\(\int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx\) [75]

Optimal result

Integrand size = 24, antiderivative size = 141 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i a^5 \cos ^5(c+d x)}{105 d}+\frac {a^5 \sin (c+d x)}{21 d}-\frac {2 a^5 \sin ^3(c+d x)}{63 d}+\frac {a^5 \sin ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d} \]

[Out]

-1/105*I*a^5*cos(d*x+c)^5/d+1/21*a^5*sin(d*x+c)/d-2/63*a^5*sin(d*x+c)^3/d+1/105*a^5*sin(d*x+c)^5/d-2/63*I*a^3*
cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2/d-2/9*I*a*cos(d*x+c)^9*(a+I*a*tan(d*x+c))^4/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3577, 3567, 2713} \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 \sin ^5(c+d x)}{105 d}-\frac {2 a^5 \sin ^3(c+d x)}{63 d}+\frac {a^5 \sin (c+d x)}{21 d}-\frac {i a^5 \cos ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d} \]

[In]

Int[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^5,x]

[Out]

((-1/105*I)*a^5*Cos[c + d*x]^5)/d + (a^5*Sin[c + d*x])/(21*d) - (2*a^5*Sin[c + d*x]^3)/(63*d) + (a^5*Sin[c + d
*x]^5)/(105*d) - (((2*I)/63)*a^3*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2)/d - (((2*I)/9)*a*Cos[c + d*x]^9*(a +
I*a*Tan[c + d*x])^4)/d

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 3567

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*((d*Sec[
e + f*x])^m/(f*m)), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 3577

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] - Dist[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]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac {1}{9} a^2 \int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx \\ & = -\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac {1}{21} a^4 \int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx \\ & = -\frac {i a^5 \cos ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}+\frac {1}{21} a^5 \int \cos ^5(c+d x) \, dx \\ & = -\frac {i a^5 \cos ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d}-\frac {a^5 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 d} \\ & = -\frac {i a^5 \cos ^5(c+d x)}{105 d}+\frac {a^5 \sin (c+d x)}{21 d}-\frac {2 a^5 \sin ^3(c+d x)}{63 d}+\frac {a^5 \sin ^5(c+d x)}{105 d}-\frac {2 i a^3 \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{63 d}-\frac {2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^4}{9 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 \sec (c+d x) (-i \cos (5 (c+d x))+\sin (5 (c+d x))) \left (678 \cos (c+d x)+475 \cos (3 (c+d x))+175 \cos (5 (c+d x))+1472 \sqrt {\cos ^2(c+d x)} \cos (5 (c+d x))-120 i \sin (c+d x)-260 i \sin (3 (c+d x))-140 i \sin (5 (c+d x))-1472 i \sqrt {\cos ^2(c+d x)} \sin (5 (c+d x))\right )}{5040 d} \]

[In]

Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*Sec[c + d*x]*((-I)*Cos[5*(c + d*x)] + Sin[5*(c + d*x)])*(678*Cos[c + d*x] + 475*Cos[3*(c + d*x)] + 175*Co
s[5*(c + d*x)] + 1472*Sqrt[Cos[c + d*x]^2]*Cos[5*(c + d*x)] - (120*I)*Sin[c + d*x] - (260*I)*Sin[3*(c + d*x)]
- (140*I)*Sin[5*(c + d*x)] - (1472*I)*Sqrt[Cos[c + d*x]^2]*Sin[5*(c + d*x)]))/(5040*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (124 ) = 248\).

