Integrand size = 26, antiderivative size = 140 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=8 a^4 (A-i B) x-\frac {8 a^4 (i A+B) \log (\cos (c+d x))}{d}-\frac {4 a^4 (A-i B) \tan (c+d x)}{d}+\frac {a (i A+B) (a+i a \tan (c+d x))^3}{3 d}+\frac {B (a+i a \tan (c+d x))^4}{4 d}+\frac {(i A+B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \]
8*a^4*(A-I*B)*x-8*a^4*(I*A+B)*ln(cos(d*x+c))/d-4*a^4*(A-I*B)*tan(d*x+c)/d+ 1/3*a*(I*A+B)*(a+I*a*tan(d*x+c))^3/d+1/4*B*(a+I*a*tan(d*x+c))^4/d+(I*A+B)* (a^2+I*a^2*tan(d*x+c))^2/d
Time = 1.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.61 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {B (a+i a \tan (c+d x))^4+\frac {4}{3} a^4 (i A+B) \left (4+24 \log (i+\tan (c+d x))+21 i \tan (c+d x)-6 \tan ^2(c+d x)-i \tan ^3(c+d x)\right )}{4 d} \]
(B*(a + I*a*Tan[c + d*x])^4 + (4*a^4*(I*A + B)*(4 + 24*Log[I + Tan[c + d*x ]] + (21*I)*Tan[c + d*x] - 6*Tan[c + d*x]^2 - I*Tan[c + d*x]^3))/3)/(4*d)
Time = 0.56 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 4010, 3042, 3959, 3042, 3959, 3042, 3958, 3042, 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.
| \(\displaystyle \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4010 |
| \(\displaystyle (A-i B) \int (i \tan (c+d x) a+a)^4dx+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-i B) \int (i \tan (c+d x) a+a)^4dx+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle (A-i B) \left (2 a \int (i \tan (c+d x) a+a)^3dx+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-i B) \left (2 a \int (i \tan (c+d x) a+a)^3dx+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3959 |
| \(\displaystyle (A-i B) \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^2dx+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-i B) \left (2 a \left (2 a \int (i \tan (c+d x) a+a)^2dx+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3958 |
| \(\displaystyle (A-i B) \left (2 a \left (2 a \left (2 i a^2 \int \tan (c+d x)dx-\frac {a^2 \tan (c+d x)}{d}+2 a^2 x\right )+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle (A-i B) \left (2 a \left (2 a \left (2 i a^2 \int \tan (c+d x)dx-\frac {a^2 \tan (c+d x)}{d}+2 a^2 x\right )+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle (A-i B) \left (2 a \left (2 a \left (-\frac {a^2 \tan (c+d x)}{d}-\frac {2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x\right )+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\right )+\frac {i a (a+i a \tan (c+d x))^3}{3 d}\right )+\frac {B (a+i a \tan (c+d x))^4}{4 d}\) |
(B*(a + I*a*Tan[c + d*x])^4)/(4*d) + (A - I*B)*(((I/3)*a*(a + I*a*Tan[c + d*x])^3)/d + 2*a*(((I/2)*a*(a + I*a*Tan[c + d*x])^2)/d + 2*a*(2*a^2*x - (( 2*I)*a^2*Log[Cos[c + d*x]])/d - (a^2*Tan[c + d*x])/d)))
3.1.28.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2) *x, x] + (Simp[b^2*(Tan[c + d*x]/d), x] + Simp[2*a*b Int[Tan[c + d*x], x] , x]) /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[2*a Int[(a + b*Tan[c + d* x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && GtQ[n , 1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Simp [(b*c + a*d)/b Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e , f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && !LtQ[m, 0]
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {a^{4} \left (-\frac {4 i B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {B \left (\tan ^{4}\left (d x +c \right )\right )}{4}-2 i A \left (\tan ^{2}\left (d x +c \right )\right )+\frac {A \left (\tan ^{3}\left (d x +c \right )\right )}{3}+8 i B \tan \left (d x +c \right )-\frac {7 B \left (\tan ^{2}\left (d x +c \right )\right )}{2}-7 A \tan \left (d x +c \right )+\frac {\left (8 i A +8 B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-8 i B +8 A \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(122\) |
