\(\int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [27]

Optimal result

Integrand size = 32, antiderivative size = 168 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-8 a^4 (i A+B) x-\frac {8 a^4 (A-i B) \log (\cos (c+d x))}{d}+\frac {4 a^4 (i A+B) \tan (c+d x)}{d}+\frac {a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac {A (a+i a \tan (c+d x))^4}{4 d}-\frac {i B (a+i a \tan (c+d x))^5}{5 a d}+\frac {(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \]

[Out]

-8*a^4*(I*A+B)*x-8*a^4*(A-I*B)*ln(cos(d*x+c))/d+4*a^4*(I*A+B)*tan(d*x+c)/d+1/3*a*(A-I*B)*(a+I*a*tan(d*x+c))^3/
d+1/4*A*(a+I*a*tan(d*x+c))^4/d-1/5*I*B*(a+I*a*tan(d*x+c))^5/a/d+(A-I*B)*(a^2+I*a^2*tan(d*x+c))^2/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3673, 3608, 3559, 3558, 3556} \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {4 a^4 (B+i A) \tan (c+d x)}{d}-\frac {8 a^4 (A-i B) \log (\cos (c+d x))}{d}-8 a^4 x (B+i A)+\frac {(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac {a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac {A (a+i a \tan (c+d x))^4}{4 d}-\frac {i B (a+i a \tan (c+d x))^5}{5 a d} \]

[In]

Int[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

-8*a^4*(I*A + B)*x - (8*a^4*(A - I*B)*Log[Cos[c + d*x]])/d + (4*a^4*(I*A + B)*Tan[c + d*x])/d + (a*(A - I*B)*(
a + I*a*Tan[c + d*x])^3)/(3*d) + (A*(a + I*a*Tan[c + d*x])^4)/(4*d) - ((I/5)*B*(a + I*a*Tan[c + d*x])^5)/(a*d)
+ ((A - I*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/d

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3558

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2)*x, x] + (Dist[2*a*b, Int[Tan[c + d
*x], x], x] + Simp[b^2*(Tan[c + d*x]/d), x]) /; FreeQ[{a, b, c, d}, x]

Rule 3559

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Dist[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] && G
tQ[n, 1]

Rule 3608

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] + Dist[(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]

Rule 3673

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[B*d*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e
+ f*x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b
*c - a*d, 0] &&  !LeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {i B (a+i a \tan (c+d x))^5}{5 a d}+\int (a+i a \tan (c+d x))^4 (-B+A \tan (c+d x)) \, dx \\ & = \frac {A (a+i a \tan (c+d x))^4}{4 d}-\frac {i B (a+i a \tan (c+d x))^5}{5 a d}-(i A+B) \int (a+i a \tan (c+d x))^4 \, dx \\ & = \frac {a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac {A (a+i a \tan (c+d x))^4}{4 d}-\frac {i B (a+i a \tan (c+d x))^5}{5 a d}-(2 a (i A+B)) \int (a+i a \tan (c+d x))^3 \, dx \\ & = \frac {a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac {A (a+i a \tan (c+d x))^4}{4 d}-\frac {i B (a+i a \tan (c+d x))^5}{5 a d}+\frac {(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 a^2 (i A+B)\right ) \int (a+i a \tan (c+d x))^2 \, dx \\ & = -8 a^4 (i A+B) x+\frac {4 a^4 (i A+B) \tan (c+d x)}{d}+\frac {a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac {A (a+i a \tan (c+d x))^4}{4 d}-\frac {i B (a+i a \tan (c+d x))^5}{5 a d}+\frac {(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (8 a^4 (A-i B)\right ) \int \tan (c+d x) \, dx \\ & = -8 a^4 (i A+B) x-\frac {8 a^4 (A-i B) \log (\cos (c+d x))}{d}+\frac {4 a^4 (i A+B) \tan (c+d x)}{d}+\frac {a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac {A (a+i a \tan (c+d x))^4}{4 d}-\frac {i B (a+i a \tan (c+d x))^5}{5 a d}+\frac {(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.71 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 \left (95 A-92 i B+480 (A-i B) \log (i+\tan (c+d x))+480 (i A+B) \tan (c+d x)-30 (7 A-8 i B) \tan ^2(c+d x)+(-80 i A-140 B) \tan ^3(c+d x)+15 (A-4 i B) \tan ^4(c+d x)+12 B \tan ^5(c+d x)\right )}{60 d} \]

