3.737 \(\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx\)

Optimal. Leaf size=251 \[ -\frac{5 (3 A+i B)}{128 a^3 c^4 f (-\tan (e+f x)+i)}-\frac{-3 B+5 i A}{128 a^3 c^4 f (-\tan (e+f x)+i)^2}+\frac{B+5 i A}{64 a^3 c^4 f (\tan (e+f x)+i)^2}+\frac{A+i B}{96 a^3 c^4 f (-\tan (e+f x)+i)^3}-\frac{2 A-i B}{48 a^3 c^4 f (\tan (e+f x)+i)^3}-\frac{B+i A}{64 a^3 c^4 f (\tan (e+f x)+i)^4}+\frac{5 x (7 A+i B)}{128 a^3 c^4}+\frac{5 A}{32 a^3 c^4 f (\tan (e+f x)+i)} \]

[Out]

(5*(7*A + I*B)*x)/(128*a^3*c^4) + (A + I*B)/(96*a^3*c^4*f*(I - Tan[e + f*x])^3) - ((5*I)*A - 3*B)/(128*a^3*c^4
*f*(I - Tan[e + f*x])^2) - (5*(3*A + I*B))/(128*a^3*c^4*f*(I - Tan[e + f*x])) - (I*A + B)/(64*a^3*c^4*f*(I + T
an[e + f*x])^4) - (2*A - I*B)/(48*a^3*c^4*f*(I + Tan[e + f*x])^3) + ((5*I)*A + B)/(64*a^3*c^4*f*(I + Tan[e + f
*x])^2) + (5*A)/(32*a^3*c^4*f*(I + Tan[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.30624, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used = {3588, 77, 203} \[ -\frac{5 (3 A+i B)}{128 a^3 c^4 f (-\tan (e+f x)+i)}-\frac{-3 B+5 i A}{128 a^3 c^4 f (-\tan (e+f x)+i)^2}+\frac{B+5 i A}{64 a^3 c^4 f (\tan (e+f x)+i)^2}+\frac{A+i B}{96 a^3 c^4 f (-\tan (e+f x)+i)^3}-\frac{2 A-i B}{48 a^3 c^4 f (\tan (e+f x)+i)^3}-\frac{B+i A}{64 a^3 c^4 f (\tan (e+f x)+i)^4}+\frac{5 x (7 A+i B)}{128 a^3 c^4}+\frac{5 A}{32 a^3 c^4 f (\tan (e+f x)+i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(7*A + I*B)*x)/(128*a^3*c^4) + (A + I*B)/(96*a^3*c^4*f*(I - Tan[e + f*x])^3) - ((5*I)*A - 3*B)/(128*a^3*c^4
*f*(I - Tan[e + f*x])^2) - (5*(3*A + I*B))/(128*a^3*c^4*f*(I - Tan[e + f*x])) - (I*A + B)/(64*a^3*c^4*f*(I + T
an[e + f*x])^4) - (2*A - I*B)/(48*a^3*c^4*f*(I + Tan[e + f*x])^3) + ((5*I)*A + B)/(64*a^3*c^4*f*(I + Tan[e + f
*x])^2) + (5*A)/(32*a^3*c^4*f*(I + Tan[e + f*x]))

Rule 3588

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]

Rule 77

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^4 (c-i c x)^5} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{A+i B}{32 a^4 c^5 (-i+x)^4}+\frac{i (5 A+3 i B)}{64 a^4 c^5 (-i+x)^3}-\frac{5 (3 A+i B)}{128 a^4 c^5 (-i+x)^2}+\frac{i A+B}{16 a^4 c^5 (i+x)^5}+\frac{2 A-i B}{16 a^4 c^5 (i+x)^4}-\frac{i (5 A-i B)}{32 a^4 c^5 (i+x)^3}-\frac{5 A}{32 a^4 c^5 (i+x)^2}+\frac{5 (7 A+i B)}{128 a^4 c^5 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{A+i B}{96 a^3 c^4 f (i-\tan (e+f x))^3}-\frac{5 i A-3 B}{128 a^3 c^4 f (i-\tan (e+f x))^2}-\frac{5 (3 A+i B)}{128 a^3 c^4 f (i-\tan (e+f x))}-\frac{i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^4}-\frac{2 A-i B}{48 a^3 c^4 f (i+\tan (e+f x))^3}+\frac{5 i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^2}+\frac{5 A}{32 a^3 c^4 f (i+\tan (e+f x))}+\frac{(5 (7 A+i B)) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 a^3 c^4 f}\\ &=\frac{5 (7 A+i B) x}{128 a^3 c^4}+\frac{A+i B}{96 a^3 c^4 f (i-\tan (e+f x))^3}-\frac{5 i A-3 B}{128 a^3 c^4 f (i-\tan (e+f x))^2}-\frac{5 (3 A+i B)}{128 a^3 c^4 f (i-\tan (e+f x))}-\frac{i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^4}-\frac{2 A-i B}{48 a^3 c^4 f (i+\tan (e+f x))^3}+\frac{5 i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^2}+\frac{5 A}{32 a^3 c^4 f (i+\tan (e+f x))}\\ \end{align*}

