\(\int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [96]

Optimal result

Integrand size = 24, antiderivative size = 136 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac {2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3578, 3569} \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{1155 d}-\frac {2 i a^2 \cos ^7(c+d x) (a+i a \tan (c+d x))^6}{231 d}-\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}-\frac {i a \cos ^9(c+d x) (a+i a \tan (c+d x))^7}{33 d} \]

[In]

Int[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(((-2*I)/1155)*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/231)*a^2*Cos[c + d*x]^7*(a + I*a*Tan[c
+ d*x])^6)/d - ((I/33)*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^7)/d - ((I/11)*Cos[c + d*x]^11*(a + I*a*Tan[c
+ d*x])^8)/d

Rule 3569

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]

Rule 3578

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*S
ec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(a*f*m)), x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {i \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8}{11 d}+\frac {1}{11} (3 a) \int \cos ^9(c+d x) (a+i a \tan (c+d x))^7 \, dx \\ & = -\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}+\frac {1}{33} \left (2 a^2\right ) \int \cos ^7(c+d x) (a+i a \tan (c+d x))^6 \, dx \\ & = -\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}+\frac {1}{231} \left (2 a^3\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \, 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.11 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \sec (c+d x) (-i \cos (6 (c+d x))+\sin (6 (c+d x))) \left (726+1111 \cos (2 (c+d x))+490 \cos (4 (c+d x))+105 \cos (6 (c+d x))+11008 \sqrt {\cos ^2(c+d x)} \cos (6 (c+d x))+649 i \sin (2 (c+d x))+490 i \sin (4 (c+d x))+105 i \sin (6 (c+d x))-11008 i \sqrt {\cos ^2(c+d x)} \sin (6 (c+d x))\right )}{18480 d} \]

[In]

Integrate[Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*Sec[c + d*x]*((-I)*Cos[6*(c + d*x)] + Sin[6*(c + d*x)])*(726 + 1111*Cos[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)*S
in[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)

Maple [B] (verified)

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}\]

[In]

int(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x)

[Out]

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*s
in(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*cos(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+12
8/63*cos(d*x+c)^2)*sin(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.46 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-105 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 385 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 495 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 231 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )}}{9240 \, d} \]

[In]

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\begin {cases} \frac {- 53760 i a^{8} d^{3} e^{11 i c} e^{11 i d x} - 197120 i a^{8} d^{3} e^{9 i c} e^{9 i d x} - 253440 i a^{8} d^{3} e^{7 i c} e^{7 i d x} - 118272 i a^{8} d^{3} e^{5 i c} e^{5 i d x}}{4730880 d^{4}} & \text {for}\: d^{4} \neq 0 \\x \left (\frac {a^{8} e^{11 i c}}{8} + \frac {3 a^{8} e^{9 i c}}{8} + \frac {3 a^{8} e^{7 i c}}{8} + \frac {a^{8} e^{5 i c}}{8}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**11*(a+I*a*tan(d*x+c))**8,x)

[Out]

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))

Maxima [B] (verification not implemented)

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 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2520 i \, a^{8} \cos \left (d x + c\right )^{11} + 24 i \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 280 i \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{8} + 1960 i \, {\left (9 \, \cos \left (d x + c\right )^{11} - 11 \, \cos \left (d x + c\right )^{9}\right )} a^{8} + 28 \, {\left (315 \, \sin \left (d x + c\right )^{11} - 1540 \, \sin \left (d x + c\right )^{9} + 2970 \, \sin \left (d x + c\right )^{7} - 2772 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 210 \, {\left (105 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 495 \, \sin \left (d x + c\right )^{7} - 231 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 140 \, {\left (63 \, \sin \left (d x + c\right )^{11} - 154 \, \sin \left (d x + c\right )^{9} + 99 \, \sin \left (d x + c\right )^{7}\right )} a^{8} + 5 \, {\left (63 \, \sin \left (d x + c\right )^{11} - 385 \, \sin \left (d x + c\right )^{9} + 990 \, \sin \left (d x + c\right )^{7} - 1386 \, \sin \left (d x + c\right )^{5} + 1155 \, \sin \left (d x + c\right )^{3} - 693 \, \sin \left (d x + c\right )\right )} a^{8} + 35 \, {\left (9 \, \sin \left (d x + c\right )^{11} - 11 \, \sin \left (d x + c\right )^{9}\right )} a^{8}}{3465 \, d} \]

[In]

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-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 - 23
1*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*co
s(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

Giac [B] (verification not implemented)

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 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^11*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