| default | \(\frac {a^{4} \left (-\frac {4 i B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {B \left (\tan ^{4}\left (d x +c \right )\right )}{4}-2 i A \left (\tan ^{2}\left (d x +c \right )\right )+\frac {A \left (\tan ^{3}\left (d x +c \right )\right )}{3}+8 i B \tan \left (d x +c \right )-\frac {7 B \left (\tan ^{2}\left (d x +c \right )\right )}{2}-7 A \tan \left (d x +c \right )+\frac {\left (8 i A +8 B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-8 i B +8 A \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(122\) |
| norman | \(\left (-8 i B \,a^{4}+8 A \,a^{4}\right ) x -\frac {\left (4 i A \,a^{4}+7 B \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {\left (-8 i B \,a^{4}+7 A \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {B \,a^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {4 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(142\) |
| parallelrisch | \(\frac {-16 i B \left (\tan ^{3}\left (d x +c \right )\right ) a^{4}+3 B \left (\tan ^{4}\left (d x +c \right )\right ) a^{4}-24 i A \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}+4 A \left (\tan ^{3}\left (d x +c \right )\right ) a^{4}-96 i B x \,a^{4} d +48 i A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}+96 A x \,a^{4} d +96 i B \tan \left (d x +c \right ) a^{4}-42 B \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}-84 A \tan \left (d x +c \right ) a^{4}+48 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}}{12 d}\) | \(156\) |
| risch | \(\frac {16 i a^{4} B c}{d}-\frac {16 a^{4} A c}{d}-\frac {4 a^{4} \left (18 i A \,{\mathrm e}^{6 i \left (d x +c \right )}+30 B \,{\mathrm e}^{6 i \left (d x +c \right )}+45 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+63 B \,{\mathrm e}^{4 i \left (d x +c \right )}+38 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+50 B \,{\mathrm e}^{2 i \left (d x +c \right )}+11 i A +14 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {8 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}-\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}\) | \(170\) |
| parts | \(A \,a^{4} x +\frac {\left (-4 i A \,a^{4}-6 B \,a^{4}\right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {\left (4 i B \,a^{4}-6 A \,a^{4}\right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {B \,a^{4} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(198\) |
1/d*a^4*(-4/3*I*B*tan(d*x+c)^3+1/4*B*tan(d*x+c)^4-2*I*A*tan(d*x+c)^2+1/3*A *tan(d*x+c)^3+8*I*B*tan(d*x+c)-7/2*B*tan(d*x+c)^2-7*A*tan(d*x+c)+1/2*(8*B+ 8*I*A)*ln(1+tan(d*x+c)^2)+(8*A-8*I*B)*arctan(tan(d*x+c)))
Time = 0.25 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.62 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (6 \, {\left (3 i \, A + 5 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 9 \, {\left (5 i \, A + 7 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (19 i \, A + 25 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (11 i \, A + 14 \, B\right )} a^{4} + 6 \, {\left ({\left (i \, A + B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, {\left (i \, A + B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, {\left (i \, A + B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, {\left (i \, A + B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, A + B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-4/3*(6*(3*I*A + 5*B)*a^4*e^(6*I*d*x + 6*I*c) + 9*(5*I*A + 7*B)*a^4*e^(4*I *d*x + 4*I*c) + 2*(19*I*A + 25*B)*a^4*e^(2*I*d*x + 2*I*c) + (11*I*A + 14*B )*a^4 + 6*((I*A + B)*a^4*e^(8*I*d*x + 8*I*c) + 4*(I*A + B)*a^4*e^(6*I*d*x + 6*I*c) + 6*(I*A + B)*a^4*e^(4*I*d*x + 4*I*c) + 4*(I*A + B)*a^4*e^(2*I*d* x + 2*I*c) + (I*A + B)*a^4)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d* x + 2*I*c) + d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (116) = 232\).
