3.65 \(\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=220 \[ -\frac{5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}-\frac{(-B+4 i A) \log (\sin (c+d x))}{a^4 d}+\frac{(4 A+i B) \cot (c+d x)}{2 a^4 d (1+i \tan (c+d x))}+\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac{5 x (13 A+3 i B)}{16 a^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4} \]

[Out]

(-5*(13*A + (3*I)*B)*x)/(16*a^4) - (5*(13*A + (3*I)*B)*Cot[c + d*x])/(16*a^4*d) - (((4*I)*A - B)*Log[Sin[c + d
*x]])/(a^4*d) + ((31*A + (9*I)*B)*Cot[c + d*x])/(48*a^4*d*(1 + I*Tan[c + d*x])^2) + ((4*A + I*B)*Cot[c + d*x])
/(2*a^4*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Cot[c + d*x])/(8*d*(a + I*a*Tan[c + d*x])^4) + ((7*A + (3*I)*B)*C
ot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^3)

________________________________________________________________________________________

Rubi [A]  time = 0.720652, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3596, 3529, 3531, 3475} \[ -\frac{5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}-\frac{(-B+4 i A) \log (\sin (c+d x))}{a^4 d}+\frac{(4 A+i B) \cot (c+d x)}{2 a^4 d (1+i \tan (c+d x))}+\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac{5 x (13 A+3 i B)}{16 a^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(-5*(13*A + (3*I)*B)*x)/(16*a^4) - (5*(13*A + (3*I)*B)*Cot[c + d*x])/(16*a^4*d) - (((4*I)*A - B)*Log[Sin[c + d
*x]])/(a^4*d) + ((31*A + (9*I)*B)*Cot[c + d*x])/(48*a^4*d*(1 + I*Tan[c + d*x])^2) + ((4*A + I*B)*Cot[c + d*x])
/(2*a^4*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Cot[c + d*x])/(8*d*(a + I*a*Tan[c + d*x])^4) + ((7*A + (3*I)*B)*C
ot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^3)

Rule 3596

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

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3531

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] + Dist[(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] && NeQ[a*c + b*d, 0]

Rule 3475

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

Rubi steps

\begin{align*} \int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^4} \, dx &=\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\int \frac{\cot ^2(c+d x) (a (9 A+i B)-5 a (i A-B) \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2}\\ &=\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot ^2(c+d x) \left (4 a^2 (17 A+3 i B)-8 a^2 (7 i A-3 B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4}\\ &=\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot ^2(c+d x) \left (12 a^3 (33 A+7 i B)-12 a^3 (31 i A-9 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{192 a^6}\\ &=\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \cot ^2(c+d x) \left (120 a^4 (13 A+3 i B)-384 a^4 (4 i A-B) \tan (c+d x)\right ) \, dx}{384 a^8}\\ &=-\frac{5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}+\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (-384 a^4 (4 i A-B)-120 a^4 (13 A+3 i B) \tan (c+d x)\right ) \, dx}{384 a^8}\\ &=-\frac{5 (13 A+3 i B) x}{16 a^4}-\frac{5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}+\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{(4 i A-B) \int \cot (c+d x) \, dx}{a^4}\\ &=-\frac{5 (13 A+3 i B) x}{16 a^4}-\frac{5 (13 A+3 i B) \cot (c+d x)}{16 a^4 d}-\frac{(4 i A-B) \log (\sin (c+d x))}{a^4 d}+\frac{(31 A+9 i B) \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}+\frac{(A+i B) \cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{(7 A+3 i B) \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{(4 A+i B) \cot (c+d x)}{2 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 7.0038, size = 1466, normalized size = 6.66 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4,x]

[Out]

