\(\int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [95]

Optimal result

Integrand size = 24, antiderivative size = 66 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d} \]

[Out]

-1/63*I*a*cos(d*x+c)^7*(a+I*a*tan(d*x+c))^7/d-1/9*I*cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3578, 3569} \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}-\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d} \]

[In]

Int[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8,x]

[Out]

((-1/63*I)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^7)/d - ((I/9)*Cos[c + d*x]^9*(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 ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}+\frac {1}{9} a \int \cos ^7(c+d x) (a+i a \tan (c+d x))^7 \, dx \\ & = -\frac {i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}-\frac {i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(66)=132\).

Time = 0.81 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.21 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \sec (c+d x) (-i \cos (5 (c+d x))+\sin (5 (c+d x))) \left (9 \cos (c+d x)+16 \cos (3 (c+d x))+7 \cos (5 (c+d x))+192 \sqrt {\cos ^2(c+d x)} \cos (5 (c+d x))+9 i \sin (c+d x)+16 i \sin (3 (c+d x))+7 i \sin (5 (c+d x))-192 i \sqrt {\cos ^2(c+d x)} \sin (5 (c+d x))\right )}{252 d} \]

[In]

Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*Sec[c + d*x]*((-I)*Cos[5*(c + d*x)] + Sin[5*(c + d*x)])*(9*Cos[c + d*x] + 16*Cos[3*(c + d*x)] + 7*Cos[5*(
c + d*x)] + 192*Sqrt[Cos[c + d*x]^2]*Cos[5*(c + d*x)] + (9*I)*Sin[c + d*x] + (16*I)*Sin[3*(c + d*x)] + (7*I)*S
in[5*(c + d*x)] - (192*I)*Sqrt[Cos[c + d*x]^2]*Sin[5*(c + d*x)]))/(252*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (58 ) = 116\).

Time = 1.59 (sec) , antiderivative size = 447, normalized size of antiderivative = 6.77

\[\frac {\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {8 i a^{8} \left (\cos ^{9}\left (d x +c \right )\right )}{9}-28 a^{8} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{9}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{63}-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{21}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{63}\right )-8 i a^{8} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{21}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105}-\frac {16 \left (\cos ^{3}\left (d x +c \right )\right )}{315}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\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 )}{105}\right )-56 i a^{8} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-28 a^{8} \left (-\frac {\left (\cos ^{8}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9}+\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 )}{63}\right )+56 i a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+\frac {a^{8} \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 )}{9}}{d}\]

[In]

int(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x)

[Out]

1/d*(1/9*a^8*sin(d*x+c)^9-8/9*I*a^8*cos(d*x+c)^9-28*a^8*(-1/9*cos(d*x+c)^4*sin(d*x+c)^5-5/63*sin(d*x+c)^3*cos(
d*x+c)^4-1/21*sin(d*x+c)*cos(d*x+c)^4+1/63*(2+cos(d*x+c)^2)*sin(d*x+c))-8*I*a^8*(-1/9*cos(d*x+c)^3*sin(d*x+c)^
6-2/21*cos(d*x+c)^3*sin(d*x+c)^4-8/105*cos(d*x+c)^3*sin(d*x+c)^2-16/315*cos(d*x+c)^3)+70*a^8*(-1/9*sin(d*x+c)^
3*cos(d*x+c)^6-1/21*sin(d*x+c)*cos(d*x+c)^6+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-56*I*a^8*(-1
/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*cos(d*x+c)^7)-28*a^8*(-1/9*cos(d*x+c)^8*sin(d*x+c)+1/63*(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*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5-4/63*cos(d*x+c)^5*sin(d
*x+c)^2-8/315*cos(d*x+c)^5)+1/9*a^8*(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))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.52 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-7 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 9 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )}}{126 \, d} \]

[In]

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/126*(-7*I*a^8*e^(9*I*d*x + 9*I*c) - 9*I*a^8*e^(7*I*d*x + 7*I*c))/d

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\begin {cases} \frac {- 14 i a^{8} d e^{9 i c} e^{9 i d x} - 18 i a^{8} d e^{7 i c} e^{7 i d x}}{252 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (\frac {a^{8} e^{9 i c}}{2} + \frac {a^{8} e^{7 i c}}{2}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**9*(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise(((-14*I*a**8*d*exp(9*I*c)*exp(9*I*d*x) - 18*I*a**8*d*exp(7*I*c)*exp(7*I*d*x))/(252*d**2), Ne(d**2, 0
)), (x*(a**8*exp(9*I*c)/2 + a**8*exp(7*I*c)/2), 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. 302 vs. \(2 (54) = 108\).

