Integrand size = 24, antiderivative size = 136 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac {2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d} \]
-2/1155*I*a^3*cos(d*x+c)^5*(a+I*a*tan(d*x+c))^5/d-2/231*I*a^2*cos(d*x+c)^7 *(a+I*a*tan(d*x+c))^6/d-1/33*I*a*cos(d*x+c)^9*(a+I*a*tan(d*x+c))^7/d-1/11* I*cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8/d
Time = 0.96 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.11 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \sec (c+d x) (-i \cos (6 (c+d x))+\sin (6 (c+d x))) \left (726+1111 \cos (2 (c+d x))+490 \cos (4 (c+d x))+105 \cos (6 (c+d x))+11008 \sqrt {\cos ^2(c+d x)} \cos (6 (c+d x))+649 i \sin (2 (c+d x))+490 i \sin (4 (c+d x))+105 i \sin (6 (c+d x))-11008 i \sqrt {\cos ^2(c+d x)} \sin (6 (c+d x))\right )}{18480 d} \]
(a^8*Sec[c + d*x]*((-I)*Cos[6*(c + d*x)] + Sin[6*(c + d*x)])*(726 + 1111*C os[2*(c + d*x)] + 490*Cos[4*(c + d*x)] + 105*Cos[6*(c + d*x)] + 11008*Sqrt [Cos[c + d*x]^2]*Cos[6*(c + d*x)] + (649*I)*Sin[2*(c + d*x)] + (490*I)*Sin [4*(c + d*x)] + (105*I)*Sin[6*(c + d*x)] - (11008*I)*Sqrt[Cos[c + d*x]^2]* Sin[6*(c + d*x)]))/(18480*d)
Time = 0.62 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
| \(\displaystyle \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^8}{\sec (c+d x)^{11}}dx\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {3}{11} a \int \cos ^9(c+d x) (i \tan (c+d x) a+a)^7dx-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \int \frac {(i \tan (c+d x) a+a)^7}{\sec (c+d x)^9}dx-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {3}{11} a \left (\frac {2}{9} a \int \cos ^7(c+d x) (i \tan (c+d x) a+a)^6dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {2}{9} a \int \frac {(i \tan (c+d x) a+a)^6}{\sec (c+d x)^7}dx-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\) |
| \(\Big \downarrow \) 3978 |
| \(\displaystyle \frac {3}{11} a \left (\frac {2}{9} a \left (\frac {1}{7} a \int \cos ^5(c+d x) (i \tan (c+d x) a+a)^5dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {2}{9} a \left (\frac {1}{7} a \int \frac {(i \tan (c+d x) a+a)^5}{\sec (c+d x)^5}dx-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{7 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\) |
| \(\Big \downarrow \) 3969 |
| \(\displaystyle \frac {3}{11} a \left (\frac {2}{9} a \left (-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{7 d}-\frac {i a \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{35 d}\right )-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{9 d}\right )-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}\) |
((-1/11*I)*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8)/d + (3*a*(((-1/9*I)*C os[c + d*x]^9*(a + I*a*Tan[c + d*x])^7)/d + (2*a*(((-1/35*I)*a*Cos[c + d*x ]^5*(a + I*a*Tan[c + d*x])^5)/d - ((I/7)*Cos[c + d*x]^7*(a + I*a*Tan[c + d *x])^6)/d))/9))/11
3.1.96.3.1 Defintions of rubi rules used
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/ (a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ [Simplify[m + n], 0]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)])^(n_), x_Symbol] :> Simp[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. 566 vs. \(2 (120 ) = 240\).