Time = 0.79 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.04

\[\frac {i a^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )-10 i a^{5} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-10 a^{5} \left (-\frac {\left (\cos ^{8}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {5 i a^{5} \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{5} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d}\]

[In]

int(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^5,x)

[Out]

1/d*(I*a^5*(-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)+5*a^5*(-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))-10*I*
a^5*(-1/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*cos(d*x+c)^7)-10*a^5*(-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))-5/9*I*a^5*cos(d*x+c)^9+1/9*a^5*(128/35+cos(d*x+c)^8+8/7*c
os(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 = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.54 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {-35 i \, a^{5} e^{\left (9 i \, d x + 9 i \, c\right )} - 180 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 378 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 420 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 315 i \, a^{5} e^{\left (i \, d x + i \, c\right )}}{5040 \, d} \]

[In]

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")

[Out]

1/5040*(-35*I*a^5*e^(9*I*d*x + 9*I*c) - 180*I*a^5*e^(7*I*d*x + 7*I*c) - 378*I*a^5*e^(5*I*d*x + 5*I*c) - 420*I*
a^5*e^(3*I*d*x + 3*I*c) - 315*I*a^5*e^(I*d*x + I*c))/d

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.36 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=\begin {cases} \frac {- 215040 i a^{5} d^{4} e^{9 i c} e^{9 i d x} - 1105920 i a^{5} d^{4} e^{7 i c} e^{7 i d x} - 2322432 i a^{5} d^{4} e^{5 i c} e^{5 i d x} - 2580480 i a^{5} d^{4} e^{3 i c} e^{3 i d x} - 1935360 i a^{5} d^{4} e^{i c} e^{i d x}}{30965760 d^{5}} & \text {for}\: d^{5} \neq 0 \\x \left (\frac {a^{5} e^{9 i c}}{16} + \frac {a^{5} e^{7 i c}}{4} + \frac {3 a^{5} e^{5 i c}}{8} + \frac {a^{5} e^{3 i c}}{4} + \frac {a^{5} e^{i c}}{16}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**9*(a+I*a*tan(d*x+c))**5,x)

[Out]

Piecewise(((-215040*I*a**5*d**4*exp(9*I*c)*exp(9*I*d*x) - 1105920*I*a**5*d**4*exp(7*I*c)*exp(7*I*d*x) - 232243
2*I*a**5*d**4*exp(5*I*c)*exp(5*I*d*x) - 2580480*I*a**5*d**4*exp(3*I*c)*exp(3*I*d*x) - 1935360*I*a**5*d**4*exp(
I*c)*exp(I*d*x))/(30965760*d**5), Ne(d**5, 0)), (x*(a**5*exp(9*I*c)/16 + a**5*exp(7*I*c)/4 + 3*a**5*exp(5*I*c)
/8 + a**5*exp(3*I*c)/4 + a**5*exp(I*c)/16), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.54 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {175 i \, a^{5} \cos \left (d x + c\right )^{9} + i \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{5} + 50 i \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{5} - 5 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{5} - 10 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{5} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{5}}{315 \, d} \]

[In]

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/315*(175*I*a^5*cos(d*x + c)^9 + I*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^5 + 50*I*(7
*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^5 - 5*(35*sin(d*x + c)^9 - 90*sin(d*x + c)^7 + 63*sin(d*x + c)^5)*a^5 -
10*(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^5 - (35*sin(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^5)/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. 1725 vs. \(2 (119) = 238\).

Time = 0.95 (sec) , antiderivative size = 1725, normalized size of antiderivative = 12.23 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")

[Out]

