Integrand size = 22, antiderivative size = 130 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {35 a^5 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {35 i a^5 \sec (c+d x)}{2 d}-\frac {7 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}-\frac {35 i \sec (c+d x) \left (a^5+i a^5 \tan (c+d x)\right )}{6 d} \]
-35/2*a^5*arctanh(sin(d*x+c))/d-35/2*I*a^5*sec(d*x+c)/d-7/3*I*a^3*sec(d*x+ c)*(a+I*a*tan(d*x+c))^2/d-2*I*a*cos(d*x+c)*(a+I*a*tan(d*x+c))^4/d-35/6*I*s ec(d*x+c)*(a^5+I*a^5*tan(d*x+c))/d
Time = 1.78 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 \cos ^2(c+d x) \left (-840 i \text {arctanh}\left (\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )\right ) \cos ^3(c+d x) (\cos (5 c)-i \sin (5 c))+(\cos (4 c-d x)-i \sin (4 c-d x)) (511 \cos (c+d x)+153 \cos (3 (c+d x))-i (49 \sin (c+d x)+57 \sin (3 (c+d x))))\right ) (-i+\tan (c+d x))^5}{24 d (\cos (d x)+i \sin (d x))^5} \]
(a^5*Cos[c + d*x]^2*((-840*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*Cos[c + d*x]^3*(Cos[5*c] - I*Sin[5*c]) + (Cos[4*c - d*x] - I*Sin[4*c - d*x])*(51 1*Cos[c + d*x] + 153*Cos[3*(c + d*x)] - I*(49*Sin[c + d*x] + 57*Sin[3*(c + d*x)])))*(-I + Tan[c + d*x])^5)/(24*d*(Cos[d*x] + I*Sin[d*x])^5)
Time = 0.64 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3977, 3042, 3979, 3042, 3979, 3042, 3967, 3042, 4257}
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.
| \(\displaystyle \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^5}{\sec (c+d x)}dx\) |
| \(\Big \downarrow \) 3977 |
| \(\displaystyle -7 a^2 \int \sec (c+d x) (i \tan (c+d x) a+a)^3dx-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -7 a^2 \int \sec (c+d x) (i \tan (c+d x) a+a)^3dx-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle -7 a^2 \left (\frac {5}{3} a \int \sec (c+d x) (i \tan (c+d x) a+a)^2dx+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}\right )-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -7 a^2 \left (\frac {5}{3} a \int \sec (c+d x) (i \tan (c+d x) a+a)^2dx+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}\right )-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3979 |
| \(\displaystyle -7 a^2 \left (\frac {5}{3} a \left (\frac {3}{2} a \int \sec (c+d x) (i \tan (c+d x) a+a)dx+\frac {i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}\right )+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}\right )-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -7 a^2 \left (\frac {5}{3} a \left (\frac {3}{2} a \int \sec (c+d x) (i \tan (c+d x) a+a)dx+\frac {i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}\right )+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}\right )-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle -7 a^2 \left (\frac {5}{3} a \left (\frac {3}{2} a \left (a \int \sec (c+d x)dx+\frac {i a \sec (c+d x)}{d}\right )+\frac {i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}\right )+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}\right )-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -7 a^2 \left (\frac {5}{3} a \left (\frac {3}{2} a \left (a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {i a \sec (c+d x)}{d}\right )+\frac {i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}\right )+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}\right )-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -7 a^2 \left (\frac {5}{3} a \left (\frac {i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{2 d}+\frac {3}{2} a \left (\frac {a \text {arctanh}(\sin (c+d x))}{d}+\frac {i a \sec (c+d x)}{d}\right )\right )+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^2}{3 d}\right )-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^4}{d}\) |
((-2*I)*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^4)/d - 7*a^2*(((I/3)*a*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^2)/d + (5*a*((3*a*((a*ArcTanh[Sin[c + d*x]] )/d + (I*a*Sec[c + d*x])/d))/2 + ((I/2)*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x]))/d))/3)
3.1.71.3.1 Defintions of rubi rules used
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] + Simp[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])
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] - Simp[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]) || (ILtQ[m, 0] & & LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]
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 - 1)/(f*(m + n - 1))), x] + Simp[a*((m + 2*n - 2)/(m + n - 1)) Int[(d*Se c[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ [2*m, 2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 7.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(-\frac {16 i a^{5} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{5} \left (87 \,{\mathrm e}^{5 i \left (d x +c \right )}+136 \,{\mathrm e}^{3 i \left (d x +c \right )}+57 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {35 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {35 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(118\) |
| derivativedivides | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-10 a^{5} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-5 i a^{5} \cos \left (d x +c \right )+a^{5} \sin \left (d x +c \right )}{d}\) | \(226\) |
| default | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-10 a^{5} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-5 i a^{5} \cos \left (d x +c \right )+a^{5} \sin \left (d x +c \right )}{d}\) | \(226\) |
-16*I/d*a^5*exp(I*(d*x+c))-1/3*I*a^5/d/(exp(2*I*(d*x+c))+1)^3*(87*exp(5*I* (d*x+c))+136*exp(3*I*(d*x+c))+57*exp(I*(d*x+c)))+35/2/d*a^5*ln(exp(I*(d*x+ c))-I)-35/2/d*a^5*ln(exp(I*(d*x+c))+I)
Time = 0.24 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.66 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {-96 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 462 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 560 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 210 i \, a^{5} e^{\left (i \, d x + i \, c\right )} - 105 \, {\left (a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 105 \, {\left (a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{5}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{6 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