Time = 1.88 (sec) , antiderivative size = 567, normalized size of antiderivative = 4.17
\[\frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{11}-\frac {7 \left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{99}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{99}-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{33}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{99}\right )-56 i a^{8} \left (-\frac {\left (\cos ^{9}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{11}-\frac {2 \left (\cos ^{9}\left (d x +c \right )\right )}{99}\right )-28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{11}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{99}-\frac {5 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{231}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{231}\right )-\frac {8 i a^{8} \left (\cos ^{11}\left (d x +c \right )\right )}{11}+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{11}-\frac {\left (\cos ^{8}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{33}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{231}\right )-8 i a^{8} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{10}\left (d x +c \right )\right )}{11}+\frac {\left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{99}\right )+56 i a^{8} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+\frac {a^{8} \left (\frac {256}{63}+\cos ^{10}\left (d x +c \right )+\frac {10 \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {80 \left (\cos ^{6}\left (d x +c \right )\right )}{63}+\frac {32 \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {128 \left (\cos ^{2}\left (d x +c \right )\right )}{63}\right ) \sin \left (d x +c \right )}{11}}{d}\]
1/d*(a^8*(-1/11*sin(d*x+c)^7*cos(d*x+c)^4-7/99*cos(d*x+c)^4*sin(d*x+c)^5-5 /99*sin(d*x+c)^3*cos(d*x+c)^4-1/33*sin(d*x+c)*cos(d*x+c)^4+1/99*(2+cos(d*x +c)^2)*sin(d*x+c))-56*I*a^8*(-1/11*cos(d*x+c)^9*sin(d*x+c)^2-2/99*cos(d*x+ c)^9)-28*a^8*(-1/11*sin(d*x+c)^5*cos(d*x+c)^6-5/99*sin(d*x+c)^3*cos(d*x+c) ^6-5/231*sin(d*x+c)*cos(d*x+c)^6+1/231*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2) *sin(d*x+c))-8/11*I*a^8*cos(d*x+c)^11+70*a^8*(-1/11*sin(d*x+c)^3*cos(d*x+c )^8-1/33*cos(d*x+c)^8*sin(d*x+c)+1/231*(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*I*a^8*(-1/11*cos(d*x+c)^5*sin(d*x+c)^6-2/ 33*sin(d*x+c)^4*cos(d*x+c)^5-8/231*cos(d*x+c)^5*sin(d*x+c)^2-16/1155*cos(d *x+c)^5)-28*a^8*(-1/11*sin(d*x+c)*cos(d*x+c)^10+1/99*(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*(-1/11*cos(d*x+c)^7*sin(d*x+c)^4-4/99*cos(d*x+c)^7*sin(d*x+c)^2-8/693*c os(d*x+c)^7)+1/11*a^8*(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))
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.46 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-105 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 385 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 495 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 231 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )}}{9240 \, d} \]
1/9240*(-105*I*a^8*e^(11*I*d*x + 11*I*c) - 385*I*a^8*e^(9*I*d*x + 9*I*c) - 495*I*a^8*e^(7*I*d*x + 7*I*c) - 231*I*a^8*e^(5*I*d*x + 5*I*c))/d
Time = 0.57 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\begin {cases} \frac {- 53760 i a^{8} d^{3} e^{11 i c} e^{11 i d x} - 197120 i a^{8} d^{3} e^{9 i c} e^{9 i d x} - 253440 i a^{8} d^{3} e^{7 i c} e^{7 i d x} - 118272 i a^{8} d^{3} e^{5 i c} e^{5 i d x}}{4730880 d^{4}} & \text {for}\: d^{4} \neq 0 \\x \left (\frac {a^{8} e^{11 i c}}{8} + \frac {3 a^{8} e^{9 i c}}{8} + \frac {3 a^{8} e^{7 i c}}{8} + \frac {a^{8} e^{5 i c}}{8}\right ) & \text {otherwise} \end {cases} \]
Piecewise(((-53760*I*a**8*d**3*exp(11*I*c)*exp(11*I*d*x) - 197120*I*a**8*d **3*exp(9*I*c)*exp(9*I*d*x) - 253440*I*a**8*d**3*exp(7*I*c)*exp(7*I*d*x) - 118272*I*a**8*d**3*exp(5*I*c)*exp(5*I*d*x))/(4730880*d**4), Ne(d**4, 0)), (x*(a**8*exp(11*I*c)/8 + 3*a**8*exp(9*I*c)/8 + 3*a**8*exp(7*I*c)/8 + a**8 *exp(5*I*c)/8), True))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (112) = 224\).
