Integrand size = 24, antiderivative size = 211 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {20 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{3003 d}-\frac {20 i a^2 \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{1287 d}-\frac {5 i a \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{143 d}-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}-\frac {8 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{9009 d}-\frac {8 i \cos ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{3003 d} \] Output:
-20/3003*I*a^3*cos(d*x+c)^7*(a+I*a*tan(d*x+c))^5/d-20/1287*I*a^2*cos(d*x+c )^9*(a+I*a*tan(d*x+c))^6/d-5/143*I*a*cos(d*x+c)^11*(a+I*a*tan(d*x+c))^7/d- 1/13*I*cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8/d-8/9009*I*a^2*cos(d*x+c)^3*(a^2 +I*a^2*tan(d*x+c))^3/d-8/3003*I*cos(d*x+c)^5*(a^2+I*a^2*tan(d*x+c))^4/d
Time = 1.66 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.80 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \sec (c+d x) (-i \cos (7 (c+d x))+\sin (7 (c+d x))) \left (44759 \cos (c+d x)+26117 \cos (3 (c+d x))+7791 \cos (5 (c+d x))+693 \cos (7 (c+d x))+275456 \sqrt {\cos ^2(c+d x)} \cos (7 (c+d x))+1001 i \sin (c+d x)+2093 i \sin (3 (c+d x))+1785 i \sin (5 (c+d x))+693 i \sin (7 (c+d x))-275456 i \sqrt {\cos ^2(c+d x)} \sin (7 (c+d x))\right )}{576576 d} \] Input:
Integrate[Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8,x]
Output:
(a^8*Sec[c + d*x]*((-I)*Cos[7*(c + d*x)] + Sin[7*(c + d*x)])*(44759*Cos[c + d*x] + 26117*Cos[3*(c + d*x)] + 7791*Cos[5*(c + d*x)] + 693*Cos[7*(c + d *x)] + 275456*Sqrt[Cos[c + d*x]^2]*Cos[7*(c + d*x)] + (1001*I)*Sin[c + d*x ] + (2093*I)*Sin[3*(c + d*x)] + (1785*I)*Sin[5*(c + d*x)] + (693*I)*Sin[7* (c + d*x)] - (275456*I)*Sqrt[Cos[c + d*x]^2]*Sin[7*(c + d*x)]))/(576576*d)
Time = 0.96 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3978, 3042, 3978, 3042, 3978, 3042, 3978, 3042, 3978, 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.
| \(\displaystyle \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^8}{\sec (c+d x)^{13}}dx\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {5}{13} a \int \cos ^{11}(c+d x) (i \tan (c+d x) a+a)^7dx-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{13} a \int \frac {(i \tan (c+d x) a+a)^7}{\sec (c+d x)^{11}}dx-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \int \cos ^9(c+d x) (i \tan (c+d x) a+a)^6dx-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \int \frac {(i \tan (c+d x) a+a)^6}{\sec (c+d x)^9}dx-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \left (\frac {1}{3} a \int \cos ^7(c+d x) (i \tan (c+d x) a+a)^5dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \left (\frac {1}{3} a \int \frac {(i \tan (c+d x) a+a)^5}{\sec (c+d x)^7}dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \left (\frac {1}{3} a \left (\frac {2}{7} a \int \cos ^5(c+d x) (i \tan (c+d x) a+a)^4dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \left (\frac {1}{3} a \left (\frac {2}{7} a \int \frac {(i \tan (c+d x) a+a)^4}{\sec (c+d x)^5}dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \left (\frac {1}{3} a \left (\frac {2}{7} a \left (\frac {1}{5} a \int \cos ^3(c+d x) (i \tan (c+d x) a+a)^3dx-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \left (\frac {1}{3} a \left (\frac {2}{7} a \left (\frac {1}{5} a \int \frac {(i \tan (c+d x) a+a)^3}{\sec (c+d x)^3}dx-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
| \(\Big \downarrow \) 3969 |
| \(\displaystyle \frac {5}{13} a \left (\frac {4}{11} a \left (\frac {1}{3} a \left (\frac {2}{7} a \left (-\frac {i \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{5 d}-\frac {i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{15 d}\right )-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^6}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^7}{11 d}\right )-\frac {i \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8}{13 d}\) |
Input:
Int[Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8,x]
Output:
((-1/13*I)*Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8)/d + (5*a*(((-1/11*I)* Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^7)/d + (4*a*(((-1/9*I)*Cos[c + d*x] ^9*(a + I*a*Tan[c + d*x])^6)/d + (a*(((-1/7*I)*Cos[c + d*x]^7*(a + I*a*Tan [c + d*x])^5)/d + (2*a*(((-1/15*I)*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x]) ^3)/d - ((I/5)*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^4)/d))/7))/3))/11))/1 3
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[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/( a*f*m)), x] + Simp[a*((m + n)/(m*d^2)) Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b ^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (187 ) = 374\).
