\(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx\) [74]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {2 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{105 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d} \]

[Out]

-2/105*I*a^2*cos(d*x+c)^3*(a+I*a*tan(d*x+c))^3/d-2/35*I*a*cos(d*x+c)^5*(a+I*a*tan(d*x+c))^4/d-1/7*I*cos(d*x+c)
^7*(a+I*a*tan(d*x+c))^5/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3578, 3569} \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {2 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{105 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d} \]

[In]

Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(((-2*I)/105)*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d - (((2*I)/35)*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d
*x])^4)/d - ((I/7)*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5)/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 ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}+\frac {1}{7} (2 a) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx \\ & = -\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}+\frac {1}{35} \left (2 a^2\right ) \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx \\ & = -\frac {2 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{105 d}-\frac {2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d}-\frac {i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 \sec (c+d x) (-i \cos (4 (c+d x))+\sin (4 (c+d x))) \left (77+92 \cos (2 (c+d x))+\left (15+416 \sqrt {\cos ^2(c+d x)}\right ) \cos (4 (c+d x))+22 i \sin (2 (c+d x))+15 i \sin (4 (c+d x))-416 i \sqrt {\cos ^2(c+d x)} \sin (4 (c+d x))\right )}{840 d} \]

[In]

Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*Sec[c + d*x]*((-I)*Cos[4*(c + d*x)] + Sin[4*(c + d*x)])*(77 + 92*Cos[2*(c + d*x)] + (15 + 416*Sqrt[Cos[c
+ d*x]^2])*Cos[4*(c + d*x)] + (22*I)*Sin[2*(c + d*x)] + (15*I)*Sin[4*(c + d*x)] - (416*I)*Sqrt[Cos[c + d*x]^2]
*Sin[4*(c + d*x)]))/(840*d)

Maple [A] (verified)

Time = 212.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.55

method	result	size
risch	\(-\frac {i a^{5} {\mathrm e}^{7 i \left (d x +c \right )}}{28 d}-\frac {i a^{5} {\mathrm e}^{5 i \left (d x +c \right )}}{10 d}-\frac {i a^{5} {\mathrm e}^{3 i \left (d x +c \right )}}{12 d}\)	\(56\)
derivativedivides	\(\frac {i a^{5} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}\right )-10 i a^{5} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {5 i a^{5} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{5} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(257\)
default	\(\frac {i a^{5} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+5 a^{5} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}\right )-10 i a^{5} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-10 a^{5} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {5 i a^{5} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{5} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(257\)

[In]

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

-1/28*I/d*a^5*exp(7*I*(d*x+c))-1/10*I/d*a^5*exp(5*I*(d*x+c))-1/12*I/d*a^5*exp(3*I*(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.48 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {-15 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 42 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 35 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )}}{420 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")

[Out]

1/420*(-15*I*a^5*e^(7*I*d*x + 7*I*c) - 42*I*a^5*e^(5*I*d*x + 5*I*c) - 35*I*a^5*e^(3*I*d*x + 3*I*c))/d

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=\begin {cases} \frac {- 120 i a^{5} d^{2} e^{7 i c} e^{7 i d x} - 336 i a^{5} d^{2} e^{5 i c} e^{5 i d x} - 280 i a^{5} d^{2} e^{3 i c} e^{3 i d x}}{3360 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (\frac {a^{5} e^{7 i c}}{4} + \frac {a^{5} e^{5 i c}}{2} + \frac {a^{5} e^{3 i c}}{4}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**5,x)

[Out]

Piecewise(((-120*I*a**5*d**2*exp(7*I*c)*exp(7*I*d*x) - 336*I*a**5*d**2*exp(5*I*c)*exp(5*I*d*x) - 280*I*a**5*d*
*2*exp(3*I*c)*exp(3*I*d*x))/(3360*d**3), Ne(d**3, 0)), (x*(a**5*exp(7*I*c)/4 + a**5*exp(5*I*c)/2 + a**5*exp(3*
I*c)/4), 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. 187 vs. \(2 (83) = 166\).

Time = 0.41 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.85 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {75 i \, a^{5} \cos \left (d x + c\right )^{7} + i \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{5} + 30 i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{5} + 10 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{5} + 15 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{5} + 3 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{5}}{105 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/105*(75*I*a^5*cos(d*x + c)^7 + I*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^5 + 30*I*(5*
cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^5 + 10*(15*sin(d*x + c)^7 - 42*sin(d*x + c)^5 + 35*sin(d*x + c)^3)*a^5 +
15*(5*sin(d*x + c)^7 - 7*sin(d*x + c)^5)*a^5 + 3*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 3
5*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. 1697 vs. \(2 (83) = 166\).