Time = 0.26 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.58 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {280 i \, a^{8} \cos \left (d x + c\right )^{9} - 35 \, a^{8} \sin \left (d x + c\right )^{9} + 56 i \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 8 i \, {\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 280 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{8} - 70 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{8} - 28 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{8} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{8} - 140 \, {\left (7 \, \sin \left (d x + c\right )^{9} - 9 \, \sin \left (d x + c\right )^{7}\right )} a^{8}}{315 \, d} \]

[In]

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/315*(280*I*a^8*cos(d*x + c)^9 - 35*a^8*sin(d*x + c)^9 + 56*I*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*co
s(d*x + c)^5)*a^8 + 8*I*(35*cos(d*x + c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 105*cos(d*x + c)^3)*a^8
+ 280*I*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^8 - 70*(35*sin(d*x + c)^9 - 90*sin(d*x + c)^7 + 63*sin(d*x +
c)^5)*a^8 - 28*(35*sin(d*x + c)^9 - 135*sin(d*x + c)^7 + 189*sin(d*x + c)^5 - 105*sin(d*x + c)^3)*a^8 - (35*si
n(d*x + c)^9 - 180*sin(d*x + c)^7 + 378*sin(d*x + c)^5 - 420*sin(d*x + c)^3 + 315*sin(d*x + c))*a^8 - 140*(7*s
in(d*x + c)^9 - 9*sin(d*x + c)^7)*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. 2451 vs. \(2 (54) = 108\).

