{12155 a^8 d}-\frac {64 \sin ^3(c+d x)}{12155 a^8 d}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac {3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac {24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac {168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac {112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac {16 i \cos (c+d x)}{2431 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )} \]
192/12155*sin(d*x+c)/a^8/d-64/12155*sin(d*x+c)^3/a^8/d+1/17*I*cos(d*x+c)/d /(a+I*a*tan(d*x+c))^8+3/85*I*cos(d*x+c)/a/d/(a+I*a*tan(d*x+c))^7+24/1105*I *cos(d*x+c)/a^2/d/(a+I*a*tan(d*x+c))^6+168/12155*I*cos(d*x+c)/a^3/d/(a+I*a *tan(d*x+c))^5+112/12155*I*cos(d*x+c)/d/(a^2+I*a^2*tan(d*x+c))^4+16/2431*I *cos(d*x+c)/a^2/d/(a^2+I*a^2*tan(d*x+c))^3+128/12155*I*cos(d*x+c)^3/d/(a^8 +I*a^8*tan(d*x+c))
Time = 1.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.51 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {i \sec ^8(c+d x) (-194480 \cos (c+d x)-148512 \cos (3 (c+d x))-89760 \cos (5 (c+d x))-58344 \cos (7 (c+d x))+5720 \cos (9 (c+d x))-24310 i \sin (c+d x)-55692 i \sin (3 (c+d x))-56100 i \sin (5 (c+d x))-51051 i \sin (7 (c+d x))+6435 i \sin (9 (c+d x)))}{3111680 a^8 d (-i+\tan (c+d x))^8} \]
((-1/3111680*I)*Sec[c + d*x]^8*(-194480*Cos[c + d*x] - 148512*Cos[3*(c + d *x)] - 89760*Cos[5*(c + d*x)] - 58344*Cos[7*(c + d*x)] + 5720*Cos[9*(c + d *x)] - (24310*I)*Sin[c + d*x] - (55692*I)*Sin[3*(c + d*x)] - (56100*I)*Sin [5*(c + d*x)] - (51051*I)*Sin[7*(c + d*x)] + (6435*I)*Sin[9*(c + d*x)]))/( a^8*d*(-I + Tan[c + d*x])^8)
Time = 1.34 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, 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.
\(\displaystyle \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sec (c+d x) (a+i a \tan (c+d x))^8}dx\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^7}dx}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^7}dx}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {8 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^6}dx}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {8 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^6}dx}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^5}dx}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^5}dx}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^4}dx}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^4}dx}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^3}dx}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^3}dx}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3983 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \int \frac {\cos (c+d x)}{(i \tan (c+d x) a+a)^2}dx}{7 a}+\frac {i \cos (c+d x)}{7 d (a+i a \tan (c+d x))^3}\right )}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \int \frac {1}{\sec (c+d x) (i \tan (c+d x) a+a)^2}dx}{7 a}+\frac {i \cos (c+d x)}{7 d (a+i a \tan (c+d x))^3}\right )}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3981 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \int \cos ^3(c+d x)dx}{5 a^2}+\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{7 a}+\frac {i \cos (c+d x)}{7 d (a+i a \tan (c+d x))^3}\right )}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (\frac {3 \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{5 a^2}+\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{7 a}+\frac {i \cos (c+d x)}{7 d (a+i a \tan (c+d x))^3}\right )}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (-\frac {3 \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{5 a^2 d}+\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{7 a}+\frac {i \cos (c+d x)}{7 d (a+i a \tan (c+d x))^3}\right )}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {9 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {5 \left (\frac {4 \left (-\frac {3 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{5 a^2 d}+\frac {2 i \cos ^3(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}\right )}{7 a}+\frac {i \cos (c+d x)}{7 d (a+i a \tan (c+d x))^3}\right )}{9 a}+\frac {i \cos (c+d x)}{9 d (a+i a \tan (c+d x))^4}\right )}{11 a}+\frac {i \cos (c+d x)}{11 d (a+i a \tan (c+d x))^5}\right )}{13 a}+\frac {i \cos (c+d x)}{13 d (a+i a \tan (c+d x))^6}\right )}{15 a}+\frac {i \cos (c+d x)}{15 d (a+i a \tan (c+d x))^7}\right )}{17 a}+\frac {i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}\) |