(((-15*I)*A + 8*B)*Cos[4*d*x]*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c +
d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (((-4*I)*A + 3*B)*Cos[6*d*x]*Sec[c + d*x]^3*(Cos[2*c]/48 -
(I/48)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a
*Tan[c + d*x])^4) + (((-36*I)*A + 13*B)*Cos[2*d*x]*Sec[c + d*x]^3*(Cos[2*c]/16 + (I/16)*Sin[2*c])*(Cos[d*x] +
I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Sec[c +
d*x]^3*((-4*I)*A*Cos[2*c] + B*Cos[2*c] + 4*A*Sin[2*c] + I*B*Sin[2*c])*((-I)*ArcTan[Tan[d*x]]*Cos[2*c] + ArcTan
[Tan[d*x]]*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a +
I*a*Tan[c + d*x])^4) + (Sec[c + d*x]^3*((-4*I)*A*Cos[2*c] + B*Cos[2*c] + 4*A*Sin[2*c] + I*B*Sin[2*c])*((Cos[2
*c]*Log[Sin[c + d*x]^2])/2 + (I/2)*Log[Sin[c + d*x]^2]*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]
))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (x*Sec[c + d*x]^3*(-12*A*Cos[c]^2 - (3*I)*
B*Cos[c]^2 + (4*I)*A*Cos[c]^2*Cot[c] - B*Cos[c]^2*Cot[c] - (12*I)*A*Cos[c]*Sin[c] + 3*B*Cos[c]*Sin[c] + 4*A*Si
n[c]^2 + I*B*Sin[c]^2 + ((-4*I)*A + B)*Cot[c]*(Cos[4*c] + I*Sin[4*c]))*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c
+ d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (((-I)*A + B)*Cos[8*d*x]*Sec[c + d*x]^
3*(Cos[4*c]/128 - (I/128)*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin
[c + d*x])*(a + I*a*Tan[c + d*x])^4) + ((13*A + (3*I)*B)*Sec[c + d*x]^3*((-5*d*x*Cos[4*c])/16 - ((5*I)/16)*d*x
*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c
+ d*x])^4) + ((36*A + (13*I)*B)*Sec[c + d*x]^3*(-Cos[2*c]/16 - (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^4*Sin
[2*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) - ((15*A + (8*I)*
B)*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^4*Sin[4*d*x]*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B*Sin[c +
d*x])*(a + I*a*Tan[c + d*x])^4) + ((4*A + (3*I)*B)*Sec[c + d*x]^3*(-Cos[2*c]/48 + (I/48)*Sin[2*c])*(Cos[d*x]
+ I*Sin[d*x])^4*Sin[6*d*x]*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4
) + ((A + I*B)*Sec[c + d*x]^3*(-Cos[4*c]/128 + (I/128)*Sin[4*c])*(Cos[d*x] + I*Sin[d*x])^4*Sin[8*d*x]*(A + B*T
an[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4) + (Csc[c]*Csc[c + d*x]*Sec[c + d*
x]^3*(Cos[d*x] + I*Sin[d*x])^4*((I/2)*A*Cos[4*c - d*x] - (I/2)*A*Cos[4*c + d*x] - (A*Sin[4*c - d*x])/2 + (A*Si
n[4*c + d*x])/2)*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^4)

________________________________________________________________________________________

Maple [A]  time = 0.122, size = 293, normalized size = 1.3 \begin{align*}{\frac{5\,A}{12\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{{\frac{i}{4}}B}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{31\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{32\,{a}^{4}d}}+{\frac{{\frac{129\,i}{32}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{{a}^{4}d}}-{\frac{7\,B}{16\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{{\frac{17\,i}{16}}A}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{{\frac{i}{8}}A}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}+{\frac{B}{8\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{49\,A}{16\,{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{15\,i}{16}}B}{{a}^{4}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{32\,{a}^{4}d}}-{\frac{{\frac{i}{32}}A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{4}d}}-{\frac{A}{{a}^{4}d\tan \left ( dx+c \right ) }}-{\frac{4\,iA\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{4}d}}+{\frac{B\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{4}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^4,x)

[Out]

5/12/d/a^4/(tan(d*x+c)-I)^3*A+1/4*I/d/a^4/(tan(d*x+c)-I)^3*B-31/32/d/a^4*ln(tan(d*x+c)-I)*B+129/32*I/d/a^4*ln(
tan(d*x+c)-I)*A-7/16/d/a^4/(tan(d*x+c)-I)^2*B+17/16*I/d/a^4/(tan(d*x+c)-I)^2*A-1/8*I/d/a^4/(tan(d*x+c)-I)^4*A+
1/8/d/a^4/(tan(d*x+c)-I)^4*B-49/16/d/a^4/(tan(d*x+c)-I)*A-15/16*I/d/a^4/(tan(d*x+c)-I)*B-1/32/d/a^4*B*ln(tan(d
*x+c)+I)-1/32*I/d/a^4*A*ln(tan(d*x+c)+I)-1/d/a^4*A/tan(d*x+c)-4*I/d/a^4*A*ln(tan(d*x+c))+1/d/a^4*B*ln(tan(d*x+
c))

