Integrand size = 24, antiderivative size = 212 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {7 a^8 \sin (c+d x)}{1287 d}-\frac {7 a^8 \sin ^3(c+d x)}{1287 d}+\frac {7 a^8 \sin ^5(c+d x)}{2145 d}-\frac {a^8 \sin ^7(c+d x)}{1287 d}-\frac {2 i a^3 \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{195 d}-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}-\frac {2 i a^2 \cos ^{11}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{715 d}-\frac {2 i \cos ^9(c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{1287 d} \] Output:
7/1287*a^8*sin(d*x+c)/d-7/1287*a^8*sin(d*x+c)^3/d+7/2145*a^8*sin(d*x+c)^5/ d-1/1287*a^8*sin(d*x+c)^7/d-2/195*I*a^3*cos(d*x+c)^13*(a+I*a*tan(d*x+c))^5 /d-2/15*I*a*cos(d*x+c)^15*(a+I*a*tan(d*x+c))^7/d-2/715*I*a^2*cos(d*x+c)^11 *(a^2+I*a^2*tan(d*x+c))^3/d-2/1287*I*cos(d*x+c)^9*(a^8+I*a^8*tan(d*x+c))/d
Time = 1.96 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.82 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \sec (c+d x) (-i \cos (8 (c+d x))+\sin (8 (c+d x))) \left (28600+48256 \cos (2 (c+d x))+28896 \cos (4 (c+d x))+12672 \cos (6 (c+d x))+3432 \cos (8 (c+d x))+317440 \sqrt {\cos ^2(c+d x)} \cos (8 (c+d x))-10946 i \sin (2 (c+d x))-13146 i \sin (4 (c+d x))-8778 i \sin (6 (c+d x))-3003 i \sin (8 (c+d x))-317440 i \sqrt {\cos ^2(c+d x)} \sin (8 (c+d x))\right )}{823680 d} \] Input:
Integrate[Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^8,x]
Output:
(a^8*Sec[c + d*x]*((-I)*Cos[8*(c + d*x)] + Sin[8*(c + d*x)])*(28600 + 4825 6*Cos[2*(c + d*x)] + 28896*Cos[4*(c + d*x)] + 12672*Cos[6*(c + d*x)] + 343 2*Cos[8*(c + d*x)] + 317440*Sqrt[Cos[c + d*x]^2]*Cos[8*(c + d*x)] - (10946 *I)*Sin[2*(c + d*x)] - (13146*I)*Sin[4*(c + d*x)] - (8778*I)*Sin[6*(c + d* x)] - (3003*I)*Sin[8*(c + d*x)] - (317440*I)*Sqrt[Cos[c + d*x]^2]*Sin[8*(c + d*x)]))/(823680*d)
Time = 0.79 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {3042, 3977, 3042, 3977, 3042, 3977, 3042, 3977, 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.
| \(\displaystyle \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^8}{\sec (c+d x)^{15}}dx\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle \frac {1}{15} a^2 \int \cos ^{13}(c+d x) (i \tan (c+d x) a+a)^6dx-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{15} a^2 \int \frac {(i \tan (c+d x) a+a)^6}{\sec (c+d x)^{13}}dx-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \int \cos ^{11}(c+d x) (i \tan (c+d x) a+a)^4dx-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \int \frac {(i \tan (c+d x) a+a)^4}{\sec (c+d x)^{11}}dx-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \left (\frac {5}{11} a^2 \int \cos ^9(c+d x) (i \tan (c+d x) a+a)^2dx-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{11 d}\right )-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \left (\frac {5}{11} a^2 \int \frac {(i \tan (c+d x) a+a)^2}{\sec (c+d x)^9}dx-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{11 d}\right )-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \left (\frac {5}{11} a^2 \left (\frac {7}{9} a^2 \int \cos ^7(c+d x)dx-\frac {2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}\right )-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{11 d}\right )-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \left (\frac {5}{11} a^2 \left (\frac {7}{9} a^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^7dx-\frac {2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}\right )-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{11 d}\right )-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \left (\frac {5}{11} a^2 \left (-\frac {7 a^2 \int \left (-\sin ^6(c+d x)+3 \sin ^4(c+d x)-3 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{9 d}-\frac {2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}\right )-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{11 d}\right )-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{15} a^2 \left (\frac {3}{13} a^2 \left (\frac {5}{11} a^2 \left (-\frac {7 a^2 \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{9 d}-\frac {2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}\right )-\frac {2 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^3}{11 d}\right )-\frac {2 i a \cos ^{13}(c+d x) (a+i a \tan (c+d x))^5}{13 d}\right )-\frac {2 i a \cos ^{15}(c+d x) (a+i a \tan (c+d x))^7}{15 d}\) |
Input:
Int[Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^8,x]
Output:
(((-2*I)/15)*a*Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^7)/d + (a^2*((((-2*I )/13)*a*Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^5)/d + (3*a^2*((((-2*I)/11) *a*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^3)/d + (5*a^2*((-7*a^2*(-Sin[c + d*x] + Sin[c + d*x]^3 - (3*Sin[c + d*x]^5)/5 + Sin[c + d*x]^7/7))/(9*d) - (((2*I)/9)*Cos[c + d*x]^9*(a^2 + I*a^2*Tan[c + d*x]))/d))/11))/13))/15
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*b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^( n - 1)/(f*m)), x] - Simp[b^2*((m + 2*n - 2)/(d^2*m)) Int[(d*Sec[e + f*x]) ^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILtQ[m, 0] & & LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (188 ) = 376\).
Time = 1.68 (sec) , antiderivative size = 667, normalized size of antiderivative = 3.15
\[\frac {a^{8} \left (-\frac {\sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{8}}{15}-\frac {7 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{8}}{195}-\frac {7 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{8}}{429}-\frac {7 \cos \left (d x +c \right )^{8} \sin \left (d x +c \right )}{1287}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{1287}\right )-56 i a^{8} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{13}}{15}-\frac {2 \cos \left (d x +c \right )^{13}}{195}\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{10}}{15}-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{10}}{39}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{10}}{143}+\frac {\left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{1287}\right )+56 i a^{8} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{11}}{15}-\frac {4 \cos \left (d x +c \right )^{11} \sin \left (d x +c \right )^{2}}{195}-\frac {8 \cos \left (d x +c \right )^{11}}{2145}\right )+70 a^{8} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{12}}{15}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{12}}{65}+\frac {\left (\frac {256}{63}+\cos \left (d x +c \right )^{10}+\frac {10 \cos \left (d x +c \right )^{8}}{9}+\frac {80 \cos \left (d x +c \right )^{6}}{63}+\frac {32 \cos \left (d x +c \right )^{4}}{21}+\frac {128 \cos \left (d x +c \right )^{2}}{63}\right ) \sin \left (d x +c \right )}{715}\right )-8 i a^{8} \left (-\frac {\sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{9}}{15}-\frac {2 \cos \left (d x +c \right )^{9} \sin \left (d x +c \right )^{4}}{65}-\frac {8 \cos \left (d x +c \right )^{9} \sin \left (d x +c \right )^{2}}{715}-\frac {16 \cos \left (d x +c \right )^{9}}{6435}\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{14}}{15}+\frac {\left (\frac {1024}{231}+\cos \left (d x +c \right )^{12}+\frac {12 \cos \left (d x +c \right )^{10}}{11}+\frac {40 \cos \left (d x +c \right )^{8}}{33}+\frac {320 \cos \left (d x +c \right )^{6}}{231}+\frac {128 \cos \left (d x +c \right )^{4}}{77}+\frac {512 \cos \left (d x +c \right )^{2}}{231}\right ) \sin \left (d x +c \right )}{195}\right )-\frac {8 i a^{8} \cos \left (d x +c \right )^{15}}{15}+\frac {a^{8} \left (\frac {2048}{429}+\cos \left (d x +c \right )^{14}+\frac {14 \cos \left (d x +c \right )^{12}}{13}+\frac {168 \cos \left (d x +c \right )^{10}}{143}+\frac {560 \cos \left (d x +c \right )^{8}}{429}+\frac {640 \cos \left (d x +c \right )^{6}}{429}+\frac {256 \cos \left (d x +c \right )^{4}}{143}+\frac {1024 \cos \left (d x +c \right )^{2}}{429}\right ) \sin \left (d x +c \right )}{15}}{d}\]
Input:
int(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x)
Output:
1/d*(a^8*(-1/15*sin(d*x+c)^7*cos(d*x+c)^8-7/195*sin(d*x+c)^5*cos(d*x+c)^8- 7/429*sin(d*x+c)^3*cos(d*x+c)^8-7/1287*cos(d*x+c)^8*sin(d*x+c)+1/1287*(16/ 5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))-56*I*a^8*(-1 /15*sin(d*x+c)^2*cos(d*x+c)^13-2/195*cos(d*x+c)^13)-28*a^8*(-1/15*sin(d*x+ c)^5*cos(d*x+c)^10-1/39*sin(d*x+c)^3*cos(d*x+c)^10-1/143*sin(d*x+c)*cos(d* x+c)^10+1/1287*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64 /35*cos(d*x+c)^2)*sin(d*x+c))+56*I*a^8*(-1/15*sin(d*x+c)^4*cos(d*x+c)^11-4 /195*cos(d*x+c)^11*sin(d*x+c)^2-8/2145*cos(d*x+c)^11)+70*a^8*(-1/15*sin(d* x+c)^3*cos(d*x+c)^12-1/65*sin(d*x+c)*cos(d*x+c)^12+1/715*(256/63+cos(d*x+c )^10+10/9*cos(d*x+c)^8+80/63*cos(d*x+c)^6+32/21*cos(d*x+c)^4+128/63*cos(d* x+c)^2)*sin(d*x+c))-8*I*a^8*(-1/15*sin(d*x+c)^6*cos(d*x+c)^9-2/65*cos(d*x+ c)^9*sin(d*x+c)^4-8/715*cos(d*x+c)^9*sin(d*x+c)^2-16/6435*cos(d*x+c)^9)-28 *a^8*(-1/15*sin(d*x+c)*cos(d*x+c)^14+1/195*(1024/231+cos(d*x+c)^12+12/11*c os(d*x+c)^10+40/33*cos(d*x+c)^8+320/231*cos(d*x+c)^6+128/77*cos(d*x+c)^4+5 12/231*cos(d*x+c)^2)*sin(d*x+c))-8/15*I*a^8*cos(d*x+c)^15+1/15*a^8*(2048/4 29+cos(d*x+c)^14+14/13*cos(d*x+c)^12+168/143*cos(d*x+c)^10+560/429*cos(d*x +c)^8+640/429*cos(d*x+c)^6+256/143*cos(d*x+c)^4+1024/429*cos(d*x+c)^2)*sin (d*x+c))
Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.56 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-429 i \, a^{8} e^{\left (15 i \, d x + 15 i \, c\right )} - 3465 i \, a^{8} e^{\left (13 i \, d x + 13 i \, c\right )} - 12285 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 25025 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 32175 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 27027 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 15015 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} - 6435 i \, a^{8} e^{\left (i \, d x + i \, c\right )}}{823680 \, d} \] Input:
integrate(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
Output:
1/823680*(-429*I*a^8*e^(15*I*d*x + 15*I*c) - 3465*I*a^8*e^(13*I*d*x + 13*I *c) - 12285*I*a^8*e^(11*I*d*x + 11*I*c) - 25025*I*a^8*e^(9*I*d*x + 9*I*c) - 32175*I*a^8*e^(7*I*d*x + 7*I*c) - 27027*I*a^8*e^(5*I*d*x + 5*I*c) - 1501 5*I*a^8*e^(3*I*d*x + 3*I*c) - 6435*I*a^8*e^(I*d*x + I*c))/d