Time = 0.44 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=- \frac {8 i a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 44 i A a^{4} - 56 B a^{4} + \left (- 152 i A a^{4} e^{2 i c} - 200 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 180 i A a^{4} e^{4 i c} - 252 B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (- 72 i A a^{4} e^{6 i c} - 120 B a^{4} e^{6 i c}\right ) e^{6 i d x}}{3 d e^{8 i c} e^{8 i d x} + 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} + 12 d e^{2 i c} e^{2 i d x} + 3 d} \]
-8*I*a**4*(A - I*B)*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-44*I*A*a**4 - 56 *B*a**4 + (-152*I*A*a**4*exp(2*I*c) - 200*B*a**4*exp(2*I*c))*exp(2*I*d*x) + (-180*I*A*a**4*exp(4*I*c) - 252*B*a**4*exp(4*I*c))*exp(4*I*d*x) + (-72*I *A*a**4*exp(6*I*c) - 120*B*a**4*exp(6*I*c))*exp(6*I*d*x))/(3*d*exp(8*I*c)* exp(8*I*d*x) + 12*d*exp(6*I*c)*exp(6*I*d*x) + 18*d*exp(4*I*c)*exp(4*I*d*x) + 12*d*exp(2*I*c)*exp(2*I*d*x) + 3*d)
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {3 \, B a^{4} \tan \left (d x + c\right )^{4} + 4 \, {\left (A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 6 \, {\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 96 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} - 48 \, {\left (-i \, A - B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, {\left (7 \, A - 8 i \, B\right )} a^{4} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(3*B*a^4*tan(d*x + c)^4 + 4*(A - 4*I*B)*a^4*tan(d*x + c)^3 - 6*(4*I*A + 7*B)*a^4*tan(d*x + c)^2 + 96*(d*x + c)*(A - I*B)*a^4 - 48*(-I*A - B)*a^ 4*log(tan(d*x + c)^2 + 1) - 12*(7*A - 8*I*B)*a^4*tan(d*x + c))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (120) = 240\).
Time = 0.54 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.91 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (6 i \, A a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 6 \, B a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 i \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 36 i \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 36 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 i \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 i \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 30 \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 45 i \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 63 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 38 i \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 50 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i \, A a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 6 \, B a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 11 i \, A a^{4} + 14 \, B a^{4}\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-4/3*(6*I*A*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 6*B*a^4 *e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 24*I*A*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 24*B*a^4*e^(6*I*d*x + 6*I*c)*log(e ^(2*I*d*x + 2*I*c) + 1) + 36*I*A*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 36*B*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 24*I*A*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 24*B*a^4*e^( 2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 18*I*A*a^4*e^(6*I*d*x + 6* I*c) + 30*B*a^4*e^(6*I*d*x + 6*I*c) + 45*I*A*a^4*e^(4*I*d*x + 4*I*c) + 63* B*a^4*e^(4*I*d*x + 4*I*c) + 38*I*A*a^4*e^(2*I*d*x + 2*I*c) + 50*B*a^4*e^(2 *I*d*x + 2*I*c) + 6*I*A*a^4*log(e^(2*I*d*x + 2*I*c) + 1) + 6*B*a^4*log(e^( 2*I*d*x + 2*I*c) + 1) + 11*I*A*a^4 + 14*B*a^4)/(d*e^(8*I*d*x + 8*I*c) + 4* d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*x + 2*I*c) + d)
Time = 7.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {B\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (8\,B\,a^4+A\,a^4\,8{}\mathrm {i}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4\,\left (A-B\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {B\,a^4}{2}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )}{2}\right )}{d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-3\,a^4\,\left (A-B\,1{}\mathrm {i}\right )+a^4\,\left (B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}+B\,a^4\,1{}\mathrm {i}+a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {B\,a^4\,1{}\mathrm {i}}{3}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}\right )}{d} \]
(log(tan(c + d*x) + 1i)*(A*a^4*8i + 8*B*a^4))/d - (tan(c + d*x)^3*((B*a^4* 1i)/3 + (a^4*(A*1i + 3*B)*1i)/3))/d - (tan(c + d*x)^2*((a^4*(A - B*1i)*3i) /2 + (B*a^4)/2 + (a^4*(A*1i + 3*B))/2))/d + (tan(c + d*x)*(a^4*(A*3i + B)* 1i - 3*a^4*(A - B*1i) + B*a^4*1i + a^4*(A*1i + 3*B)*1i))/d + (B*a^4*tan(c + d*x)^4)/(4*d)