Time = 0.94 (sec) , antiderivative size = 1697, normalized size of antiderivative = 16.80 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")

[Out]

-1/3440640*(7357770*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 58862160*a^5*e^(14*I*d*x + 6*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 206017560*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 412035120*a^5*e^(10
*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 412035120*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 20
6017560*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 58862160*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x +
I*c) + 1) + 515043900*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 7357770*a^5*e^(-8*I*c)*log(I*e^(I*d*x + I*
c) + 1) + 7390425*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 59123400*a^5*e^(14*I*d*x + 6*I*c)*log(
I*e^(I*d*x + I*c) - 1) + 206931900*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 413863800*a^5*e^(10*I
*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 413863800*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 2069
31900*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 59123400*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 517329750*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 7390425*a^5*e^(-8*I*c)*log(I*e^(I*d*x + I*c)
- 1) - 7357770*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 58862160*a^5*e^(14*I*d*x + 6*I*c)*log(-
I*e^(I*d*x + I*c) + 1) - 206017560*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 412035120*a^5*e^(10*
I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 412035120*a^5*e^(6*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2
06017560*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 58862160*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*
x + I*c) + 1) - 515043900*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 7357770*a^5*e^(-8*I*c)*log(-I*e^(I*d*x
+ I*c) + 1) - 7390425*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 59123400*a^5*e^(14*I*d*x + 6*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 206931900*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 413863800*a^5
*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 413863800*a^5*e^(6*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) -
1) - 206931900*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 59123400*a^5*e^(2*I*d*x - 6*I*c)*log(-I*
e^(I*d*x + I*c) - 1) - 517329750*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 7390425*a^5*e^(-8*I*c)*log(-I*e
^(I*d*x + I*c) - 1) + 32655*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 261240*a^5*e^(14*I*d*x + 6*
I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 914340*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 1828680*a^5*e
^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 1828680*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-I*c))
+ 914340*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 261240*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x)
+ e^(-I*c)) + 2285850*a^5*e^(8*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) + 32655*a^5*e^(-8*I*c)*log(I*e^(I*d*x) + e^(
-I*c)) - 32655*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 261240*a^5*e^(14*I*d*x + 6*I*c)*log(-I*
e^(I*d*x) + e^(-I*c)) - 914340*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 1828680*a^5*e^(10*I*d*x
+ 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 1828680*a^5*e^(6*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 914340
*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 261240*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-
I*c)) - 2285850*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) - 32655*a^5*e^(-8*I*c)*log(-I*e^(I*d*x) + e^(-I*c
)) + 122880*I*a^5*e^(23*I*d*x + 15*I*c) + 1327104*I*a^5*e^(21*I*d*x + 13*I*c) + 6479872*I*a^5*e^(19*I*d*x + 11
*I*c) + 18808832*I*a^5*e^(17*I*d*x + 9*I*c) + 35897344*I*a^5*e^(15*I*d*x + 7*I*c) + 47022080*I*a^5*e^(13*I*d*x
+ 5*I*c) + 42778624*I*a^5*e^(11*I*d*x + 3*I*c) + 26673152*I*a^5*e^(9*I*d*x + I*c) + 10903552*I*a^5*e^(7*I*d*x
- I*c) + 2637824*I*a^5*e^(5*I*d*x - 3*I*c) + 286720*I*a^5*e^(3*I*d*x - 5*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 = 4.71 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.84 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {2\,a^5\,\left (105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,210{}\mathrm {i}-455\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,350{}\mathrm {i}+273\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,56{}\mathrm {i}-23\right )}{105\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,7{}\mathrm {i}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,35{}\mathrm {i}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,21{}\mathrm {i}+7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]

[In]

int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^5,x)

[Out]

-(2*a^5*(tan(c/2 + (d*x)/2)*56i + 273*tan(c/2 + (d*x)/2)^2 - tan(c/2 + (d*x)/2)^3*350i - 455*tan(c/2 + (d*x)/2
)^4 + tan(c/2 + (d*x)/2)^5*210i + 105*tan(c/2 + (d*x)/2)^6 - 23))/(105*d*(7*tan(c/2 + (d*x)/2) - tan(c/2 + (d*
x)/2)^2*21i - 35*tan(c/2 + (d*x)/2)^3 + tan(c/2 + (d*x)/2)^4*35i + 21*tan(c/2 + (d*x)/2)^5 - tan(c/2 + (d*x)/2
)^6*7i - tan(c/2 + (d*x)/2)^7 + 1i))