Time = 2.91 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.92
\[\frac {a^{8} \left (-\frac {\sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{6}}{13}-\frac {7 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{6}}{143}-\frac {35 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{1287}-\frac {5 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{429}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{429}\right )+56 i a^{8} \left (-\frac {\cos \left (d x +c \right )^{9} \sin \left (d x +c \right )^{4}}{13}-\frac {4 \cos \left (d x +c \right )^{9} \sin \left (d x +c \right )^{2}}{143}-\frac {8 \cos \left (d x +c \right )^{9}}{1287}\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{8}}{13}-\frac {5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{8}}{143}-\frac {5 \cos \left (d x +c \right )^{8} \sin \left (d x +c \right )}{429}+\frac {5 \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 )}{3003}\right )-\frac {8 i a^{8} \cos \left (d x +c \right )^{13}}{13}+70 a^{8} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{10}}{13}-\frac {3 \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 )}{429}\right )-8 i a^{8} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{6}}{13}-\frac {6 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{4}}{143}-\frac {8 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{429}-\frac {16 \cos \left (d x +c \right )^{7}}{3003}\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{12}}{13}+\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 )}{143}\right )-56 i a^{8} \left (-\frac {\cos \left (d x +c \right )^{11} \sin \left (d x +c \right )^{2}}{13}-\frac {2 \cos \left (d x +c \right )^{11}}{143}\right )+\frac {a^{8} \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 )}{13}}{d}\]
Input:
int(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x)
Output:
1/d*(a^8*(-1/13*sin(d*x+c)^7*cos(d*x+c)^6-7/143*sin(d*x+c)^5*cos(d*x+c)^6- 35/1287*sin(d*x+c)^3*cos(d*x+c)^6-5/429*sin(d*x+c)*cos(d*x+c)^6+1/429*(8/3 +cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+56*I*a^8*(-1/13*cos(d*x+c)^9*s in(d*x+c)^4-4/143*cos(d*x+c)^9*sin(d*x+c)^2-8/1287*cos(d*x+c)^9)-28*a^8*(- 1/13*sin(d*x+c)^5*cos(d*x+c)^8-5/143*sin(d*x+c)^3*cos(d*x+c)^8-5/429*cos(d *x+c)^8*sin(d*x+c)+5/3003*(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))-8/13*I*a^8*cos(d*x+c)^13+70*a^8*(-1/13*sin(d*x+c)^3*cos( d*x+c)^10-3/143*sin(d*x+c)*cos(d*x+c)^10+1/429*(128/35+cos(d*x+c)^8+8/7*co s(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c))-8*I*a^8*(-1/ 13*cos(d*x+c)^7*sin(d*x+c)^6-6/143*cos(d*x+c)^7*sin(d*x+c)^4-8/429*cos(d*x +c)^7*sin(d*x+c)^2-16/3003*cos(d*x+c)^7)-28*a^8*(-1/13*sin(d*x+c)*cos(d*x+ c)^12+1/143*(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))-56*I*a^8*(-1/13*cos(d*x+c )^11*sin(d*x+c)^2-2/143*cos(d*x+c)^11)+1/13*a^8*(1024/231+cos(d*x+c)^12+12 /11*cos(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+512/231*cos(d*x+c)^2)*sin(d*x+c))
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.43 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-693 i \, a^{8} e^{\left (13 i \, d x + 13 i \, c\right )} - 4095 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 10010 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 12870 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 9009 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 3003 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )}}{288288 \, d} \] Input:
integrate(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
Output:
1/288288*(-693*I*a^8*e^(13*I*d*x + 13*I*c) - 4095*I*a^8*e^(11*I*d*x + 11*I *c) - 10010*I*a^8*e^(9*I*d*x + 9*I*c) - 12870*I*a^8*e^(7*I*d*x + 7*I*c) - 9009*I*a^8*e^(5*I*d*x + 5*I*c) - 3003*I*a^8*e^(3*I*d*x + 3*I*c))/d
