Integrand size = 32, antiderivative size = 162 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=-\frac {(15 i A-B) x}{16 a^4}+\frac {A \log (\sin (c+d x))}{a^4 d}+\frac {7 A+i B}{16 a^4 d (1+i \tan (c+d x))^2}+\frac {15 A+i B}{16 a^4 d (1+i \tan (c+d x))}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}+\frac {3 A+i B}{12 a d (a+i a \tan (c+d x))^3} \]
-1/16*(15*I*A-B)*x/a^4+A*ln(sin(d*x+c))/a^4/d+1/16*(7*A+I*B)/a^4/d/(1+I*ta n(d*x+c))^2+1/16*(15*A+I*B)/a^4/d/(1+I*tan(d*x+c))+1/8*(A+I*B)/d/(a+I*a*ta n(d*x+c))^4+1/12*(3*A+I*B)/a/d/(a+I*a*tan(d*x+c))^3
Time = 1.72 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.94 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=\frac {-3 (31 A+i B) \log (i-\tan (c+d x))+96 A \log (\tan (c+d x))-3 (A-i B) \log (i+\tan (c+d x))+\frac {12 (A+i B)}{(-i+\tan (c+d x))^4}+\frac {24 i A-8 B}{(-i+\tan (c+d x))^3}-\frac {6 (7 A+i B)}{(-i+\tan (c+d x))^2}+\frac {6 (-15 i A+B)}{-i+\tan (c+d x)}}{96 a^4 d} \]
(-3*(31*A + I*B)*Log[I - Tan[c + d*x]] + 96*A*Log[Tan[c + d*x]] - 3*(A - I *B)*Log[I + Tan[c + d*x]] + (12*(A + I*B))/(-I + Tan[c + d*x])^4 + ((24*I) *A - 8*B)/(-I + Tan[c + d*x])^3 - (6*(7*A + I*B))/(-I + Tan[c + d*x])^2 + (6*((-15*I)*A + B))/(-I + Tan[c + d*x]))/(96*a^4*d)
Time = 1.17 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.18, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4079, 27, 3042, 4079, 27, 3042, 4079, 27, 3042, 4079, 3042, 4014, 3042, 25, 3956}
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.
| \(\displaystyle \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x) (a+i a \tan (c+d x))^4}dx\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\int \frac {4 \cot (c+d x) (2 a A-a (i A-B) \tan (c+d x))}{(i \tan (c+d x) a+a)^3}dx}{8 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot (c+d x) (2 a A-a (i A-B) \tan (c+d x))}{(i \tan (c+d x) a+a)^3}dx}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a A-a (i A-B) \tan (c+d x)}{\tan (c+d x) (i \tan (c+d x) a+a)^3}dx}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int \frac {3 \cot (c+d x) \left (4 a^2 A-a^2 (3 i A-B) \tan (c+d x)\right )}{(i \tan (c+d x) a+a)^2}dx}{6 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\cot (c+d x) \left (4 a^2 A-a^2 (3 i A-B) \tan (c+d x)\right )}{(i \tan (c+d x) a+a)^2}dx}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {4 a^2 A-a^2 (3 i A-B) \tan (c+d x)}{\tan (c+d x) (i \tan (c+d x) a+a)^2}dx}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 \cot (c+d x) \left (8 a^3 A-a^3 (7 i A-B) \tan (c+d x)\right )}{i \tan (c+d x) a+a}dx}{4 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cot (c+d x) \left (8 a^3 A-a^3 (7 i A-B) \tan (c+d x)\right )}{i \tan (c+d x) a+a}dx}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {8 a^3 A-a^3 (7 i A-B) \tan (c+d x)}{\tan (c+d x) (i \tan (c+d x) a+a)}dx}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \cot (c+d x) \left (16 a^4 A-a^4 (15 i A-B) \tan (c+d x)\right )dx}{2 a^2}+\frac {a^3 (15 A+i B)}{2 d (a+i a \tan (c+d x))}}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {16 a^4 A-a^4 (15 i A-B) \tan (c+d x)}{\tan (c+d x)}dx}{2 a^2}+\frac {a^3 (15 A+i B)}{2 d (a+i a \tan (c+d x))}}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {\frac {\frac {16 a^4 A \int \cot (c+d x)dx-a^4 x (-B+15 i A)}{2 a^2}+\frac {a^3 (15 A+i B)}{2 d (a+i a \tan (c+d x))}}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {16 a^4 A \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-a^4 x (-B+15 i A)}{2 a^2}+\frac {a^3 (15 A+i B)}{2 d (a+i a \tan (c+d x))}}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {-16 a^4 A \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\left (a^4 x (-B+15 i A)\right )}{2 a^2}+\frac {a^3 (15 A+i B)}{2 d (a+i a \tan (c+d x))}}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 (15 A+i B)}{2 d (a+i a \tan (c+d x))}+\frac {\frac {16 a^4 A \log (-\sin (c+d x))}{d}-a^4 x (-B+15 i A)}{2 a^2}}{2 a^2}+\frac {7 A+i B}{4 d (1+i \tan (c+d x))^2}}{2 a^2}+\frac {a (3 A+i B)}{6 d (a+i a \tan (c+d x))^3}}{2 a^2}+\frac {A+i B}{8 d (a+i a \tan (c+d x))^4}\) |