Time = 1.62 (sec) , antiderivative size = 2451, normalized size of antiderivative = 37.14 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/66060288*(1419343317*a^8*e^(24*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 17032119804*a^8*e^(22*I*d*x + 10
*I*c)*log(I*e^(I*d*x + I*c) + 1) + 93676658922*a^8*e^(20*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 312255529
740*a^8*e^(18*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 702574941915*a^8*e^(16*I*d*x + 4*I*c)*log(I*e^(I*d*x
+ I*c) + 1) + 1124119907064*a^8*e^(14*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1124119907064*a^8*e^(10*I*d
*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 702574941915*a^8*e^(8*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 312
255529740*a^8*e^(6*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 93676658922*a^8*e^(4*I*d*x - 8*I*c)*log(I*e^(I*
d*x + I*c) + 1) + 17032119804*a^8*e^(2*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1311473224908*a^8*e^(12*I*
d*x)*log(I*e^(I*d*x + I*c) + 1) + 1419343317*a^8*e^(-12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1419097050*a^8*e^(24
*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 17029164600*a^8*e^(22*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1)
+ 93660405300*a^8*e^(20*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 312201351000*a^8*e^(18*I*d*x + 6*I*c)*log
(I*e^(I*d*x + I*c) - 1) + 702453039750*a^8*e^(16*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1123924863600*a^8
*e^(14*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1123924863600*a^8*e^(10*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c
) - 1) + 702453039750*a^8*e^(8*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 312201351000*a^8*e^(6*I*d*x - 6*I*c
)*log(I*e^(I*d*x + I*c) - 1) + 93660405300*a^8*e^(4*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 17029164600*a^
8*e^(2*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1311245674200*a^8*e^(12*I*d*x)*log(I*e^(I*d*x + I*c) - 1)
+ 1419097050*a^8*e^(-12*I*c)*log(I*e^(I*d*x + I*c) - 1) - 1419343317*a^8*e^(24*I*d*x + 12*I*c)*log(-I*e^(I*d*x
+ I*c) + 1) - 17032119804*a^8*e^(22*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 93676658922*a^8*e^(20*I*d*x
+ 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 312255529740*a^8*e^(18*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 70
2574941915*a^8*e^(16*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1124119907064*a^8*e^(14*I*d*x + 2*I*c)*log(-
I*e^(I*d*x + I*c) + 1) - 1124119907064*a^8*e^(10*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 702574941915*a^8
*e^(8*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 312255529740*a^8*e^(6*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c)
+ 1) - 93676658922*a^8*e^(4*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 17032119804*a^8*e^(2*I*d*x - 10*I*c)
*log(-I*e^(I*d*x + I*c) + 1) - 1311473224908*a^8*e^(12*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 1419343317*a^8*e^(
-12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1419097050*a^8*e^(24*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1702
9164600*a^8*e^(22*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 93660405300*a^8*e^(20*I*d*x + 8*I*c)*log(-I*e^
(I*d*x + I*c) - 1) - 312201351000*a^8*e^(18*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 702453039750*a^8*e^(1
6*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1123924863600*a^8*e^(14*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) -
1) - 1123924863600*a^8*e^(10*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 702453039750*a^8*e^(8*I*d*x - 4*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 312201351000*a^8*e^(6*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 93660405300
*a^8*e^(4*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 17029164600*a^8*e^(2*I*d*x - 10*I*c)*log(-I*e^(I*d*x +
I*c) - 1) - 1311245674200*a^8*e^(12*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 1419097050*a^8*e^(-12*I*c)*log(-I*e^(
I*d*x + I*c) - 1) - 246267*a^8*e^(24*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2955204*a^8*e^(22*I*d*x + 1
0*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 16253622*a^8*e^(20*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 54178740*
a^8*e^(18*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 121902165*a^8*e^(16*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^
(-I*c)) - 195043464*a^8*e^(14*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 195043464*a^8*e^(10*I*d*x - 2*I*c)*
log(I*e^(I*d*x) + e^(-I*c)) - 121902165*a^8*e^(8*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 54178740*a^8*e^(
6*I*d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 16253622*a^8*e^(4*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) -
2955204*a^8*e^(2*I*d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 227550708*a^8*e^(12*I*d*x)*log(I*e^(I*d*x) + e^
(-I*c)) - 246267*a^8*e^(-12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 246267*a^8*e^(24*I*d*x + 12*I*c)*log(-I*e^(I*d*
x) + e^(-I*c)) + 2955204*a^8*e^(22*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 16253622*a^8*e^(20*I*d*x + 8
*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 54178740*a^8*e^(18*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 12190216
5*a^8*e^(16*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 195043464*a^8*e^(14*I*d*x + 2*I*c)*log(-I*e^(I*d*x)
+ e^(-I*c)) + 195043464*a^8*e^(10*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 121902165*a^8*e^(8*I*d*x - 4*I
*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 54178740*a^8*e^(6*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 16253622*a^
8*e^(4*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2955204*a^8*e^(2*I*d*x - 10*I*c)*log(-I*e^(I*d*x) + e^(-I
*c)) + 227550708*a^8*e^(12*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) + 246267*a^8*e^(-12*I*c)*log(-I*e^(I*d*x) + e^(
-I*c)) - 3670016*I*a^8*e^(33*I*d*x + 21*I*c) - 48758784*I*a^8*e^(31*I*d*x + 19*I*c) - 298844160*I*a^8*e^(29*I*
d*x + 17*I*c) - 1118830592*I*a^8*e^(27*I*d*x + 15*I*c) - 2854748160*I*a^8*e^(25*I*d*x + 13*I*c) - 5242355712*I
*a^8*e^(23*I*d*x + 11*I*c) - 7128219648*I*a^8*e^(21*I*d*x + 9*I*c) - 7266631680*I*a^8*e^(19*I*d*x + 7*I*c) - 5
553782784*I*a^8*e^(17*I*d*x + 5*I*c) - 3143106560*I*a^8*e^(15*I*d*x + 3*I*c) - 1280311296*I*a^8*e^(13*I*d*x +
I*c) - 355467264*I*a^8*e^(11*I*d*x - I*c) - 60293120*I*a^8*e^(9*I*d*x - 3*I*c) - 4718592*I*a^8*e^(7*I*d*x - 5*
I*c))/(d*e^(24*I*d*x + 12*I*c) + 12*d*e^(22*I*d*x + 10*I*c) + 66*d*e^(20*I*d*x + 8*I*c) + 220*d*e^(18*I*d*x +
6*I*c) + 495*d*e^(16*I*d*x + 4*I*c) + 792*d*e^(14*I*d*x + 2*I*c) + 792*d*e^(10*I*d*x - 2*I*c) + 495*d*e^(8*I*d
*x - 4*I*c) + 220*d*e^(6*I*d*x - 6*I*c) + 66*d*e^(4*I*d*x - 8*I*c) + 12*d*e^(2*I*d*x - 10*I*c) + 924*d*e^(12*I
*d*x) + d*e^(-12*I*c))

Mupad [B] (verification not implemented)

Time = 4.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.56 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {2\,a^8\,\left (\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,9{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,7{}\mathrm {i}}{4}\right )}{63\,d} \]

[In]

int(cos(c + d*x)^9*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

-(2*a^8*((exp(c*7i + d*x*7i)*9i)/4 + (exp(c*9i + d*x*9i)*7i)/4))/(63*d)