[In]

Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(a^4*(95*A - (92*I)*B + 480*(A - I*B)*Log[I + Tan[c + d*x]] + 480*(I*A + B)*Tan[c + d*x] - 30*(7*A - (8*I)*B)*
Tan[c + d*x]^2 + ((-80*I)*A - 140*B)*Tan[c + d*x]^3 + 15*(A - (4*I)*B)*Tan[c + d*x]^4 + 12*B*Tan[c + d*x]^5))/
(60*d)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {a^{4} \left (-i B \left (\tan ^{4}\left (d x +c \right )\right )+\frac {B \left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {4 i A \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {A \left (\tan ^{4}\left (d x +c \right )\right )}{4}+4 i B \left (\tan ^{2}\left (d x +c \right )\right )-\frac {7 B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+8 i A \tan \left (d x +c \right )-\frac {7 A \left (\tan ^{2}\left (d x +c \right )\right )}{2}+8 B \tan \left (d x +c \right )+\frac {\left (-8 i B +8 A \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-8 i A -8 B \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(145\)
default	\(\frac {a^{4} \left (-i B \left (\tan ^{4}\left (d x +c \right )\right )+\frac {B \left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {4 i A \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {A \left (\tan ^{4}\left (d x +c \right )\right )}{4}+4 i B \left (\tan ^{2}\left (d x +c \right )\right )-\frac {7 B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+8 i A \tan \left (d x +c \right )-\frac {7 A \left (\tan ^{2}\left (d x +c \right )\right )}{2}+8 B \tan \left (d x +c \right )+\frac {\left (-8 i B +8 A \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-8 i A -8 B \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(145\)
norman	\(\left (-8 i A \,a^{4}-8 B \,a^{4}\right ) x -\frac {\left (4 i A \,a^{4}+7 B \,a^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {\left (-8 i B \,a^{4}+7 A \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {8 \left (i A \,a^{4}+B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {B \,a^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {4 \left (-i B \,a^{4}+A \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(168\)
parallelrisch	\(-\frac {60 i B \left (\tan ^{4}\left (d x +c \right )\right ) a^{4}-12 B \,a^{4} \left (\tan ^{5}\left (d x +c \right )\right )+80 i A \left (\tan ^{3}\left (d x +c \right )\right ) a^{4}-15 A \left (\tan ^{4}\left (d x +c \right )\right ) a^{4}+480 i A x \,a^{4} d -240 i B \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}+140 B \left (\tan ^{3}\left (d x +c \right )\right ) a^{4}-480 i A \tan \left (d x +c \right ) a^{4}+210 A \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}+240 i B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}+480 B x \,a^{4} d -240 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}-480 B \tan \left (d x +c \right ) a^{4}}{60 d}\)	\(185\)
risch	\(\frac {16 a^{4} B c}{d}+\frac {16 i a^{4} A c}{d}+\frac {4 i a^{4} \left (150 i A \,{\mathrm e}^{8 i \left (d x +c \right )}+210 B \,{\mathrm e}^{8 i \left (d x +c \right )}+465 i A \,{\mathrm e}^{6 i \left (d x +c \right )}+555 B \,{\mathrm e}^{6 i \left (d x +c \right )}+565 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+655 B \,{\mathrm e}^{4 i \left (d x +c \right )}+320 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+365 B \,{\mathrm e}^{2 i \left (d x +c \right )}+70 i A +79 B \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}-\frac {8 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}\)	\(196\)
parts	\(\frac {\left (-4 i A \,a^{4}-6 B \,a^{4}\right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (4 i B \,a^{4}-6 A \,a^{4}\right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}}{2 d}+\frac {B \,a^{4} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(236\)