Time = 0.66 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.14 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\begin {cases} \frac {- 17439916032 i a^{8} d^{5} e^{13 i c} e^{13 i d x} - 103054049280 i a^{8} d^{5} e^{11 i c} e^{11 i d x} - 251909898240 i a^{8} d^{5} e^{9 i c} e^{9 i d x} - 323884154880 i a^{8} d^{5} e^{7 i c} e^{7 i d x} - 226718908416 i a^{8} d^{5} e^{5 i c} e^{5 i d x} - 75572969472 i a^{8} d^{5} e^{3 i c} e^{3 i d x}}{7255005069312 d^{6}} & \text {for}\: d^{6} \neq 0 \\x \left (\frac {a^{8} e^{13 i c}}{32} + \frac {5 a^{8} e^{11 i c}}{32} + \frac {5 a^{8} e^{9 i c}}{16} + \frac {5 a^{8} e^{7 i c}}{16} + \frac {5 a^{8} e^{5 i c}}{32} + \frac {a^{8} e^{3 i c}}{32}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**13*(a+I*a*tan(d*x+c))**8,x)
Output:
Piecewise(((-17439916032*I*a**8*d**5*exp(13*I*c)*exp(13*I*d*x) - 103054049 280*I*a**8*d**5*exp(11*I*c)*exp(11*I*d*x) - 251909898240*I*a**8*d**5*exp(9 *I*c)*exp(9*I*d*x) - 323884154880*I*a**8*d**5*exp(7*I*c)*exp(7*I*d*x) - 22 6718908416*I*a**8*d**5*exp(5*I*c)*exp(5*I*d*x) - 75572969472*I*a**8*d**5*e xp(3*I*c)*exp(3*I*d*x))/(7255005069312*d**6), Ne(d**6, 0)), (x*(a**8*exp(1 3*I*c)/32 + 5*a**8*exp(11*I*c)/32 + 5*a**8*exp(9*I*c)/16 + 5*a**8*exp(7*I* c)/16 + 5*a**8*exp(5*I*c)/32 + a**8*exp(3*I*c)/32), True))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (175) = 350\).
Time = 0.04 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.92 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {5544 i \, a^{8} \cos \left (d x + c\right )^{13} + 24 i \, {\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} a^{8} + 392 i \, {\left (99 \, \cos \left (d x + c\right )^{13} - 234 \, \cos \left (d x + c\right )^{11} + 143 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 3528 i \, {\left (11 \, \cos \left (d x + c\right )^{13} - 13 \, \cos \left (d x + c\right )^{11}\right )} a^{8} - 42 \, {\left (1155 \, \sin \left (d x + c\right )^{13} - 5460 \, \sin \left (d x + c\right )^{11} + 10010 \, \sin \left (d x + c\right )^{9} - 8580 \, \sin \left (d x + c\right )^{7} + 3003 \, \sin \left (d x + c\right )^{5}\right )} a^{8} - 28 \, {\left (693 \, \sin \left (d x + c\right )^{13} - 4095 \, \sin \left (d x + c\right )^{11} + 10010 \, \sin \left (d x + c\right )^{9} - 12870 \, \sin \left (d x + c\right )^{7} + 9009 \, \sin \left (d x + c\right )^{5} - 3003 \, \sin \left (d x + c\right )^{3}\right )} a^{8} - 84 \, {\left (231 \, \sin \left (d x + c\right )^{13} - 819 \, \sin \left (d x + c\right )^{11} + 1001 \, \sin \left (d x + c\right )^{9} - 429 \, \sin \left (d x + c\right )^{7}\right )} a^{8} - 3 \, {\left (231 \, \sin \left (d x + c\right )^{13} - 1638 \, \sin \left (d x + c\right )^{11} + 5005 \, \sin \left (d x + c\right )^{9} - 8580 \, \sin \left (d x + c\right )^{7} + 9009 \, \sin \left (d x + c\right )^{5} - 6006 \, \sin \left (d x + c\right )^{3} + 3003 \, \sin \left (d x + c\right )\right )} a^{8} - 7 \, {\left (99 \, \sin \left (d x + c\right )^{13} - 234 \, \sin \left (d x + c\right )^{11} + 143 \, \sin \left (d x + c\right )^{9}\right )} a^{8}}{9009 \, d} \] Input:
integrate(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
Output:
-1/9009*(5544*I*a^8*cos(d*x + c)^13 + 24*I*(231*cos(d*x + c)^13 - 819*cos( d*x + c)^11 + 1001*cos(d*x + c)^9 - 429*cos(d*x + c)^7)*a^8 + 392*I*(99*co s(d*x + c)^13 - 234*cos(d*x + c)^11 + 143*cos(d*x + c)^9)*a^8 + 3528*I*(11 *cos(d*x + c)^13 - 13*cos(d*x + c)^11)*a^8 - 42*(1155*sin(d*x + c)^13 - 54 60*sin(d*x + c)^11 + 10010*sin(d*x + c)^9 - 8580*sin(d*x + c)^7 + 3003*sin (d*x + c)^5)*a^8 - 28*(693*sin(d*x + c)^13 - 4095*sin(d*x + c)^11 + 10010* sin(d*x + c)^9 - 12870*sin(d*x + c)^7 + 9009*sin(d*x + c)^5 - 3003*sin(d*x + c)^3)*a^8 - 84*(231*sin(d*x + c)^13 - 819*sin(d*x + c)^11 + 1001*sin(d* x + c)^9 - 429*sin(d*x + c)^7)*a^8 - 3*(231*sin(d*x + c)^13 - 1638*sin(d*x + c)^11 + 5005*sin(d*x + c)^9 - 8580*sin(d*x + c)^7 + 9009*sin(d*x + c)^5 - 6006*sin(d*x + c)^3 + 3003*sin(d*x + c))*a^8 - 7*(99*sin(d*x + c)^13 - 234*sin(d*x + c)^11 + 143*sin(d*x + c)^9)*a^8)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2891 vs. \(2 (175) = 350\).