(A + I*B)/(8*d*(a + I*a*Tan[c + d*x])^4) + ((a*(3*A + I*B))/(6*d*(a + I*a* Tan[c + d*x])^3) + ((7*A + I*B)/(4*d*(1 + I*Tan[c + d*x])^2) + ((-(a^4*((1 5*I)*A - B)*x) + (16*a^4*A*Log[-Sin[c + d*x]])/d)/(2*a^2) + (a^3*(15*A + I *B))/(2*d*(a + I*a*Tan[c + d*x])))/(2*a^2))/(2*a^2))/(2*a^2)
3.1.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Time = 0.20 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(\frac {x B}{16 a^{4}}-\frac {31 i x A}{16 a^{4}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 d \,a^{4}}+\frac {13 \,{\mathrm e}^{-2 i \left (d x +c \right )} A}{16 d \,a^{4}}+\frac {3 i B \,{\mathrm e}^{-4 i \left (d x +c \right )}}{32 d \,a^{4}}+\frac {{\mathrm e}^{-4 i \left (d x +c \right )} A}{4 d \,a^{4}}+\frac {i {\mathrm e}^{-6 i \left (d x +c \right )} B}{24 d \,a^{4}}+\frac {{\mathrm e}^{-6 i \left (d x +c \right )} A}{16 d \,a^{4}}+\frac {i {\mathrm e}^{-8 i \left (d x +c \right )} B}{128 d \,a^{4}}+\frac {{\mathrm e}^{-8 i \left (d x +c \right )} A}{128 d \,a^{4}}-\frac {2 i A c}{a^{4} d}+\frac {A \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}\) | \(196\) |
| derivativedivides | \(-\frac {i B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {i B}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {B \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {A \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{4}}-\frac {15 i A \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {7 A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {15 i A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {i A}{4 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {B}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {A}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}\) | \(234\) |
| default | \(-\frac {i B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {B}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {i B}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {B \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {A \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{4}}-\frac {15 i A \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{4}}-\frac {7 A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {15 i A}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}+\frac {i A}{4 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {B}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {A}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}\) | \(234\) |
1/16*x/a^4*B-31/16*I*x/a^4*A+1/8*I/d/a^4*exp(-2*I*(d*x+c))*B+13/16/d/a^4*e xp(-2*I*(d*x+c))*A+3/32*I*B/d/a^4*exp(-4*I*(d*x+c))+1/4/d/a^4*exp(-4*I*(d* x+c))*A+1/24*I/d/a^4*exp(-6*I*(d*x+c))*B+1/16/d/a^4*exp(-6*I*(d*x+c))*A+1/ 128*I/d/a^4*exp(-8*I*(d*x+c))*B+1/128/d/a^4*exp(-8*I*(d*x+c))*A-2*I/a^4*A/ d*c+1/a^4*A/d*ln(exp(2*I*(d*x+c))-1)
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.75 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=-\frac {{\left (24 \, {\left (31 i \, A - B\right )} d x e^{\left (8 i \, d x + 8 i \, c\right )} - 384 \, A e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 24 \, {\left (13 \, A + 2 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - 12 \, {\left (8 \, A + 3 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 8 \, {\left (3 \, A + 2 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 \, A - 3 i \, B\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{384 \, a^{4} d} \]
-1/384*(24*(31*I*A - B)*d*x*e^(8*I*d*x + 8*I*c) - 384*A*e^(8*I*d*x + 8*I*c )*log(e^(2*I*d*x + 2*I*c) - 1) - 24*(13*A + 2*I*B)*e^(6*I*d*x + 6*I*c) - 1 2*(8*A + 3*I*B)*e^(4*I*d*x + 4*I*c) - 8*(3*A + 2*I*B)*e^(2*I*d*x + 2*I*c) - 3*A - 3*I*B)*e^(-8*I*d*x - 8*I*c)/(a^4*d)