Mathematica [A]  time = 3.21447, size = 267, normalized size = 1.06 \[ \frac{\sec ^3(e+f x) (-\cos (4 (e+f x))-i \sin (4 (e+f x))) (60 (A (-7-14 i f x)+B (2 f x+i)) \cos (e+f x)+18 (7 A+9 i B) \cos (3 (e+f x))-420 i A \sin (e+f x)-840 A f x \sin (e+f x)-378 i A \sin (3 (e+f x))-70 i A \sin (5 (e+f x))-7 i A \sin (7 (e+f x))+14 A \cos (5 (e+f x))+A \cos (7 (e+f x))-60 B \sin (e+f x)-120 i B f x \sin (e+f x)+54 B \sin (3 (e+f x))+10 B \sin (5 (e+f x))+B \sin (7 (e+f x))+50 i B \cos (5 (e+f x))+7 i B \cos (7 (e+f x)))}{3072 a^3 c^4 f (\tan (e+f x)-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]^3*(-Cos[4*(e + f*x)] - I*Sin[4*(e + f*x)])*(60*(A*(-7 - (14*I)*f*x) + B*(I + 2*f*x))*Cos[e + f*x
] + 18*(7*A + (9*I)*B)*Cos[3*(e + f*x)] + 14*A*Cos[5*(e + f*x)] + (50*I)*B*Cos[5*(e + f*x)] + A*Cos[7*(e + f*x
)] + (7*I)*B*Cos[7*(e + f*x)] - (420*I)*A*Sin[e + f*x] - 60*B*Sin[e + f*x] - 840*A*f*x*Sin[e + f*x] - (120*I)*
B*f*x*Sin[e + f*x] - (378*I)*A*Sin[3*(e + f*x)] + 54*B*Sin[3*(e + f*x)] - (70*I)*A*Sin[5*(e + f*x)] + 10*B*Sin
[5*(e + f*x)] - (7*I)*A*Sin[7*(e + f*x)] + B*Sin[7*(e + f*x)]))/(3072*a^3*c^4*f*(-I + Tan[e + f*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.076, size = 397, normalized size = 1.6 \begin{align*}{\frac{15\,A}{128\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{5\,i}{128}}A}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{{\frac{35\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) A}{f{a}^{3}{c}^{4}}}+{\frac{3\,B}{128\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{A}{96\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}+{\frac{{\frac{5\,i}{128}}B}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{5\,i}{64}}A}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+{\frac{5\,\ln \left ( \tan \left ( fx+e \right ) -i \right ) B}{256\,f{a}^{3}{c}^{4}}}-{\frac{{\frac{i}{64}}A}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-{\frac{B}{64\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{5\,A}{32\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) }}-{\frac{{\frac{i}{96}}B}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}-{\frac{5\,\ln \left ( \tan \left ( fx+e \right ) +i \right ) B}{256\,f{a}^{3}{c}^{4}}}-{\frac{{\frac{35\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) A}{f{a}^{3}{c}^{4}}}-{\frac{A}{24\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}+{\frac{{\frac{i}{48}}B}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}+{\frac{B}{64\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x)

[Out]

15/128/f/a^3/c^4/(tan(f*x+e)-I)*A-5/128*I/f/a^3/c^4/(tan(f*x+e)-I)^2*A+35/256*I/f/a^3/c^4*ln(tan(f*x+e)+I)*A+3
/128/f/a^3/c^4/(tan(f*x+e)-I)^2*B-1/96/f/a^3/c^4/(tan(f*x+e)-I)^3*A+5/128*I/f/a^3/c^4/(tan(f*x+e)-I)*B+5/64*I/
f/a^3/c^4/(tan(f*x+e)+I)^2*A+5/256/f/a^3/c^4*ln(tan(f*x+e)-I)*B-1/64*I/f/a^3/c^4/(tan(f*x+e)+I)^4*A-1/64/f/a^3
/c^4/(tan(f*x+e)+I)^4*B+5/32*A/a^3/c^4/f/(tan(f*x+e)+I)-1/96*I/f/a^3/c^4/(tan(f*x+e)-I)^3*B-5/256/f/a^3/c^4*ln
(tan(f*x+e)+I)*B-35/256*I/f/a^3/c^4*ln(tan(f*x+e)-I)*A-1/24/f/a^3/c^4/(tan(f*x+e)+I)^3*A+1/48*I/f/a^3/c^4/(tan
(f*x+e)+I)^3*B+1/64/f/a^3/c^4/(tan(f*x+e)+I)^2*B