((I/17)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (9*(((I/15)*Cos[c + d *x])/(d*(a + I*a*Tan[c + d*x])^7) + (8*(((I/13)*Cos[c + d*x])/(d*(a + I*a* Tan[c + d*x])^6) + (7*(((I/11)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^5) + (6*(((I/9)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^4) + (5*(((I/7)*Cos[c + d*x])/(d*(a + I*a*Tan[c + d*x])^3) + (4*((-3*(-Sin[c + d*x] + Sin[c + d *x]^3/3))/(5*a^2*d) + (((2*I)/5)*Cos[c + d*x]^3)/(d*(a^2 + I*a^2*Tan[c + d *x]))))/(7*a)))/(9*a)))/(11*a)))/(13*a)))/(15*a)))/(17*a)
3.2.83.3.1 Defintions of rubi rules used
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]
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]
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 = 0.73 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {3 i {\mathrm e}^{-3 i \left (d x +c \right )}}{128 a^{8} d}+\frac {21 i {\mathrm e}^{-5 i \left (d x +c \right )}}{640 a^{8} d}+\frac {9 i {\mathrm e}^{-7 i \left (d x +c \right )}}{256 a^{8} d}+\frac {7 i {\mathrm e}^{-9 i \left (d x +c \right )}}{256 a^{8} d}+\frac {21 i {\mathrm e}^{-11 i \left (d x +c \right )}}{1408 a^{8} d}+\frac {9 i {\mathrm e}^{-13 i \left (d x +c \right )}}{1664 a^{8} d}+\frac {3 i {\mathrm e}^{-15 i \left (d x +c \right )}}{2560 a^{8} d}+\frac {i {\mathrm e}^{-17 i \left (d x +c \right )}}{8704 a^{8} d}+\frac {i \cos \left (d x +c \right )}{64 a^{8} d}+\frac {5 \sin \left (d x +c \right )}{256 a^{8} d}\) | \(175\) |
derivativedivides | \(\frac {\frac {2}{512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+512 i}+\frac {38218 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {7937 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1793 i}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{16}}+\frac {1568 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}-\frac {10241 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {13313 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {5384 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {256}{17 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{17}}-\frac {2752}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}+\frac {42800}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {77908}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {6847}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {12799}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {57083}{80 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {4351}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {511}{256 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{8} d}\) | \(306\) |
default | \(\frac {\frac {2}{512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+512 i}+\frac {38218 i}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {7937 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1793 i}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{16}}+\frac {1568 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{14}}-\frac {10241 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {13313 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {5384 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {256}{17 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{17}}-\frac {2752}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{15}}+\frac {42800}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {77908}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {6847}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {12799}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {57083}{80 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {4351}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {511}{256 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{8} d}\) | \(306\) |