Time = 0.45 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.22 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=\frac {A \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{4} d} + \begin {cases} \frac {\left (\left (24576 A a^{12} d^{3} e^{12 i c} + 24576 i B a^{12} d^{3} e^{12 i c}\right ) e^{- 8 i d x} + \left (196608 A a^{12} d^{3} e^{14 i c} + 131072 i B a^{12} d^{3} e^{14 i c}\right ) e^{- 6 i d x} + \left (786432 A a^{12} d^{3} e^{16 i c} + 294912 i B a^{12} d^{3} e^{16 i c}\right ) e^{- 4 i d x} + \left (2555904 A a^{12} d^{3} e^{18 i c} + 393216 i B a^{12} d^{3} e^{18 i c}\right ) e^{- 2 i d x}\right ) e^{- 20 i c}}{3145728 a^{16} d^{4}} & \text {for}\: a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (- \frac {- 31 i A + B}{16 a^{4}} + \frac {\left (- 31 i A e^{8 i c} - 26 i A e^{6 i c} - 16 i A e^{4 i c} - 6 i A e^{2 i c} - i A + B e^{8 i c} + 4 B e^{6 i c} + 6 B e^{4 i c} + 4 B e^{2 i c} + B\right ) e^{- 8 i c}}{16 a^{4}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (- 31 i A + B\right )}{16 a^{4}} \]
A*log(exp(2*I*d*x) - exp(-2*I*c))/(a**4*d) + Piecewise((((24576*A*a**12*d* *3*exp(12*I*c) + 24576*I*B*a**12*d**3*exp(12*I*c))*exp(-8*I*d*x) + (196608 *A*a**12*d**3*exp(14*I*c) + 131072*I*B*a**12*d**3*exp(14*I*c))*exp(-6*I*d* x) + (786432*A*a**12*d**3*exp(16*I*c) + 294912*I*B*a**12*d**3*exp(16*I*c)) *exp(-4*I*d*x) + (2555904*A*a**12*d**3*exp(18*I*c) + 393216*I*B*a**12*d**3 *exp(18*I*c))*exp(-2*I*d*x))*exp(-20*I*c)/(3145728*a**16*d**4), Ne(a**16*d **4*exp(20*I*c), 0)), (x*(-(-31*I*A + B)/(16*a**4) + (-31*I*A*exp(8*I*c) - 26*I*A*exp(6*I*c) - 16*I*A*exp(4*I*c) - 6*I*A*exp(2*I*c) - I*A + B*exp(8* I*c) + 4*B*exp(6*I*c) + 6*B*exp(4*I*c) + 4*B*exp(2*I*c) + B)*exp(-8*I*c)/( 16*a**4)), True)) + x*(-31*I*A + B)/(16*a**4)
Exception generated. \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=\text {Exception raised: RuntimeError} \]
Time = 0.91 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.02 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=-\frac {\frac {12 \, {\left (A - i \, B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} + \frac {12 \, {\left (31 \, A + i \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} - \frac {384 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{4}} - \frac {775 \, A \tan \left (d x + c\right )^{4} + 25 i \, B \tan \left (d x + c\right )^{4} - 3460 i \, A \tan \left (d x + c\right )^{3} + 124 \, B \tan \left (d x + c\right )^{3} - 5898 \, A \tan \left (d x + c\right )^{2} - 246 i \, B \tan \left (d x + c\right )^{2} + 4612 i \, A \tan \left (d x + c\right ) - 252 \, B \tan \left (d x + c\right ) + 1447 \, A + 153 i \, B}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \]
-1/384*(12*(A - I*B)*log(tan(d*x + c) + I)/a^4 + 12*(31*A + I*B)*log(tan(d *x + c) - I)/a^4 - 384*A*log(tan(d*x + c))/a^4 - (775*A*tan(d*x + c)^4 + 2 5*I*B*tan(d*x + c)^4 - 3460*I*A*tan(d*x + c)^3 + 124*B*tan(d*x + c)^3 - 58 98*A*tan(d*x + c)^2 - 246*I*B*tan(d*x + c)^2 + 4612*I*A*tan(d*x + c) - 252 *B*tan(d*x + c) + 1447*A + 153*I*B)/(a^4*(tan(d*x + c) - I)^4))/d
Time = 7.30 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.21 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx=\frac {\frac {7\,A}{4\,a^4}-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-\frac {B}{16\,a^4}+\frac {A\,15{}\mathrm {i}}{16\,a^4}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {13\,A}{4\,a^4}+\frac {B\,1{}\mathrm {i}}{4\,a^4}\right )+\frac {B\,1{}\mathrm {i}}{3\,a^4}+\mathrm {tan}\left (c+d\,x\right )\,\left (-\frac {19\,B}{48\,a^4}+\frac {A\,63{}\mathrm {i}}{16\,a^4}\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,4{}\mathrm {i}+1\right )}+\frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (31\,A+B\,1{}\mathrm {i}\right )}{32\,a^4\,d} \]
((7*A)/(4*a^4) - tan(c + d*x)^3*((A*15i)/(16*a^4) - B/(16*a^4)) - tan(c + d*x)^2*((13*A)/(4*a^4) + (B*1i)/(4*a^4)) + (B*1i)/(3*a^4) + tan(c + d*x)*( (A*63i)/(16*a^4) - (19*B)/(48*a^4)))/(d*(tan(c + d*x)*4i - 6*tan(c + d*x)^ 2 - tan(c + d*x)^3*4i + tan(c + d*x)^4 + 1)) + (A*log(tan(c + d*x)))/(a^4* d) + (log(tan(c + d*x) + 1i)*(A*1i + B)*1i)/(32*a^4*d) - (log(tan(c + d*x) - 1i)*(31*A + B*1i))/(32*a^4*d)