________________________________________________________________________________________

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(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.78027, size = 578, normalized size = 2.63 \begin{align*} -\frac{24 \,{\left (129 \, A + 31 i \, B\right )} d x e^{\left (10 i \, d x + 10 i \, c\right )} -{\left (24 \,{\left (129 \, A + 31 i \, B\right )} d x - 1632 i \, A + 312 \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} -{\left (684 i \, A - 216 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} -{\left (148 i \, A - 72 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} -{\left (29 i \, A - 21 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left ({\left (-1536 i \, A + 384 \, B\right )} e^{\left (10 i \, d x + 10 i \, c\right )} +{\left (1536 i \, A - 384 \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 3 i \, A + 3 \, B}{384 \,{\left (a^{4} d e^{\left (10 i \, d x + 10 i \, c\right )} - a^{4} d e^{\left (8 i \, d x + 8 i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(24*(129*A + 31*I*B)*d*x*e^(10*I*d*x + 10*I*c) - (24*(129*A + 31*I*B)*d*x - 1632*I*A + 312*B)*e^(8*I*d*
x + 8*I*c) - (684*I*A - 216*B)*e^(6*I*d*x + 6*I*c) - (148*I*A - 72*B)*e^(4*I*d*x + 4*I*c) - (29*I*A - 21*B)*e^
(2*I*d*x + 2*I*c) - ((-1536*I*A + 384*B)*e^(10*I*d*x + 10*I*c) + (1536*I*A - 384*B)*e^(8*I*d*x + 8*I*c))*log(e
^(2*I*d*x + 2*I*c) - 1) - 3*I*A + 3*B)/(a^4*d*e^(10*I*d*x + 10*I*c) - a^4*d*e^(8*I*d*x + 8*I*c))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.30358, size = 278, normalized size = 1.26 \begin{align*} \frac{\frac{12 \,{\left (-i \, A - B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} - \frac{12 \,{\left (-129 i \, A + 31 \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} - \frac{384 \,{\left (4 i \, A - B\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{384 \,{\left (-4 i \, A \tan \left (d x + c\right ) + B \tan \left (d x + c\right ) + A\right )}}{a^{4} \tan \left (d x + c\right )} - \frac{3225 i \, A \tan \left (d x + c\right )^{4} - 775 \, B \tan \left (d x + c\right )^{4} + 14076 \, A \tan \left (d x + c\right )^{3} + 3460 i \, B \tan \left (d x + c\right )^{3} - 23286 i \, A \tan \left (d x + c\right )^{2} + 5898 \, B \tan \left (d x + c\right )^{2} - 17404 \, A \tan \left (d x + c\right ) - 4612 i \, B \tan \left (d x + c\right ) + 5017 i \, A - 1447 \, B}{a^{4}{\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(12*(-I*A - B)*log(tan(d*x + c) + I)/a^4 - 12*(-129*I*A + 31*B)*log(tan(d*x + c) - I)/a^4 - 384*(4*I*A -
B)*log(abs(tan(d*x + c)))/a^4 - 384*(-4*I*A*tan(d*x + c) + B*tan(d*x + c) + A)/(a^4*tan(d*x + c)) - (3225*I*A
*tan(d*x + c)^4 - 775*B*tan(d*x + c)^4 + 14076*A*tan(d*x + c)^3 + 3460*I*B*tan(d*x + c)^3 - 23286*I*A*tan(d*x
+ c)^2 + 5898*B*tan(d*x + c)^2 - 17404*A*tan(d*x + c) - 4612*I*B*tan(d*x + c) + 5017*I*A - 1447*B)/(a^4*(tan(d
*x + c) - I)^4))/d