[In]

int(tan(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*a^4*(-I*B*tan(d*x+c)^4+1/5*B*tan(d*x+c)^5-4/3*I*A*tan(d*x+c)^3+1/4*A*tan(d*x+c)^4+4*I*B*tan(d*x+c)^2-7/3*B
*tan(d*x+c)^3+8*I*A*tan(d*x+c)-7/2*A*tan(d*x+c)^2+8*B*tan(d*x+c)+1/2*(8*A-8*I*B)*ln(1+tan(d*x+c)^2)+(-8*B-8*I*
A)*arctan(tan(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.66 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (30 \, {\left (5 \, A - 7 i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 15 \, {\left (31 \, A - 37 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 5 \, {\left (113 \, A - 131 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, {\left (64 \, A - 73 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (70 \, A - 79 i \, B\right )} a^{4} + 30 \, {\left ({\left (A - i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, {\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, {\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, {\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, {\left (A - i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (A - i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-4/15*(30*(5*A - 7*I*B)*a^4*e^(8*I*d*x + 8*I*c) + 15*(31*A - 37*I*B)*a^4*e^(6*I*d*x + 6*I*c) + 5*(113*A - 131*
I*B)*a^4*e^(4*I*d*x + 4*I*c) + 5*(64*A - 73*I*B)*a^4*e^(2*I*d*x + 2*I*c) + (70*A - 79*I*B)*a^4 + 30*((A - I*B)
*a^4*e^(10*I*d*x + 10*I*c) + 5*(A - I*B)*a^4*e^(8*I*d*x + 8*I*c) + 10*(A - I*B)*a^4*e^(6*I*d*x + 6*I*c) + 10*(
A - I*B)*a^4*e^(4*I*d*x + 4*I*c) + 5*(A - I*B)*a^4*e^(2*I*d*x + 2*I*c) + (A - I*B)*a^4)*log(e^(2*I*d*x + 2*I*c
) + 1))/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) + 10*d*e^(4*I*d*x + 4*I*
c) + 5*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (138) = 276\).

Time = 0.50 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.73 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=- \frac {8 a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 280 A a^{4} + 316 i B a^{4} + \left (- 1280 A a^{4} e^{2 i c} + 1460 i B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 2260 A a^{4} e^{4 i c} + 2620 i B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (- 1860 A a^{4} e^{6 i c} + 2220 i B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (- 600 A a^{4} e^{8 i c} + 840 i B a^{4} e^{8 i c}\right ) e^{8 i d x}}{15 d e^{10 i c} e^{10 i d x} + 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} + 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} + 15 d} \]

[In]

integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

-8*a**4*(A - I*B)*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-280*A*a**4 + 316*I*B*a**4 + (-1280*A*a**4*exp(2*I*c) +
1460*I*B*a**4*exp(2*I*c))*exp(2*I*d*x) + (-2260*A*a**4*exp(4*I*c) + 2620*I*B*a**4*exp(4*I*c))*exp(4*I*d*x) +
(-1860*A*a**4*exp(6*I*c) + 2220*I*B*a**4*exp(6*I*c))*exp(6*I*d*x) + (-600*A*a**4*exp(8*I*c) + 840*I*B*a**4*exp
(8*I*c))*exp(8*I*d*x))/(15*d*exp(10*I*c)*exp(10*I*d*x) + 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6
*I*d*x) + 150*d*exp(4*I*c)*exp(4*I*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) + 15*d)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {12 \, B a^{4} \tan \left (d x + c\right )^{5} + 15 \, {\left (A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} - 20 \, {\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 30 \, {\left (7 \, A - 8 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 480 \, {\left (d x + c\right )} {\left (i \, A + B\right )} a^{4} + 240 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 480 \, {\left (-i \, A - B\right )} a^{4} \tan \left (d x + c\right )}{60 \, d} \]

[In]

integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/60*(12*B*a^4*tan(d*x + c)^5 + 15*(A - 4*I*B)*a^4*tan(d*x + c)^4 - 20*(4*I*A + 7*B)*a^4*tan(d*x + c)^3 - 30*(
7*A - 8*I*B)*a^4*tan(d*x + c)^2 - 480*(d*x + c)*(I*A + B)*a^4 + 240*(A - I*B)*a^4*log(tan(d*x + c)^2 + 1) - 48
0*(-I*A - B)*a^4*tan(d*x + c))/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. 504 vs. \(2 (142) = 284\).

