\(\int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx\) [717]

Optimal result

Integrand size = 41, antiderivative size = 158 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=\frac {6 (A+3 i B) c^4 x}{a^2}+\frac {6 (i A-3 B) c^4 \log (\cos (e+f x))}{a^2 f}-\frac {4 (i A-B) c^4}{a^2 f (i-\tan (e+f x))^2}+\frac {4 (3 A+5 i B) c^4}{a^2 f (i-\tan (e+f x))}-\frac {(A+6 i B) c^4 \tan (e+f x)}{a^2 f}-\frac {B c^4 \tan ^2(e+f x)}{2 a^2 f} \]

[Out]

6*(A+3*I*B)*c^4*x/a^2+6*(I*A-3*B)*c^4*ln(cos(f*x+e))/a^2/f-4*(I*A-B)*c^4/a^2/f/(I-tan(f*x+e))^2+4*(3*A+5*I*B)*
c^4/a^2/f/(I-tan(f*x+e))-(A+6*I*B)*c^4*tan(f*x+e)/a^2/f-1/2*B*c^4*tan(f*x+e)^2/a^2/f

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 78} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=-\frac {c^4 (A+6 i B) \tan (e+f x)}{a^2 f}+\frac {4 c^4 (3 A+5 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac {4 c^4 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}+\frac {6 c^4 (-3 B+i A) \log (\cos (e+f x))}{a^2 f}+\frac {6 c^4 x (A+3 i B)}{a^2}-\frac {B c^4 \tan ^2(e+f x)}{2 a^2 f} \]

[In]

Int[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x])^2,x]

[Out]

(6*(A + (3*I)*B)*c^4*x)/a^2 + (6*(I*A - 3*B)*c^4*Log[Cos[e + f*x]])/(a^2*f) - (4*(I*A - B)*c^4)/(a^2*f*(I - Ta
n[e + f*x])^2) + (4*(3*A + (5*I)*B)*c^4)/(a^2*f*(I - Tan[e + f*x])) - ((A + (6*I)*B)*c^4*Tan[e + f*x])/(a^2*f)
- (B*c^4*Tan[e + f*x]^2)/(2*a^2*f)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 3669

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a*(c/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x
], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^3}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (-\frac {(A+6 i B) c^3}{a^3}-\frac {B c^3 x}{a^3}+\frac {8 i (A+i B) c^3}{a^3 (-i+x)^3}+\frac {4 (3 A+5 i B) c^3}{a^3 (-i+x)^2}+\frac {6 (-i A+3 B) c^3}{a^3 (-i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {6 (A+3 i B) c^4 x}{a^2}+\frac {6 (i A-3 B) c^4 \log (\cos (e+f x))}{a^2 f}-\frac {4 (i A-B) c^4}{a^2 f (i-\tan (e+f x))^2}+\frac {4 (3 A+5 i B) c^4}{a^2 f (i-\tan (e+f x))}-\frac {(A+6 i B) c^4 \tan (e+f x)}{a^2 f}-\frac {B c^4 \tan ^2(e+f x)}{2 a^2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=\frac {c^4 \left (\frac {2 (A+3 i B) (i+\tan (e+f x))^3}{(a+i a \tan (e+f x))^2}+\frac {B (i+\tan (e+f x))^4}{(a+i a \tan (e+f x))^2}+\frac {12 (-i A+3 B) \left (\log (i-\tan (e+f x))+\frac {-2-4 i \tan (e+f x)}{(-i+\tan (e+f x))^2}\right )}{a^2}\right )}{2 f} \]

[In]

Integrate[((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x])^2,x]

[Out]

(c^4*((2*(A + (3*I)*B)*(I + Tan[e + f*x])^3)/(a + I*a*Tan[e + f*x])^2 + (B*(I + Tan[e + f*x])^4)/(a + I*a*Tan[
e + f*x])^2 + (12*((-I)*A + 3*B)*(Log[I - Tan[e + f*x]] + (-2 - (4*I)*Tan[e + f*x])/(-I + Tan[e + f*x])^2))/a^
2))/(2*f)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.51