Time = 1.34 (sec) , antiderivative size = 2891, normalized size of antiderivative = 13.70 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
Output:
1/151145938944*(1945052766657*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I *c) + 1) + 27230738733198*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 176999801765787*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1 ) + 707999207063148*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1946997819423657*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 389 3995638847314*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 584099 3458270971*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 584099345 8270971*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 389399563884 7314*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 194699781942365 7*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 707999207063148*a^8 *e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 176999801765787*a^8*e^(4 *I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 27230738733198*a^8*e^(2*I*d* x - 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 6675421095166824*a^8*e^(14*I*d*x) *log(I*e^(I*d*x + I*c) + 1) + 1945052766657*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1944080407269*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I* c) - 1) + 27217125701766*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 176911317061479*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 707645268245916*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1 946024487676269*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 3892 048975352538*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5838...
Time = 1.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.44 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {a^8\,\left (\frac {{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,1{}\mathrm {i}}{96}+\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{32}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,5{}\mathrm {i}}{112}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,5{}\mathrm {i}}{144}+\frac {{\mathrm {e}}^{c\,11{}\mathrm {i}+d\,x\,11{}\mathrm {i}}\,5{}\mathrm {i}}{352}+\frac {{\mathrm {e}}^{c\,13{}\mathrm {i}+d\,x\,13{}\mathrm {i}}\,1{}\mathrm {i}}{416}\right )}{d} \] Input:
int(cos(c + d*x)^13*(a + a*tan(c + d*x)*1i)^8,x)
Output:
-(a^8*((exp(c*3i + d*x*3i)*1i)/96 + (exp(c*5i + d*x*5i)*1i)/32 + (exp(c*7i + d*x*7i)*5i)/112 + (exp(c*9i + d*x*9i)*5i)/144 + (exp(c*11i + d*x*11i)*5 i)/352 + (exp(c*13i + d*x*13i)*1i)/416))/d
Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.91 \[ \int \cos ^{13}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^{8} \left (-88704 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{12} i +374976 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10} i -624400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} i +511624 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} i -207672 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} i +35416 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} i -1240 \cos \left (d x +c \right ) i +88704 \sin \left (d x +c \right )^{13}-419328 \sin \left (d x +c \right )^{11}+800800 \sin \left (d x +c \right )^{9}-782496 \sin \left (d x +c \right )^{7}+405405 \sin \left (d x +c \right )^{5}-102102 \sin \left (d x +c \right )^{3}+9009 \sin \left (d x +c \right )+1240 i \right )}{9009 d} \] Input:
int(cos(d*x+c)^13*(a+I*a*tan(d*x+c))^8,x)
Output:
(a**8*( - 88704*cos(c + d*x)*sin(c + d*x)**12*i + 374976*cos(c + d*x)*sin( c + d*x)**10*i - 624400*cos(c + d*x)*sin(c + d*x)**8*i + 511624*cos(c + d* x)*sin(c + d*x)**6*i - 207672*cos(c + d*x)*sin(c + d*x)**4*i + 35416*cos(c + d*x)*sin(c + d*x)**2*i - 1240*cos(c + d*x)*i + 88704*sin(c + d*x)**13 - 419328*sin(c + d*x)**11 + 800800*sin(c + d*x)**9 - 782496*sin(c + d*x)**7 + 405405*sin(c + d*x)**5 - 102102*sin(c + d*x)**3 + 9009*sin(c + d*x) + 1 240*i))/(9009*d)