3/128*I/a^8/d*exp(-3*I*(d*x+c))+21/640*I/a^8/d*exp(-5*I*(d*x+c))+9/256*I/a ^8/d*exp(-7*I*(d*x+c))+7/256*I/a^8/d*exp(-9*I*(d*x+c))+21/1408*I/a^8/d*exp (-11*I*(d*x+c))+9/1664*I/a^8/d*exp(-13*I*(d*x+c))+3/2560*I/a^8/d*exp(-15*I *(d*x+c))+1/8704*I/a^8/d*exp(-17*I*(d*x+c))+1/64*I/a^8/d*cos(d*x+c)+5/256* sin(d*x+c)/a^8/d
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.44 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (-12155 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 109395 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 145860 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 204204 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 218790 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 170170 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 92820 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 33660 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7293 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 715 i\right )} e^{\left (-17 i \, d x - 17 i \, c\right )}}{6223360 \, a^{8} d} \]
1/6223360*(-12155*I*e^(18*I*d*x + 18*I*c) + 109395*I*e^(16*I*d*x + 16*I*c) + 145860*I*e^(14*I*d*x + 14*I*c) + 204204*I*e^(12*I*d*x + 12*I*c) + 21879 0*I*e^(10*I*d*x + 10*I*c) + 170170*I*e^(8*I*d*x + 8*I*c) + 92820*I*e^(6*I* d*x + 6*I*c) + 33660*I*e^(4*I*d*x + 4*I*c) + 7293*I*e^(2*I*d*x + 2*I*c) + 715*I)*e^(-17*I*d*x - 17*I*c)/(a^8*d)
Time = 0.58 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\begin {cases} \frac {\left (- 143500911498201343931187200 i a^{72} d^{9} e^{82 i c} e^{i d x} + 1291508203483812095380684800 i a^{72} d^{9} e^{80 i c} e^{- i d x} + 1722010937978416127174246400 i a^{72} d^{9} e^{78 i c} e^{- 3 i d x} + 2410815313169782578043944960 i a^{72} d^{9} e^{76 i c} e^{- 5 i d x} + 2583016406967624190761369600 i a^{72} d^{9} e^{74 i c} e^{- 7 i d x} + 2009012760974818815036620800 i a^{72} d^{9} e^{72 i c} e^{- 9 i d x} + 1095825142349901171838156800 i a^{72} d^{9} e^{70 i c} e^{- 11 i d x} + 397387139533480644732518400 i a^{72} d^{9} e^{68 i c} e^{- 13 i d x} + 86100546898920806358712320 i a^{72} d^{9} e^{66 i c} e^{- 15 i d x} + 8441230088129490819481600 i a^{72} d^{9} e^{64 i c} e^{- 17 i d x}\right ) e^{- 81 i c}}{73472466687079088092767846400 a^{80} d^{10}} & \text {for}\: a^{80} d^{10} e^{81 i c} \neq 0 \\\frac {x \left (e^{18 i c} + 9 e^{16 i c} + 36 e^{14 i c} + 84 e^{12 i c} + 126 e^{10 i c} + 126 e^{8 i c} + 84 e^{6 i c} + 36 e^{4 i c} + 9 e^{2 i c} + 1\right ) e^{- 17 i c}}{512 a^{8}} & \text {otherwise} \end {cases} \]
Piecewise(((-143500911498201343931187200*I*a**72*d**9*exp(82*I*c)*exp(I*d* x) + 1291508203483812095380684800*I*a**72*d**9*exp(80*I*c)*exp(-I*d*x) + 1 722010937978416127174246400*I*a**72*d**9*exp(78*I*c)*exp(-3*I*d*x) + 24108 15313169782578043944960*I*a**72*d**9*exp(76*I*c)*exp(-5*I*d*x) + 258301640 6967624190761369600*I*a**72*d**9*exp(74*I*c)*exp(-7*I*d*x) + 2009012760974 818815036620800*I*a**72*d**9*exp(72*I*c)*exp(-9*I*d*x) + 10958251423499011 71838156800*I*a**72*d**9*exp(70*I*c)*exp(-11*I*d*x) + 39738713953348064473 2518400*I*a**72*d**9*exp(68*I*c)*exp(-13*I*d*x) + 861005468989208063587123 20*I*a**72*d**9*exp(66*I*c)*exp(-15*I*d*x) + 8441230088129490819481600*I*a **72*d**9*exp(64*I*c)*exp(-17*I*d*x))*exp(-81*I*c)/(7347246668707908809276 7846400*a**80*d**10), Ne(a**80*d**10*exp(81*I*c), 0)), (x*(exp(18*I*c) + 9 *exp(16*I*c) + 36*exp(14*I*c) + 84*exp(12*I*c) + 126*exp(10*I*c) + 126*exp (8*I*c) + 84*exp(6*I*c) + 36*exp(4*I*c) + 9*exp(2*I*c) + 1)*exp(-17*I*c)/( 512*a**8), True))
Exception generated. \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Exception raised: RuntimeError} \]