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.05392, size = 455, normalized size = 1.81 \begin{align*} \frac{{\left (120 \,{\left (7 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-3 i \, A - 3 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} +{\left (-28 i \, A - 20 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-126 i \, A - 54 \, B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-420 i \, A - 60 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (252 i \, A - 108 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (42 i \, A - 30 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, A - 4 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{3072 \, a^{3} c^{4} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

1/3072*(120*(7*A + I*B)*f*x*e^(6*I*f*x + 6*I*e) + (-3*I*A - 3*B)*e^(14*I*f*x + 14*I*e) + (-28*I*A - 20*B)*e^(1
2*I*f*x + 12*I*e) + (-126*I*A - 54*B)*e^(10*I*f*x + 10*I*e) + (-420*I*A - 60*B)*e^(8*I*f*x + 8*I*e) + (252*I*A
- 108*B)*e^(4*I*f*x + 4*I*e) + (42*I*A - 30*B)*e^(2*I*f*x + 2*I*e) + 4*I*A - 4*B)*e^(-6*I*f*x - 6*I*e)/(a^3*c
^4*f)

________________________________________________________________________________________

Sympy [A]  time = 7.96462, size = 607, normalized size = 2.42 \begin{align*} \begin{cases} \frac{\left (\left (13510798882111488 i A a^{18} c^{24} f^{6} e^{6 i e} - 13510798882111488 B a^{18} c^{24} f^{6} e^{6 i e}\right ) e^{- 6 i f x} + \left (141863388262170624 i A a^{18} c^{24} f^{6} e^{8 i e} - 101330991615836160 B a^{18} c^{24} f^{6} e^{8 i e}\right ) e^{- 4 i f x} + \left (851180329573023744 i A a^{18} c^{24} f^{6} e^{10 i e} - 364791569817010176 B a^{18} c^{24} f^{6} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 1418633882621706240 i A a^{18} c^{24} f^{6} e^{14 i e} - 202661983231672320 B a^{18} c^{24} f^{6} e^{14 i e}\right ) e^{2 i f x} + \left (- 425590164786511872 i A a^{18} c^{24} f^{6} e^{16 i e} - 182395784908505088 B a^{18} c^{24} f^{6} e^{16 i e}\right ) e^{4 i f x} + \left (- 94575592174780416 i A a^{18} c^{24} f^{6} e^{18 i e} - 67553994410557440 B a^{18} c^{24} f^{6} e^{18 i e}\right ) e^{6 i f x} + \left (- 10133099161583616 i A a^{18} c^{24} f^{6} e^{20 i e} - 10133099161583616 B a^{18} c^{24} f^{6} e^{20 i e}\right ) e^{8 i f x}\right ) e^{- 12 i e}}{10376293541461622784 a^{21} c^{28} f^{7}} & \text{for}\: 10376293541461622784 a^{21} c^{28} f^{7} e^{12 i e} \neq 0 \\x \left (- \frac{35 A + 5 i B}{128 a^{3} c^{4}} + \frac{\left (A e^{14 i e} + 7 A e^{12 i e} + 21 A e^{10 i e} + 35 A e^{8 i e} + 35 A e^{6 i e} + 21 A e^{4 i e} + 7 A e^{2 i e} + A - i B e^{14 i e} - 5 i B e^{12 i e} - 9 i B e^{10 i e} - 5 i B e^{8 i e} + 5 i B e^{6 i e} + 9 i B e^{4 i e} + 5 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{128 a^{3} c^{4}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (35 A + 5 i B\right )}{128 a^{3} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**4,x)

[Out]