method	result	size
derivativedivides	\(-\frac {B \,c^{4} \tan \left (f x +e \right )^{2}}{2 a^{2} f}-\frac {c^{4} A \tan \left (f x +e \right )}{f \,a^{2}}-\frac {6 i c^{4} B \tan \left (f x +e \right )}{f \,a^{2}}-\frac {20 i c^{4} B}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}-\frac {12 c^{4} A}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {6 c^{4} A \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{2}}-\frac {3 i c^{4} A \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{f \,a^{2}}+\frac {18 i c^{4} B \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{2}}+\frac {9 c^{4} B \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{f \,a^{2}}-\frac {4 i c^{4} A}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {4 c^{4} B}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}\)	\(239\)
default	\(-\frac {B \,c^{4} \tan \left (f x +e \right )^{2}}{2 a^{2} f}-\frac {c^{4} A \tan \left (f x +e \right )}{f \,a^{2}}-\frac {6 i c^{4} B \tan \left (f x +e \right )}{f \,a^{2}}-\frac {20 i c^{4} B}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}-\frac {12 c^{4} A}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {6 c^{4} A \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{2}}-\frac {3 i c^{4} A \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{f \,a^{2}}+\frac {18 i c^{4} B \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{2}}+\frac {9 c^{4} B \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{f \,a^{2}}-\frac {4 i c^{4} A}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {4 c^{4} B}{f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}\)	\(239\)
risch	\(\frac {8 c^{4} {\mathrm e}^{-2 i \left (f x +e \right )} B}{a^{2} f}-\frac {4 i c^{4} {\mathrm e}^{-2 i \left (f x +e \right )} A}{a^{2} f}-\frac {c^{4} {\mathrm e}^{-4 i \left (f x +e \right )} B}{a^{2} f}+\frac {i c^{4} {\mathrm e}^{-4 i \left (f x +e \right )} A}{a^{2} f}+\frac {36 i c^{4} B x}{a^{2}}+\frac {12 c^{4} A x}{a^{2}}+\frac {36 i c^{4} B e}{f \,a^{2}}+\frac {12 c^{4} A e}{f \,a^{2}}+\frac {2 c^{4} \left (-i A \,{\mathrm e}^{2 i \left (f x +e \right )}+5 B \,{\mathrm e}^{2 i \left (f x +e \right )}-i A +6 B \right )}{f \,a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}-\frac {18 c^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) B}{f \,a^{2}}+\frac {6 i c^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) A}{f \,a^{2}}\)	\(242\)
norman	\(\frac {\frac {-8 i c^{4} A +17 c^{4} B}{a f}+\frac {6 \left (3 i c^{4} B +c^{4} A \right ) x}{a}-\frac {\left (6 i c^{4} B +c^{4} A \right ) \tan \left (f x +e \right )^{5}}{a f}+\frac {\left (-32 i c^{4} A +51 c^{4} B \right ) \tan \left (f x +e \right )^{2}}{2 a f}+\frac {12 \left (3 i c^{4} B +c^{4} A \right ) x \tan \left (f x +e \right )^{2}}{a}+\frac {6 \left (3 i c^{4} B +c^{4} A \right ) x \tan \left (f x +e \right )^{4}}{a}-\frac {\left (18 i c^{4} B +5 c^{4} A \right ) \tan \left (f x +e \right )}{a f}-\frac {2 \left (16 i c^{4} B +7 c^{4} A \right ) \tan \left (f x +e \right )^{3}}{a f}-\frac {c^{4} B \tan \left (f x +e \right )^{6}}{2 a f}}{a \left (1+\tan \left (f x +e \right )^{2}\right )^{2}}+\frac {3 \left (-i c^{4} A +3 c^{4} B \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{a^{2} f}\)	\(283\)

[In]

int((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*B*c^4*tan(f*x+e)^2/a^2/f-1/f*c^4/a^2*A*tan(f*x+e)-6*I/f*c^4/a^2*B*tan(f*x+e)-20*I/f*c^4/a^2/(-I+tan(f*x+e
))*B-12/f*c^4/a^2/(-I+tan(f*x+e))*A+6/f*c^4/a^2*A*arctan(tan(f*x+e))-3*I/f*c^4/a^2*A*ln(1+tan(f*x+e)^2)+18*I/f
*c^4/a^2*B*arctan(tan(f*x+e))+9/f*c^4/a^2*B*ln(1+tan(f*x+e)^2)-4*I/f*c^4/a^2/(-I+tan(f*x+e))^2*A+4/f*c^4/a^2/(
-I+tan(f*x+e))^2*B

