3.58 \(\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=216 \[ -\frac{(7 A+3 i B) \cot ^2(c+d x)}{2 a^3 d}+\frac{5 (-5 B+11 i A) \cot (c+d x)}{8 a^3 d}-\frac{(7 A+3 i B) \log (\sin (c+d x))}{a^3 d}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{5 x (-5 B+11 i A)}{8 a^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3} \]

[Out]

(5*((11*I)*A - 5*B)*x)/(8*a^3) + (5*((11*I)*A - 5*B)*Cot[c + d*x])/(8*a^3*d) - ((7*A + (3*I)*B)*Cot[c + d*x]^2
)/(2*a^3*d) - ((7*A + (3*I)*B)*Log[Sin[c + d*x]])/(a^3*d) + ((A + I*B)*Cot[c + d*x]^2)/(6*d*(a + I*a*Tan[c + d
*x])^3) + ((13*A + (7*I)*B)*Cot[c + d*x]^2)/(24*a*d*(a + I*a*Tan[c + d*x])^2) + (5*(11*A + (5*I)*B)*Cot[c + d*
x]^2)/(24*d*(a^3 + I*a^3*Tan[c + d*x]))

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Rubi [A]  time = 0.598347, antiderivative size = 216, 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} \[ -\frac{(7 A+3 i B) \cot ^2(c+d x)}{2 a^3 d}+\frac{5 (-5 B+11 i A) \cot (c+d x)}{8 a^3 d}-\frac{(7 A+3 i B) \log (\sin (c+d x))}{a^3 d}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{5 x (-5 B+11 i A)}{8 a^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(5*((11*I)*A - 5*B)*x)/(8*a^3) + (5*((11*I)*A - 5*B)*Cot[c + d*x])/(8*a^3*d) - ((7*A + (3*I)*B)*Cot[c + d*x]^2
)/(2*a^3*d) - ((7*A + (3*I)*B)*Log[Sin[c + d*x]])/(a^3*d) + ((A + I*B)*Cot[c + d*x]^2)/(6*d*(a + I*a*Tan[c + d
*x])^3) + ((13*A + (7*I)*B)*Cot[c + d*x]^2)/(24*a*d*(a + I*a*Tan[c + d*x])^2) + (5*(11*A + (5*I)*B)*Cot[c + d*
x]^2)/(24*d*(a^3 + I*a^3*Tan[c + d*x]))

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 ^3(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot ^3(c+d x) (2 a (4 A+i B)-5 a (i A-B) \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\cot ^3(c+d x) \left (2 a^2 (29 A+11 i B)-4 a^2 (13 i A-7 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot ^3(c+d x) \left (48 a^3 (7 A+3 i B)-30 a^3 (11 i A-5 B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{(7 A+3 i B) \cot ^2(c+d x)}{2 a^3 d}+\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot ^2(c+d x) \left (-30 a^3 (11 i A-5 B)-48 a^3 (7 A+3 i B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=\frac{5 (11 i A-5 B) \cot (c+d x)}{8 a^3 d}-\frac{(7 A+3 i B) \cot ^2(c+d x)}{2 a^3 d}+\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (-48 a^3 (7 A+3 i B)+30 a^3 (11 i A-5 B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=\frac{5 (11 i A-5 B) x}{8 a^3}+\frac{5 (11 i A-5 B) \cot (c+d x)}{8 a^3 d}-\frac{(7 A+3 i B) \cot ^2(c+d x)}{2 a^3 d}+\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{(7 A+3 i B) \int \cot (c+d x) \, dx}{a^3}\\ &=\frac{5 (11 i A-5 B) x}{8 a^3}+\frac{5 (11 i A-5 B) \cot (c+d x)}{8 a^3 d}-\frac{(7 A+3 i B) \cot ^2(c+d x)}{2 a^3 d}-\frac{(7 A+3 i B) \log (\sin (c+d x))}{a^3 d}+\frac{(A+i B) \cot ^2(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(13 A+7 i B) \cot ^2(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{5 (11 A+5 i B) \cot ^2(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 7.25086, size = 1448, normalized size = 6.7 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]