Time = 0.43 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.61 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2520 i \, a^{8} \cos \left (d x + c\right )^{11} + 24 i \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 280 i \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{8} + 1960 i \, {\left (9 \, \cos \left (d x + c\right )^{11} - 11 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 28 \, {\left (315 \, \sin \left (d x + c\right )^{11} - 1540 \, \sin \left (d x + c\right )^{9} + 2970 \, \sin \left (d x + c\right )^{7} - 2772 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 210 \, {\left (105 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 495 \, \sin \left (d x + c\right )^{7} - 231 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 140 \, {\left (63 \, \sin \left (d x + c\right )^{11} - 154 \, \sin \left (d x + c\right )^{9} + 99 \, \sin \left (d x + c\right )^{7}\right )} a^{8} + 5 \, {\left (63 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 990 \, \sin \left (d x + c\right )^{7} - 1386 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3} - 693 \, \sin \left (d x + c\right )\right )} a^{8} + 35 \, {\left (9 \, \sin \left (d x + c\right )^{11} - 11 \, \sin \left (d x + c\right )^{9}\right )} a^{8}}{3465 \, d} \]
-1/3465*(2520*I*a^8*cos(d*x + c)^11 + 24*I*(105*cos(d*x + c)^11 - 385*cos( d*x + c)^9 + 495*cos(d*x + c)^7 - 231*cos(d*x + c)^5)*a^8 + 280*I*(63*cos( d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^8 + 1960*I*(9*cos( d*x + c)^11 - 11*cos(d*x + c)^9)*a^8 + 28*(315*sin(d*x + c)^11 - 1540*sin( d*x + c)^9 + 2970*sin(d*x + c)^7 - 2772*sin(d*x + c)^5 + 1155*sin(d*x + c) ^3)*a^8 + 210*(105*sin(d*x + c)^11 - 385*sin(d*x + c)^9 + 495*sin(d*x + c) ^7 - 231*sin(d*x + c)^5)*a^8 + 140*(63*sin(d*x + c)^11 - 154*sin(d*x + c)^ 9 + 99*sin(d*x + c)^7)*a^8 + 5*(63*sin(d*x + c)^11 - 385*sin(d*x + c)^9 + 990*sin(d*x + c)^7 - 1386*sin(d*x + c)^5 + 1155*sin(d*x + c)^3 - 693*sin(d *x + c))*a^8 + 35*(9*sin(d*x + c)^11 - 11*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. 2863 vs. \(2 (112) = 224\).
Time = 1.73 (sec) , antiderivative size = 2863, normalized size of antiderivative = 21.05 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]
1/4844421120*(82027951005*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1148391314070*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 7464543541455*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 298 58174165820*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82109978 956005*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1642199579120 10*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 246329936868015*a ^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 246329936868015*a^8*e ^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 164219957912010*a^8*e^(10 *I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82109978956005*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 29858174165820*a^8*e^(6*I*d*x - 8*I *c)*log(I*e^(I*d*x + I*c) + 1) + 7464543541455*a^8*e^(4*I*d*x - 10*I*c)*lo g(I*e^(I*d*x + I*c) + 1) + 1148391314070*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^ (I*d*x + I*c) + 1) + 281519927849160*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c ) + 1) + 82027951005*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82004266 575*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1148059732050*a ^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 7462388258325*a^8*e^ (24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 29849553033300*a^8*e^(22* I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 82086270841575*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 164172541683150*a^8*e^(18*I*d*x + 4 *I*c)*log(I*e^(I*d*x + I*c) - 1) + 246258812524725*a^8*e^(16*I*d*x + 2*...
Time = 4.81 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.48 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {a^8\,\left (\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,3{}\mathrm {i}}{56}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^{c\,11{}\mathrm {i}+d\,x\,11{}\mathrm {i}}\,1{}\mathrm {i}}{88}\right )}{d} \]