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.58 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=\frac {12 \, {\left (A + 3 i \, B\right )} c^{4} f x e^{\left (8 i \, f x + 8 i \, e\right )} - 2 \, {\left (i \, A - 3 \, B\right )} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A - B\right )} c^{4} + 6 \, {\left (4 \, {\left (A + 3 i \, B\right )} c^{4} f x - {\left (i \, A - 3 \, B\right )} c^{4}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, {\left (4 \, {\left (A + 3 i \, B\right )} c^{4} f x - 3 \, {\left (i \, A - 3 \, B\right )} c^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 6 \, {\left ({\left (-i \, A + 3 \, B\right )} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, {\left (-i \, A + 3 \, B\right )} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-i \, A + 3 \, B\right )} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}} \]

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

(12*(A + 3*I*B)*c^4*f*x*e^(8*I*f*x + 8*I*e) - 2*(I*A - 3*B)*c^4*e^(2*I*f*x + 2*I*e) + (I*A - B)*c^4 + 6*(4*(A
+ 3*I*B)*c^4*f*x - (I*A - 3*B)*c^4)*e^(6*I*f*x + 6*I*e) + 3*(4*(A + 3*I*B)*c^4*f*x - 3*(I*A - 3*B)*c^4)*e^(4*I
*f*x + 4*I*e) - 6*((-I*A + 3*B)*c^4*e^(8*I*f*x + 8*I*e) + 2*(-I*A + 3*B)*c^4*e^(6*I*f*x + 6*I*e) + (-I*A + 3*B
)*c^4*e^(4*I*f*x + 4*I*e))*log(e^(2*I*f*x + 2*I*e) + 1))/(a^2*f*e^(8*I*f*x + 8*I*e) + 2*a^2*f*e^(6*I*f*x + 6*I
*e) + a^2*f*e^(4*I*f*x + 4*I*e))

Sympy [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.39 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=\frac {- 2 i A c^{4} + 12 B c^{4} + \left (- 2 i A c^{4} e^{2 i e} + 10 B c^{4} e^{2 i e}\right ) e^{2 i f x}}{a^{2} f e^{4 i e} e^{4 i f x} + 2 a^{2} f e^{2 i e} e^{2 i f x} + a^{2} f} + \begin {cases} \frac {\left (\left (i A a^{2} c^{4} f e^{2 i e} - B a^{2} c^{4} f e^{2 i e}\right ) e^{- 4 i f x} + \left (- 4 i A a^{2} c^{4} f e^{4 i e} + 8 B a^{2} c^{4} f e^{4 i e}\right ) e^{- 2 i f x}\right ) e^{- 6 i e}}{a^{4} f^{2}} & \text {for}\: a^{4} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac {12 A c^{4} + 36 i B c^{4}}{a^{2}} + \frac {\left (12 A c^{4} e^{4 i e} - 8 A c^{4} e^{2 i e} + 4 A c^{4} + 36 i B c^{4} e^{4 i e} - 16 i B c^{4} e^{2 i e} + 4 i B c^{4}\right ) e^{- 4 i e}}{a^{2}}\right ) & \text {otherwise} \end {cases} + \frac {6 i c^{4} \left (A + 3 i B\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{2} f} + \frac {x \left (12 A c^{4} + 36 i B c^{4}\right )}{a^{2}} \]

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**4/(a+I*a*tan(f*x+e))**2,x)

[Out]

(-2*I*A*c**4 + 12*B*c**4 + (-2*I*A*c**4*exp(2*I*e) + 10*B*c**4*exp(2*I*e))*exp(2*I*f*x))/(a**2*f*exp(4*I*e)*ex
p(4*I*f*x) + 2*a**2*f*exp(2*I*e)*exp(2*I*f*x) + a**2*f) + Piecewise((((I*A*a**2*c**4*f*exp(2*I*e) - B*a**2*c**
4*f*exp(2*I*e))*exp(-4*I*f*x) + (-4*I*A*a**2*c**4*f*exp(4*I*e) + 8*B*a**2*c**4*f*exp(4*I*e))*exp(-2*I*f*x))*ex
p(-6*I*e)/(a**4*f**2), Ne(a**4*f**2*exp(6*I*e), 0)), (x*(-(12*A*c**4 + 36*I*B*c**4)/a**2 + (12*A*c**4*exp(4*I*
e) - 8*A*c**4*exp(2*I*e) + 4*A*c**4 + 36*I*B*c**4*exp(4*I*e) - 16*I*B*c**4*exp(2*I*e) + 4*I*B*c**4)*exp(-4*I*e
)/a**2), True)) + 6*I*c**4*(A + 3*I*B)*log(exp(2*I*f*x) + exp(-2*I*e))/(a**2*f) + x*(12*A*c**4 + 36*I*B*c**4)/
a**2

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

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. 423 vs. \(2 (136) = 272\).