Time = 0.63 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.00 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (30 \, A a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 30 i \, B a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 150 \, 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 ) - 150 i \, 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 ) + 300 \, 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 ) - 300 i \, 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 ) + 300 \, 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 ) - 300 i \, 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 ) + 150 \, 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 ) - 150 i \, 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 ) + 150 \, A a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 210 i \, B a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 465 \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 555 i \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 565 \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 655 i \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 320 \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 365 i \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 30 \, A a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 30 i \, B a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 70 \, A a^{4} - 79 i \, B a^{4}\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate(tan(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-4/15*(30*A*a^4*e^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 30*I*B*a^4*e^(10*I*d*x + 10*I*c)*log(e^(2
*I*d*x + 2*I*c) + 1) + 150*A*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 150*I*B*a^4*e^(8*I*d*x + 8
*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 300*A*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 300*I*B*a^4*
e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 300*A*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1)
- 300*I*B*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 150*A*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x
+ 2*I*c) + 1) - 150*I*B*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 150*A*a^4*e^(8*I*d*x + 8*I*c) -
210*I*B*a^4*e^(8*I*d*x + 8*I*c) + 465*A*a^4*e^(6*I*d*x + 6*I*c) - 555*I*B*a^4*e^(6*I*d*x + 6*I*c) + 565*A*a^4
*e^(4*I*d*x + 4*I*c) - 655*I*B*a^4*e^(4*I*d*x + 4*I*c) + 320*A*a^4*e^(2*I*d*x + 2*I*c) - 365*I*B*a^4*e^(2*I*d*
x + 2*I*c) + 30*A*a^4*log(e^(2*I*d*x + 2*I*c) + 1) - 30*I*B*a^4*log(e^(2*I*d*x + 2*I*c) + 1) + 70*A*a^4 - 79*I
*B*a^4)/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) + 10*d*e^(4*I*d*x + 4*I*
c) + 5*d*e^(2*I*d*x + 2*I*c) + d)

Mupad [B] (verification not implemented)

Time = 7.50 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.43 \[ \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {3\,a^4\,\left (A-B\,1{}\mathrm {i}\right )}{2}+\frac {a^4\,\left (B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {B\,a^4\,1{}\mathrm {i}}{2}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}\right )}{d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A\,a^4\,1{}\mathrm {i}+a^4\,\left (A-B\,1{}\mathrm {i}\right )\,3{}\mathrm {i}+a^4\,\left (B+A\,3{}\mathrm {i}\right )+B\,a^4+a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {B\,a^4\,1{}\mathrm {i}}{4}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (8\,A\,a^4-B\,a^4\,8{}\mathrm {i}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a^4\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+\frac {B\,a^4}{3}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )}{3}\right )}{d}+\frac {B\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d} \]

[In]

int(tan(c + d*x)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

(tan(c + d*x)^2*((a^4*(A*3i + B)*1i)/2 - (3*a^4*(A - B*1i))/2 + (B*a^4*1i)/2 + (a^4*(A*1i + 3*B)*1i)/2))/d + (
tan(c + d*x)*(A*a^4*1i + a^4*(A - B*1i)*3i + a^4*(A*3i + B) + B*a^4 + a^4*(A*1i + 3*B)))/d - (tan(c + d*x)^4*(
(B*a^4*1i)/4 + (a^4*(A*1i + 3*B)*1i)/4))/d + (log(tan(c + d*x) + 1i)*(8*A*a^4 - B*a^4*8i))/d - (tan(c + d*x)^3
*(a^4*(A - B*1i)*1i + (B*a^4)/3 + (a^4*(A*1i + 3*B))/3))/d + (B*a^4*tan(c + d*x)^5)/(5*d)