-1/41287680*(69853455*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 558827640*a^5*e^(14*I*d*x + 6*I*c)
*log(I*e^(I*d*x + I*c) + 1) + 1955896740*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3911793480*a^5*
e^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3911793480*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1
) + 1955896740*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 558827640*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^
(I*d*x + I*c) + 1) + 4889741850*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 69853455*a^5*e^(-8*I*c)*log(I*e^(
I*d*x + I*c) + 1) + 70703325*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 565626600*a^5*e^(14*I*d*x +
6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1979693100*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 39593862
00*a^5*e^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 3959386200*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 1979693100*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 565626600*a^5*e^(2*I*d*x - 6*I*c)*l
og(I*e^(I*d*x + I*c) - 1) + 4949232750*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 70703325*a^5*e^(-8*I*c)*lo
g(I*e^(I*d*x + I*c) - 1) - 69853455*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 558827640*a^5*e^(14
*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1955896740*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1)
- 3911793480*a^5*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 3911793480*a^5*e^(6*I*d*x - 2*I*c)*log(-I*
e^(I*d*x + I*c) + 1) - 1955896740*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 558827640*a^5*e^(2*I*d
*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4889741850*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 69853455*a^
5*e^(-8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 70703325*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 565
626600*a^5*e^(14*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1979693100*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*
d*x + I*c) - 1) - 3959386200*a^5*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3959386200*a^5*e^(6*I*d*x
- 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1979693100*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 565626
600*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 4949232750*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) -
1) - 70703325*a^5*e^(-8*I*c)*log(-I*e^(I*d*x + I*c) - 1) + 849870*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e
^(-I*c)) + 6798960*a^5*e^(14*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 23796360*a^5*e^(12*I*d*x + 4*I*c)*lo
g(I*e^(I*d*x) + e^(-I*c)) + 47592720*a^5*e^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 47592720*a^5*e^(6*
I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 23796360*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 67
98960*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 59490900*a^5*e^(8*I*d*x)*log(I*e^(I*d*x) + e^(-I*c
)) + 849870*a^5*e^(-8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 849870*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^
(-I*c)) - 6798960*a^5*e^(14*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 23796360*a^5*e^(12*I*d*x + 4*I*c)*lo
g(-I*e^(I*d*x) + e^(-I*c)) - 47592720*a^5*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 47592720*a^5*e^(
6*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 23796360*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c))
- 6798960*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 59490900*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x) + e
^(-I*c)) - 849870*a^5*e^(-8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 286720*I*a^5*e^(25*I*d*x + 17*I*c) + 3768320*I
*a^5*e^(23*I*d*x + 15*I*c) + 22921216*I*a^5*e^(21*I*d*x + 13*I*c) + 85557248*I*a^5*e^(19*I*d*x + 11*I*c) + 219
455488*I*a^5*e^(17*I*d*x + 9*I*c) + 409665536*I*a^5*e^(15*I*d*x + 7*I*c) + 572293120*I*a^5*e^(13*I*d*x + 5*I*c
) + 602341376*I*a^5*e^(11*I*d*x + 3*I*c) + 472096768*I*a^5*e^(9*I*d*x + I*c) + 267091968*I*a^5*e^(7*I*d*x - I*
c) + 102875136*I*a^5*e^(5*I*d*x - 3*I*c) + 24084480*I*a^5*e^(3*I*d*x - 5*I*c) + 2580480*I*a^5*e^(I*d*x - 7*I*c
))/(d*e^(16*I*d*x + 8*I*c) + 8*d*e^(14*I*d*x + 6*I*c) + 28*d*e^(12*I*d*x + 4*I*c) + 56*d*e^(10*I*d*x + 2*I*c)
+ 56*d*e^(6*I*d*x - 2*I*c) + 28*d*e^(4*I*d*x - 4*I*c) + 8*d*e^(2*I*d*x - 6*I*c) + 70*d*e^(8*I*d*x) + d*e^(-8*I
*c))

Mupad [B] (verification not implemented)

Time = 4.86 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.56 \[ \int \cos ^9(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {a^5\,\left (\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{16}+\frac {{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,1{}\mathrm {i}}{12}+\frac {{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,3{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,1{}\mathrm {i}}{28}+\frac {{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{144}\right )}{d} \]

[In]

int(cos(c + d*x)^9*(a + a*tan(c + d*x)*1i)^5,x)

[Out]

-(a^5*((exp(c*1i + d*x*1i)*1i)/16 + (exp(c*3i + d*x*3i)*1i)/12 + (exp(c*5i + d*x*5i)*3i)/40 + (exp(c*7i + d*x*
7i)*1i)/28 + (exp(c*9i + d*x*9i)*1i)/144))/d