Time = 0.73 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.48 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\begin {cases} \frac {- 10867748850798428160 i a^{8} d^{7} e^{15 i c} e^{15 i d x} - 87777971487218073600 i a^{8} d^{7} e^{13 i c} e^{13 i d x} - 311212808000136806400 i a^{8} d^{7} e^{11 i c} e^{11 i d x} - 633952016296574976000 i a^{8} d^{7} e^{9 i c} e^{9 i d x} - 815081163809882112000 i a^{8} d^{7} e^{7 i c} e^{7 i d x} - 684668177600300974080 i a^{8} d^{7} e^{5 i c} e^{5 i d x} - 380371209777944985600 i a^{8} d^{7} e^{3 i c} e^{3 i d x} - 163016232761976422400 i a^{8} d^{7} e^{i c} e^{i d x}}{20866077793532982067200 d^{8}} & \text {for}\: d^{8} \neq 0 \\x \left (\frac {a^{8} e^{15 i c}}{128} + \frac {7 a^{8} e^{13 i c}}{128} + \frac {21 a^{8} e^{11 i c}}{128} + \frac {35 a^{8} e^{9 i c}}{128} + \frac {35 a^{8} e^{7 i c}}{128} + \frac {21 a^{8} e^{5 i c}}{128} + \frac {7 a^{8} e^{3 i c}}{128} + \frac {a^{8} e^{i c}}{128}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**15*(a+I*a*tan(d*x+c))**8,x)
Output:
Piecewise(((-10867748850798428160*I*a**8*d**7*exp(15*I*c)*exp(15*I*d*x) - 87777971487218073600*I*a**8*d**7*exp(13*I*c)*exp(13*I*d*x) - 3112128080001 36806400*I*a**8*d**7*exp(11*I*c)*exp(11*I*d*x) - 633952016296574976000*I*a **8*d**7*exp(9*I*c)*exp(9*I*d*x) - 815081163809882112000*I*a**8*d**7*exp(7 *I*c)*exp(7*I*d*x) - 684668177600300974080*I*a**8*d**7*exp(5*I*c)*exp(5*I* d*x) - 380371209777944985600*I*a**8*d**7*exp(3*I*c)*exp(3*I*d*x) - 1630162 32761976422400*I*a**8*d**7*exp(I*c)*exp(I*d*x))/(20866077793532982067200*d **8), Ne(d**8, 0)), (x*(a**8*exp(15*I*c)/128 + 7*a**8*exp(13*I*c)/128 + 21 *a**8*exp(11*I*c)/128 + 35*a**8*exp(9*I*c)/128 + 35*a**8*exp(7*I*c)/128 + 21*a**8*exp(5*I*c)/128 + 7*a**8*exp(3*I*c)/128 + a**8*exp(I*c)/128), True) )
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (180) = 360\).
Time = 0.06 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.14 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {3432 i \, a^{8} \cos \left (d x + c\right )^{15} + 8 i \, {\left (429 \, \cos \left (d x + c\right )^{15} - 1485 \, \cos \left (d x + c\right )^{13} + 1755 \, \cos \left (d x + c\right )^{11} - 715 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 168 i \, {\left (143 \, \cos \left (d x + c\right )^{15} - 330 \, \cos \left (d x + c\right )^{13} + 195 \, \cos \left (d x + c\right )^{11}\right )} a^{8} + 1848 i \, {\left (13 \, \cos \left (d x + c\right )^{15} - 15 \, \cos \left (d x + c\right )^{13}\right )} a^{8} + 4 \, {\left (3003 \, \sin \left (d x + c\right )^{15} - 13860 \, \sin \left (d x + c\right )^{13} + 24570 \, \sin \left (d x + c\right )^{11} - 20020 \, \sin \left (d x + c\right )^{9} + 6435 \, \sin \left (d x + c\right )^{7}\right )} a^{8} + 10 \, {\left (3003 \, \sin \left (d x + c\right )^{15} - 17325 \, \sin \left (d x + c\right )^{13} + 40950 \, \sin \left (d x + c\right )^{11} - 50050 \, \sin \left (d x + c\right )^{9} + 32175 \, \sin \left (d x + c\right )^{7} - 9009 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 4 \, {\left (3003 \, \sin \left (d x + c\right )^{15} - 20790 \, \sin \left (d x + c\right )^{13} + 61425 \, \sin \left (d x + c\right )^{11} - 100100 \, \sin \left (d x + c\right )^{9} + 96525 \, \sin \left (d x + c\right )^{7} - 54054 \, \sin \left (d x + c\right )^{5} + 15015 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + {\left (429 \, \sin \left (d x + c\right )^{15} - 1485 \, \sin \left (d x + c\right )^{13} + 1755 \, \sin \left (d x + c\right )^{11} - 715 \, \sin \left (d x + c\right )^{9}\right )} a^{8} + {\left (429 \, \sin \left (d x + c\right )^{15} - 3465 \, \sin \left (d x + c\right )^{13} + 12285 \, \sin \left (d x + c\right )^{11} - 25025 \, \sin \left (d x + c\right )^{9} + 32175 \, \sin \left (d x + c\right )^{7} - 27027 \, \sin \left (d x + c\right )^{5} + 15015 \, \sin \left (d x + c\right )^{3} - 6435 \, \sin \left (d x + c\right )\right )} a^{8}}{6435 \, d} \] Input:
integrate(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
Output:
-1/6435*(3432*I*a^8*cos(d*x + c)^15 + 8*I*(429*cos(d*x + c)^15 - 1485*cos( d*x + c)^13 + 1755*cos(d*x + c)^11 - 715*cos(d*x + c)^9)*a^8 + 168*I*(143* cos(d*x + c)^15 - 330*cos(d*x + c)^13 + 195*cos(d*x + c)^11)*a^8 + 1848*I* (13*cos(d*x + c)^15 - 15*cos(d*x + c)^13)*a^8 + 4*(3003*sin(d*x + c)^15 - 13860*sin(d*x + c)^13 + 24570*sin(d*x + c)^11 - 20020*sin(d*x + c)^9 + 643 5*sin(d*x + c)^7)*a^8 + 10*(3003*sin(d*x + c)^15 - 17325*sin(d*x + c)^13 + 40950*sin(d*x + c)^11 - 50050*sin(d*x + c)^9 + 32175*sin(d*x + c)^7 - 900 9*sin(d*x + c)^5)*a^8 + 4*(3003*sin(d*x + c)^15 - 20790*sin(d*x + c)^13 + 61425*sin(d*x + c)^11 - 100100*sin(d*x + c)^9 + 96525*sin(d*x + c)^7 - 540 54*sin(d*x + c)^5 + 15015*sin(d*x + c)^3)*a^8 + (429*sin(d*x + c)^15 - 148 5*sin(d*x + c)^13 + 1755*sin(d*x + c)^11 - 715*sin(d*x + c)^9)*a^8 + (429* sin(d*x + c)^15 - 3465*sin(d*x + c)^13 + 12285*sin(d*x + c)^11 - 25025*sin (d*x + c)^9 + 32175*sin(d*x + c)^7 - 27027*sin(d*x + c)^5 + 15015*sin(d*x + c)^3 - 6435*sin(d*x + c))*a^8)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2919 vs. \(2 (180) = 360\).
Time = 1.41 (sec) , antiderivative size = 2919, normalized size of antiderivative = 13.77 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
Output:
1/863691079680*(5682101344920*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I *c) + 1) + 79549418828880*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 517071222387720*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1 ) + 2068284889550880*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5687783446264920*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 11 375566892529840*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1706 3350338794760*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 170633 50338794760*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 11375566 892529840*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5687783446 264920*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 20682848895508 80*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 517071222387720*a^ 8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 79549418828880*a^8*e^( 2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 19500971815765440*a^8*e^(14 *I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 5682101344920*a^8*e^(-14*I*c)*log(I*e ^(I*d*x + I*c) + 1) + 5674116082635*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d *x + I*c) - 1) + 79437625156890*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 516344563519785*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I* c) - 1) + 2065378254079140*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5679790198717635*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1 ) + 11359580397435270*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - ...