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) + 2
9858174165820*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 82109978956005*a^8*e^(20*I*d*x + 6*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 164219957912010*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2463299368680
15*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)*log(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) + 82004266575*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 11480
59732050*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*I*c)*log(I*e^(I*d*x + I*c) - 1) + 246258812524725*a^8*e^(12*I*d
*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 164172541683150*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) +
82086270841575*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 29849553033300*a^8*e^(6*I*d*x - 8*I*c)*lo
g(I*e^(I*d*x + I*c) - 1) + 7462388258325*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1148059732050*a
^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 281438642885400*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) -
1) + 82004266575*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) - 1) - 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) - 29858174165820*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*
c) + 1) - 82109978956005*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 164219957912010*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) - 2985817
4165820*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)*log(-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) - 820042665
75*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*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 246258812524725*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1)
- 164172541683150*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 82086270841575*a^8*e^(8*I*d*x - 6*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 29849553033300*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 746238825
8325*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1148059732050*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I
*d*x + I*c) - 1) - 281438642885400*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 82004266575*a^8*e^(-14*I*c)*
log(-I*e^(I*d*x + I*c) - 1) - 23684430*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 331582020*a^8*e
^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2155283130*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x) + e^(-
I*c)) - 8621132520*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23708114430*a^8*e^(20*I*d*x + 6*I*c)
*log(I*e^(I*d*x) + e^(-I*c)) - 47416228860*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 71124343290*
a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 71124343290*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x) +
e^(-I*c)) - 47416228860*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23708114430*a^8*e^(8*I*d*x - 6*
I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 8621132520*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2155283130
*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 331582020*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e
^(-I*c)) - 81284963760*a^8*e^(14*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) - 23684430*a^8*e^(-14*I*c)*log(I*e^(I*d*x)
+ e^(-I*c)) + 23684430*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 331582020*a^8*e^(26*I*d*x + 1
2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2155283130*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 8621
132520*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 23708114430*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(
I*d*x) + e^(-I*c)) + 47416228860*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 71124343290*a^8*e^(16
*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 71124343290*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c
)) + 47416228860*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 23708114430*a^8*e^(8*I*d*x - 6*I*c)*l
og(-I*e^(I*d*x) + e^(-I*c)) + 8621132520*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2155283130*a^8
*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 331582020*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x) + e^(
-I*c)) + 81284963760*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) + 23684430*a^8*e^(-14*I*c)*log(-I*e^(I*d*x)
+ e^(-I*c)) - 55050240*I*a^8*e^(39*I*d*x + 25*I*c) - 972554240*I*a^8*e^(37*I*d*x + 23*I*c) - 8095006720*I*a^8
*e^(35*I*d*x + 21*I*c) - 42161143808*I*a^8*e^(33*I*d*x + 19*I*c) - 153891110912*I*a^8*e^(31*I*d*x + 17*I*c) -
417750581248*I*a^8*e^(29*I*d*x + 15*I*c) - 873287647232*I*a^8*e^(27*I*d*x + 13*I*c) - 1435886419968*I*a^8*e^(2
5*I*d*x + 11*I*c) - 1879877615616*I*a^8*e^(23*I*d*x + 9*I*c) - 1970745114624*I*a^8*e^(21*I*d*x + 7*I*c) - 1654
208331776*I*a^8*e^(19*I*d*x + 5*I*c) - 1105350098944*I*a^8*e^(17*I*d*x + 3*I*c) - 580728651776*I*a^8*e^(15*I*d
*x + I*c) - 234836983808*I*a^8*e^(13*I*d*x - I*c) - 70581747712*I*a^8*e^(11*I*d*x - 3*I*c) - 14856224768*I*a^8
*e^(9*I*d*x - 5*I*c) - 1955069952*I*a^8*e^(7*I*d*x - 7*I*c) - 121110528*I*a^8*e^(5*I*d*x - 9*I*c))/(d*e^(28*I*
d*x + 14*I*c) + 14*d*e^(26*I*d*x + 12*I*c) + 91*d*e^(24*I*d*x + 10*I*c) + 364*d*e^(22*I*d*x + 8*I*c) + 1001*d*
e^(20*I*d*x + 6*I*c) + 2002*d*e^(18*I*d*x + 4*I*c) + 3003*d*e^(16*I*d*x + 2*I*c) + 3003*d*e^(12*I*d*x - 2*I*c)
+ 2002*d*e^(10*I*d*x - 4*I*c) + 1001*d*e^(8*I*d*x - 6*I*c) + 364*d*e^(6*I*d*x - 8*I*c) + 91*d*e^(4*I*d*x - 10
*I*c) + 14*d*e^(2*I*d*x - 12*I*c) + 3432*d*e^(14*I*d*x) + d*e^(-14*I*c))

Mupad [B] (verification not implemented)

Time = 4.81 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.48 \[ \int \cos ^{11}(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {a^8\,\left (\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,3{}\mathrm {i}}{56}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^{c\,11{}\mathrm {i}+d\,x\,11{}\mathrm {i}}\,1{}\mathrm {i}}{88}\right )}{d} \]

[In]

int(cos(c + d*x)^11*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

-(a^8*((exp(c*5i + d*x*5i)*1i)/40 + (exp(c*7i + d*x*7i)*3i)/56 + (exp(c*9i + d*x*9i)*1i)/24 + (exp(c*11i + d*x
*11i)*1i)/88))/d