[Out]

((9*A + (7*I)*B)*Cos[4*d*x]*Sec[c + d*x]^2*(-Cos[c]/32 + (I/32)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c
+ d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((39*A + (23*I)*B)*Cos[2*d*x]*Sec[c
+ d*x]^2*(-Cos[c]/16 - (I/16)*Sin[c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*
Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c + d*x]^2*(7*A*Cos[(3*c)/2] + (3*I)*B*Cos[(3*c)/2] + (7*I)*A*S
in[(3*c)/2] - 3*B*Sin[(3*c)/2])*(I*ArcTan[Tan[d*x]]*Cos[(3*c)/2] - ArcTan[Tan[d*x]]*Sin[(3*c)/2])*(Cos[d*x] +
I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Sec[c +
d*x]^2*(7*A*Cos[(3*c)/2] + (3*I)*B*Cos[(3*c)/2] + (7*I)*A*Sin[(3*c)/2] - 3*B*Sin[(3*c)/2])*(-(Cos[(3*c)/2]*Log
[Sin[c + d*x]^2])/2 - (I/2)*Log[Sin[c + d*x]^2]*Sin[(3*c)/2])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/
(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (x*Sec[c + d*x]^2*((14*I)*A*Cos[c] - 6*B*Cos[
c] + 7*A*Cos[c]*Cot[c] + (3*I)*B*Cos[c]*Cot[c] - 7*A*Sin[c] - (3*I)*B*Sin[c] + (7*A + (3*I)*B)*Cot[c]*(-Cos[3*
c] - I*Sin[3*c]))*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/((A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*
Tan[c + d*x])^3) + ((A + I*B)*Cos[6*d*x]*Sec[c + d*x]^2*(-Cos[3*c]/48 + (I/48)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x
])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + (Csc[c + d*x]^2*Se
c[c + d*x]^2*(-(A*Cos[3*c])/2 - (I/2)*A*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c
+ d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3) + ((11*A + (5*I)*B)*Sec[c + d*x]^2*(((5*I)/8)*d*x*Cos[3*c]
- (5*d*x*Sin[3*c])/8)*(Cos[d*x] + I*Sin[d*x])^3*(A + B*Tan[c + d*x]))/(d*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a
+ I*a*Tan[c + d*x])^3) + ((39*A + (23*I)*B)*Sec[c + d*x]^2*((I/16)*Cos[c] - Sin[c]/16)*(Cos[d*x] + I*Sin[d*x])
^3*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])^3) + ((9*A + (
7*I)*B)*Sec[c + d*x]^2*((I/32)*Cos[c] + Sin[c]/32)*(Cos[d*x] + I*Sin[d*x])^3*Sin[4*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])^3) + ((A + I*B)*Sec[c + d*x]^2*((I/48)*Cos[3*c] +
Sin[3*c]/48)*(Cos[d*x] + I*Sin[d*x])^3*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])^3) + (Csc[c/2]*Csc[c + d*x]*Sec[c/2]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((3*A*Cos[
3*c - d*x])/2 + (I/2)*B*Cos[3*c - d*x] - (3*A*Cos[3*c + d*x])/2 - (I/2)*B*Cos[3*c + d*x] + ((3*I)/2)*A*Sin[3*c
- d*x] - (B*Sin[3*c - d*x])/2 - ((3*I)/2)*A*Sin[3*c + d*x] + (B*Sin[3*c + d*x])/2)*(A + B*Tan[c + d*x]))/(2*d
*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Tan[c + d*x])^3)