Time = 2.39 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.05 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {2\,a^8\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )\,\left (-\frac {44779\,{\sin \left (c+d\,x\right )}^2}{32}+\frac {\sin \left (c+d\,x\right )\,32175{}\mathrm {i}}{128}-\frac {26075\,{\sin \left (2\,c+2\,d\,x\right )}^2}{16}-\frac {\sin \left (2\,c+2\,d\,x\right )\,3575{}\mathrm {i}}{8}+\frac {114583\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}-\frac {57925\,{\sin \left (3\,c+3\,d\,x\right )}^2}{32}+\frac {\sin \left (3\,c+3\,d\,x\right )\,84227{}\mathrm {i}}{128}+\frac {116585\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2}{64}+\frac {119315\,{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2}{64}+\frac {122285\,{\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}^2}{64}-\sin \left (4\,c+4\,d\,x\right )\,754{}\mathrm {i}+\frac {\sin \left (5\,c+5\,d\,x\right )\,111527{}\mathrm {i}}{128}-\frac {\sin \left (6\,c+6\,d\,x\right )\,7187{}\mathrm {i}}{8}+\frac {\sin \left (7\,c+7\,d\,x\right )\,121427{}\mathrm {i}}{128}-952\right )}{6435\,d\,\left (-{\sin \left (\frac {15\,c}{4}+\frac {15\,d\,x}{4}\right )}^2\,2{}\mathrm {i}+\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )+1{}\mathrm {i}\right )} \] Input:
int(cos(c + d*x)^15*(a + a*tan(c + d*x)*1i)^8,x)
Output:
(2*a^8*(2*sin(c/4 + (d*x)/4)^2 - 1)*((sin(c + d*x)*32175i)/128 - (sin(2*c + 2*d*x)*3575i)/8 + (sin(3*c + 3*d*x)*84227i)/128 - sin(4*c + 4*d*x)*754i + (sin(5*c + 5*d*x)*111527i)/128 - (sin(6*c + 6*d*x)*7187i)/8 + (sin(7*c + 7*d*x)*121427i)/128 - (26075*sin(2*c + 2*d*x)^2)/16 + (114583*sin(c/2 + ( d*x)/2)^2)/64 - (57925*sin(3*c + 3*d*x)^2)/32 + (116585*sin((3*c)/2 + (3*d *x)/2)^2)/64 + (119315*sin((5*c)/2 + (5*d*x)/2)^2)/64 + (122285*sin((7*c)/ 2 + (7*d*x)/2)^2)/64 - (44779*sin(c + d*x)^2)/32 - 952))/(6435*d*(sin((15* c)/2 + (15*d*x)/2) - sin((15*c)/4 + (15*d*x)/4)^2*2i + 1i))
Time = 0.16 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03 \[ \int \cos ^{15}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^{8} \left (54912 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{14} i -289344 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{12} i +629712 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10} i -724600 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} i +466240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} i -161232 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} i +25264 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} i -952 \cos \left (d x +c \right ) i -54912 \sin \left (d x +c \right )^{15}+316800 \sin \left (d x +c \right )^{13}-767520 \sin \left (d x +c \right )^{11}+1006720 \sin \left (d x +c \right )^{9}-765765 \sin \left (d x +c \right )^{7}+333333 \sin \left (d x +c \right )^{5}-75075 \sin \left (d x +c \right )^{3}+6435 \sin \left (d x +c \right )+952 i \right )}{6435 d} \] Input:
int(cos(d*x+c)^15*(a+I*a*tan(d*x+c))^8,x)
Output:
(a**8*(54912*cos(c + d*x)*sin(c + d*x)**14*i - 289344*cos(c + d*x)*sin(c + d*x)**12*i + 629712*cos(c + d*x)*sin(c + d*x)**10*i - 724600*cos(c + d*x) *sin(c + d*x)**8*i + 466240*cos(c + d*x)*sin(c + d*x)**6*i - 161232*cos(c + d*x)*sin(c + d*x)**4*i + 25264*cos(c + d*x)*sin(c + d*x)**2*i - 952*cos( c + d*x)*i - 54912*sin(c + d*x)**15 + 316800*sin(c + d*x)**13 - 767520*sin (c + d*x)**11 + 1006720*sin(c + d*x)**9 - 765765*sin(c + d*x)**7 + 333333* sin(c + d*x)**5 - 75075*sin(c + d*x)**3 + 6435*sin(c + d*x) + 952*i))/(643 5*d)