Piecewise((((13510798882111488*I*A*a**18*c**24*f**6*exp(6*I*e) - 13510798882111488*B*a**18*c**24*f**6*exp(6*I*
e))*exp(-6*I*f*x) + (141863388262170624*I*A*a**18*c**24*f**6*exp(8*I*e) - 101330991615836160*B*a**18*c**24*f**
6*exp(8*I*e))*exp(-4*I*f*x) + (851180329573023744*I*A*a**18*c**24*f**6*exp(10*I*e) - 364791569817010176*B*a**1
8*c**24*f**6*exp(10*I*e))*exp(-2*I*f*x) + (-1418633882621706240*I*A*a**18*c**24*f**6*exp(14*I*e) - 20266198323
1672320*B*a**18*c**24*f**6*exp(14*I*e))*exp(2*I*f*x) + (-425590164786511872*I*A*a**18*c**24*f**6*exp(16*I*e) -
182395784908505088*B*a**18*c**24*f**6*exp(16*I*e))*exp(4*I*f*x) + (-94575592174780416*I*A*a**18*c**24*f**6*ex
p(18*I*e) - 67553994410557440*B*a**18*c**24*f**6*exp(18*I*e))*exp(6*I*f*x) + (-10133099161583616*I*A*a**18*c**
24*f**6*exp(20*I*e) - 10133099161583616*B*a**18*c**24*f**6*exp(20*I*e))*exp(8*I*f*x))*exp(-12*I*e)/(1037629354
1461622784*a**21*c**28*f**7), Ne(10376293541461622784*a**21*c**28*f**7*exp(12*I*e), 0)), (x*(-(35*A + 5*I*B)/(
128*a**3*c**4) + (A*exp(14*I*e) + 7*A*exp(12*I*e) + 21*A*exp(10*I*e) + 35*A*exp(8*I*e) + 35*A*exp(6*I*e) + 21*
A*exp(4*I*e) + 7*A*exp(2*I*e) + A - I*B*exp(14*I*e) - 5*I*B*exp(12*I*e) - 9*I*B*exp(10*I*e) - 5*I*B*exp(8*I*e)
+ 5*I*B*exp(6*I*e) + 9*I*B*exp(4*I*e) + 5*I*B*exp(2*I*e) + I*B)*exp(-6*I*e)/(128*a**3*c**4)), True)) + x*(35*
A + 5*I*B)/(128*a**3*c**4)

________________________________________________________________________________________

Giac [A]  time = 1.50037, size = 366, normalized size = 1.46 \begin{align*} \frac{\frac{12 \,{\left (35 i \, A - 5 \, B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{4}} - \frac{12 \,{\left (35 i \, A - 5 \, B\right )} \log \left (-i \, \tan \left (f x + e\right ) - 1\right )}{a^{3} c^{4}} + \frac{2 \,{\left (385 \, A \tan \left (f x + e\right )^{3} + 55 i \, B \tan \left (f x + e\right )^{3} - 1335 i \, A \tan \left (f x + e\right )^{2} + 225 \, B \tan \left (f x + e\right )^{2} - 1575 \, A \tan \left (f x + e\right ) - 321 i \, B \tan \left (f x + e\right ) + 641 i \, A - 167 \, B\right )}}{a^{3} c^{4}{\left (i \, \tan \left (f x + e\right ) + 1\right )}^{3}} + \frac{-875 i \, A \tan \left (f x + e\right )^{4} + 125 \, B \tan \left (f x + e\right )^{4} + 3980 \, A \tan \left (f x + e\right )^{3} + 500 i \, B \tan \left (f x + e\right )^{3} + 6930 i \, A \tan \left (f x + e\right )^{2} - 702 \, B \tan \left (f x + e\right )^{2} - 5548 \, A \tan \left (f x + e\right ) - 340 i \, B \tan \left (f x + e\right ) - 1771 i \, A - 35 \, B}{a^{3} c^{4}{\left (\tan \left (f x + e\right ) + i\right )}^{4}}}{3072 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

1/3072*(12*(35*I*A - 5*B)*log(tan(f*x + e) + I)/(a^3*c^4) - 12*(35*I*A - 5*B)*log(-I*tan(f*x + e) - 1)/(a^3*c^
4) + 2*(385*A*tan(f*x + e)^3 + 55*I*B*tan(f*x + e)^3 - 1335*I*A*tan(f*x + e)^2 + 225*B*tan(f*x + e)^2 - 1575*A
*tan(f*x + e) - 321*I*B*tan(f*x + e) + 641*I*A - 167*B)/(a^3*c^4*(I*tan(f*x + e) + 1)^3) + (-875*I*A*tan(f*x +
e)^4 + 125*B*tan(f*x + e)^4 + 3980*A*tan(f*x + e)^3 + 500*I*B*tan(f*x + e)^3 + 6930*I*A*tan(f*x + e)^2 - 702*
B*tan(f*x + e)^2 - 5548*A*tan(f*x + e) - 340*I*B*tan(f*x + e) - 1771*I*A - 35*B)/(a^3*c^4*(tan(f*x + e) + I)^4
))/f