1/6*(-96*I*a^5*e^(7*I*d*x + 7*I*c) - 462*I*a^5*e^(5*I*d*x + 5*I*c) - 560*I *a^5*e^(3*I*d*x + 3*I*c) - 210*I*a^5*e^(I*d*x + I*c) - 105*(a^5*e^(6*I*d*x + 6*I*c) + 3*a^5*e^(4*I*d*x + 4*I*c) + 3*a^5*e^(2*I*d*x + 2*I*c) + a^5)*l og(e^(I*d*x + I*c) + I) + 105*(a^5*e^(6*I*d*x + 6*I*c) + 3*a^5*e^(4*I*d*x + 4*I*c) + 3*a^5*e^(2*I*d*x + 2*I*c) + a^5)*log(e^(I*d*x + I*c) - I))/(d*e ^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d )
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.52 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {35 a^{5} \left (\frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{2} - \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{2}\right )}{d} + \frac {- 87 i a^{5} e^{5 i c} e^{5 i d x} - 136 i a^{5} e^{3 i c} e^{3 i d x} - 57 i a^{5} e^{i c} e^{i d x}}{3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} + 3 d} + \begin {cases} - \frac {16 i a^{5} e^{i c} e^{i d x}}{d} & \text {for}\: d \neq 0 \\16 a^{5} x e^{i c} & \text {otherwise} \end {cases} \]
35*a**5*(log(exp(I*d*x) - I*exp(-I*c))/2 - log(exp(I*d*x) + I*exp(-I*c))/2 )/d + (-87*I*a**5*exp(5*I*c)*exp(5*I*d*x) - 136*I*a**5*exp(3*I*c)*exp(3*I* d*x) - 57*I*a**5*exp(I*c)*exp(I*d*x))/(3*d*exp(6*I*c)*exp(6*I*d*x) + 9*d*e xp(4*I*c)*exp(4*I*d*x) + 9*d*exp(2*I*c)*exp(2*I*d*x) + 3*d) + Piecewise((- 16*I*a**5*exp(I*c)*exp(I*d*x)/d, Ne(d, 0)), (16*a**5*x*exp(I*c), True))
Time = 0.32 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.33 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {15 \, a^{5} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} + 120 i \, a^{5} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 4 i \, a^{5} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} + 60 \, a^{5} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 60 i \, a^{5} \cos \left (d x + c\right ) - 12 \, a^{5} \sin \left (d x + c\right )}{12 \, d} \]
-1/12*(15*a^5*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) + 3*log(sin(d*x + c) + 1) - 3*log(sin(d*x + c) - 1) - 4*sin(d*x + c)) + 120*I*a^5*(1/cos(d*x + c) + cos(d*x + c)) + 4*I*a^5*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos( d*x + c)) + 60*a^5*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1) - 2*sin( d*x + c)) + 60*I*a^5*cos(d*x + c) - 12*a^5*sin(d*x + c))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (108) = 216\).
Time = 0.81 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.92 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {8295 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 24885 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 24885 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 18585 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 55755 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 55755 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 8295 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 24885 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 24885 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 18585 \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 55755 \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 55755 \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 24576 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 118272 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 143360 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - 53760 i \, a^{5} e^{\left (i \, d x + i \, c\right )} + 8295 \, a^{5} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 18585 \, a^{5} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 8295 \, a^{5} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 18585 \, a^{5} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right )}{1536 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
1/1536*(8295*a^5*e^(6*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 24885*a^ 5*e^(4*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 24885*a^5*e^(2*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) - 18585*a^5*e^(6*I*d*x + 6*I*c)*log(I*e^ (I*d*x + I*c) - 1) - 55755*a^5*e^(4*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) - 55755*a^5*e^(2*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) - 8295*a^5* e^(6*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 24885*a^5*e^(4*I*d*x + 4 *I*c)*log(-I*e^(I*d*x + I*c) + 1) - 24885*a^5*e^(2*I*d*x + 2*I*c)*log(-I*e ^(I*d*x + I*c) + 1) + 18585*a^5*e^(6*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) + 55755*a^5*e^(4*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) + 55755* a^5*e^(2*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 24576*I*a^5*e^(7*I*d *x + 7*I*c) - 118272*I*a^5*e^(5*I*d*x + 5*I*c) - 143360*I*a^5*e^(3*I*d*x + 3*I*c) - 53760*I*a^5*e^(I*d*x + I*c) + 8295*a^5*log(I*e^(I*d*x + I*c) + 1 ) - 18585*a^5*log(I*e^(I*d*x + I*c) - 1) - 8295*a^5*log(-I*e^(I*d*x + I*c) + 1) + 18585*a^5*log(-I*e^(I*d*x + I*c) - 1))/(d*e^(6*I*d*x + 6*I*c) + 3* d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)
Time = 8.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.71 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {35\,a^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {37\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,27{}\mathrm {i}-118\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,48{}\mathrm {i}+139\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,55{}\mathrm {i}}{3}-\frac {166\,a^5}{3}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,1{}\mathrm {i}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,3{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
- (35*a^5*atanh(tan(c/2 + (d*x)/2)))/d - (139*a^5*tan(c/2 + (d*x)/2)^2 - a ^5*tan(c/2 + (d*x)/2)^3*48i - 118*a^5*tan(c/2 + (d*x)/2)^4 + a^5*tan(c/2 + (d*x)/2)^5*27i + 37*a^5*tan(c/2 + (d*x)/2)^6 - (166*a^5)/3 + (a^5*tan(c/2 + (d*x)/2)*55i)/3)/(d*(tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2*3i - 3*t an(c/2 + (d*x)/2)^3 + tan(c/2 + (d*x)/2)^4*3i + 3*tan(c/2 + (d*x)/2)^5 - t an(c/2 + (d*x)/2)^6*1i - tan(c/2 + (d*x)/2)^7 + 1i))