Time = 1.26 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\frac {12155}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}} + \frac {6211205 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 55791450 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 303072770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 1091397450 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2909561798 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 5901218466 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 9405145178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 11877161010 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 12017308160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 9710430158 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6263238566 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3172666718 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1247921210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 365303990 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 77883902 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10498214 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 982907}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{17}}}{3111680 \, d} \]
1/3111680*(12155/(a^8*(tan(1/2*d*x + 1/2*c) + I)) + (6211205*tan(1/2*d*x + 1/2*c)^16 - 55791450*I*tan(1/2*d*x + 1/2*c)^15 - 303072770*tan(1/2*d*x + 1/2*c)^14 + 1091397450*I*tan(1/2*d*x + 1/2*c)^13 + 2909561798*tan(1/2*d*x + 1/2*c)^12 - 5901218466*I*tan(1/2*d*x + 1/2*c)^11 - 9405145178*tan(1/2*d* x + 1/2*c)^10 + 11877161010*I*tan(1/2*d*x + 1/2*c)^9 + 12017308160*tan(1/2 *d*x + 1/2*c)^8 - 9710430158*I*tan(1/2*d*x + 1/2*c)^7 - 6263238566*tan(1/2 *d*x + 1/2*c)^6 + 3172666718*I*tan(1/2*d*x + 1/2*c)^5 + 1247921210*tan(1/2 *d*x + 1/2*c)^4 - 365303990*I*tan(1/2*d*x + 1/2*c)^3 - 77883902*tan(1/2*d* x + 1/2*c)^2 + 10498214*I*tan(1/2*d*x + 1/2*c) + 982907)/(a^8*(tan(1/2*d*x + 1/2*c) - I)^17))/d
Time = 7.47 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.97 \[ \int \frac {\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {152329\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}-\frac {41121\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{32}+\frac {41121\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}-\frac {96165\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{64}+\frac {96165\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{64}-\frac {55095\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {55095\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{32}-\frac {491811\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{256}+\frac {6435\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{256}+\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,12155{}\mathrm {i}}{16}-\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,12155{}\mathrm {i}}{16}+\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,21437{}\mathrm {i}}{16}-\frac {\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,21437{}\mathrm {i}}{16}+\frac {\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,27047{}\mathrm {i}}{16}-\frac {\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,27047{}\mathrm {i}}{16}+\frac {\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,61387{}\mathrm {i}}{32}-\frac {\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )\,715{}\mathrm {i}}{32}\right )\,2{}\mathrm {i}}{12155\,a^8\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^{17}\,\left (\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )} \]
(cos(c/2 + (d*x)/2)*((cos((3*c)/2 + (3*d*x)/2)*12155i)/16 - (cos((5*c)/2 + (5*d*x)/2)*12155i)/16 + (cos((7*c)/2 + (7*d*x)/2)*21437i)/16 - (cos((9*c) /2 + (9*d*x)/2)*21437i)/16 + (cos((11*c)/2 + (11*d*x)/2)*27047i)/16 - (cos ((13*c)/2 + (13*d*x)/2)*27047i)/16 + (cos((15*c)/2 + (15*d*x)/2)*61387i)/3 2 - (cos((17*c)/2 + (17*d*x)/2)*715i)/32 + (152329*sin(c/2 + (d*x)/2))/128 - (41121*sin((3*c)/2 + (3*d*x)/2))/32 + (41121*sin((5*c)/2 + (5*d*x)/2))/ 32 - (96165*sin((7*c)/2 + (7*d*x)/2))/64 + (96165*sin((9*c)/2 + (9*d*x)/2) )/64 - (55095*sin((11*c)/2 + (11*d*x)/2))/32 + (55095*sin((13*c)/2 + (13*d *x)/2))/32 - (491811*sin((15*c)/2 + (15*d*x)/2))/256 + (6435*sin((17*c)/2 + (17*d*x)/2))/256)*2i)/(12155*a^8*d*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x) /2)*1i)^17*(cos(c/2 + (d*x)/2)*1i + sin(c/2 + (d*x)/2)))