Time = 0.78 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.68 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=\frac {\frac {6 \, {\left (i \, A c^{4} - 3 \, B c^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2}} - \frac {12 \, {\left (i \, A c^{4} - 3 \, B c^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}{a^{2}} - \frac {6 \, {\left (-i \, A c^{4} + 3 \, B c^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{a^{2}} - \frac {9 i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 27 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 18 i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 56 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 i \, A c^{4} - 27 \, B c^{4}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} - \frac {-25 i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 75 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 108 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 324 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 182 i \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 514 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 108 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 324 i \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 25 i \, A c^{4} + 75 \, B c^{4}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{4}}}{f} \]

[In]

integrate((A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^4/(a+I*a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

(6*(I*A*c^4 - 3*B*c^4)*log(tan(1/2*f*x + 1/2*e) + 1)/a^2 - 12*(I*A*c^4 - 3*B*c^4)*log(tan(1/2*f*x + 1/2*e) - I
)/a^2 - 6*(-I*A*c^4 + 3*B*c^4)*log(tan(1/2*f*x + 1/2*e) - 1)/a^2 - (9*I*A*c^4*tan(1/2*f*x + 1/2*e)^4 - 27*B*c^
4*tan(1/2*f*x + 1/2*e)^4 - 2*A*c^4*tan(1/2*f*x + 1/2*e)^3 - 12*I*B*c^4*tan(1/2*f*x + 1/2*e)^3 - 18*I*A*c^4*tan
(1/2*f*x + 1/2*e)^2 + 56*B*c^4*tan(1/2*f*x + 1/2*e)^2 + 2*A*c^4*tan(1/2*f*x + 1/2*e) + 12*I*B*c^4*tan(1/2*f*x
+ 1/2*e) + 9*I*A*c^4 - 27*B*c^4)/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*a^2) - (-25*I*A*c^4*tan(1/2*f*x + 1/2*e)^4 +
75*B*c^4*tan(1/2*f*x + 1/2*e)^4 - 108*A*c^4*tan(1/2*f*x + 1/2*e)^3 - 324*I*B*c^4*tan(1/2*f*x + 1/2*e)^3 + 182*
I*A*c^4*tan(1/2*f*x + 1/2*e)^2 - 514*B*c^4*tan(1/2*f*x + 1/2*e)^2 + 108*A*c^4*tan(1/2*f*x + 1/2*e) + 324*I*B*c
^4*tan(1/2*f*x + 1/2*e) - 25*I*A*c^4 + 75*B*c^4)/(a^2*(tan(1/2*f*x + 1/2*e) - I)^4))/f

Mupad [B] (verification not implemented)

Time = 8.48 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx=-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-\frac {18\,B\,c^4}{a^2}+\frac {A\,c^4\,6{}\mathrm {i}}{a^2}\right )}{f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {c^4\,\left (A+B\,3{}\mathrm {i}\right )}{a^2}+\frac {B\,c^4\,3{}\mathrm {i}}{a^2}\right )}{f}-\frac {\frac {\left (-6\,B\,c^4+A\,c^4\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a^2}-\frac {\left (-18\,B\,c^4+A\,c^4\,6{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,a^2}+\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {2\,\left (-18\,B\,c^4+A\,c^4\,6{}\mathrm {i}\right )}{a^2}+\frac {16\,B\,c^4}{a^2}\right )-\frac {B\,c^4\,8{}\mathrm {i}}{a^2}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}-\frac {B\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2\,a^2\,f} \]

[In]

int(((A + B*tan(e + f*x))*(c - c*tan(e + f*x)*1i)^4)/(a + a*tan(e + f*x)*1i)^2,x)

[Out]

- (log(tan(e + f*x) - 1i)*((A*c^4*6i)/a^2 - (18*B*c^4)/a^2))/f - (tan(e + f*x)*((c^4*(A + B*3i))/a^2 + (B*c^4*
3i)/a^2))/f - (((A*c^4*2i - 6*B*c^4)*1i)/(2*a^2) - ((A*c^4*6i - 18*B*c^4)*3i)/(2*a^2) + tan(e + f*x)*((2*(A*c^
4*6i - 18*B*c^4))/a^2 + (16*B*c^4)/a^2) - (B*c^4*8i)/a^2)/(f*(2*tan(e + f*x) + tan(e + f*x)^2*1i - 1i)) - (B*c
^4*tan(e + f*x)^2)/(2*a^2*f)