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Maple [A]  time = 0.13, size = 288, normalized size = 1.3 \begin{align*}{\frac{{\frac{31\,i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{17\,B}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{49\,i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{{a}^{3}d}}+{\frac{111\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{16\,{a}^{3}d}}-{\frac{{\frac{i}{6}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{B}{6\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{7\,A}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{{\frac{5\,i}{8}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,{a}^{3}d}}-{\frac{{\frac{i}{16}}B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{3}d}}-{\frac{A}{2\,{a}^{3}d \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,iA}{{a}^{3}d\tan \left ( dx+c \right ) }}-{\frac{B}{{a}^{3}d\tan \left ( dx+c \right ) }}-{\frac{3\,iB\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{3}d}}-7\,{\frac{A\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x)

[Out]

31/8*I/d/a^3/(tan(d*x+c)-I)*A-17/8/d/a^3/(tan(d*x+c)-I)*B+49/16*I/d/a^3*ln(tan(d*x+c)-I)*B+111/16/d/a^3*ln(tan
(d*x+c)-I)*A-1/6*I/d/a^3/(tan(d*x+c)-I)^3*A+1/6/d/a^3/(tan(d*x+c)-I)^3*B+7/8/d/a^3/(tan(d*x+c)-I)^2*A+5/8*I/d/
a^3/(tan(d*x+c)-I)^2*B+1/16/d/a^3*A*ln(tan(d*x+c)+I)-1/16*I/d/a^3*B*ln(tan(d*x+c)+I)-1/2/d/a^3*A/tan(d*x+c)^2+
3*I/d/a^3/tan(d*x+c)*A-1/d/a^3/tan(d*x+c)*B-3*I/d/a^3*B*ln(tan(d*x+c))-7/d/a^3*A*ln(tan(d*x+c))

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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)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.52476, size = 691, normalized size = 3.2 \begin{align*} \frac{{\left (1332 i \, A - 588 \, B\right )} d x e^{\left (10 i \, d x + 10 i \, c\right )} +{\left ({\left (-2664 i \, A + 1176 \, B\right )} d x - 618 \, A - 330 i \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left ({\left (1332 i \, A - 588 \, B\right )} d x + 1017 \, A + 447 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - 14 \,{\left (13 \, A + 7 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} -{\left (23 \, A + 17 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 96 \,{\left ({\left (7 \, A + 3 i \, B\right )} e^{\left (10 i \, d x + 10 i \, c\right )} - 2 \,{\left (7 \, A + 3 i \, B\right )} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (7 \, A + 3 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 2 \, A - 2 i \, B}{96 \,{\left (a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} - 2 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/96*((1332*I*A - 588*B)*d*x*e^(10*I*d*x + 10*I*c) + ((-2664*I*A + 1176*B)*d*x - 618*A - 330*I*B)*e^(8*I*d*x +
8*I*c) + ((1332*I*A - 588*B)*d*x + 1017*A + 447*I*B)*e^(6*I*d*x + 6*I*c) - 14*(13*A + 7*I*B)*e^(4*I*d*x + 4*I
*c) - (23*A + 17*I*B)*e^(2*I*d*x + 2*I*c) - 96*((7*A + 3*I*B)*e^(10*I*d*x + 10*I*c) - 2*(7*A + 3*I*B)*e^(8*I*d
*x + 8*I*c) + (7*A + 3*I*B)*e^(6*I*d*x + 6*I*c))*log(e^(2*I*d*x + 2*I*c) - 1) - 2*A - 2*I*B)/(a^3*d*e^(10*I*d*
x + 10*I*c) - 2*a^3*d*e^(8*I*d*x + 8*I*c) + a^3*d*e^(6*I*d*x + 6*I*c))

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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)**3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.52209, size = 286, normalized size = 1.32 \begin{align*} \frac{\frac{6 \,{\left (111 \, A + 49 i \, B\right )} \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{6 \,{\left (A - i \, B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac{96 \,{\left (7 \, A + 3 i \, B\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} + \frac{48 \,{\left (21 \, A \tan \left (d x + c\right )^{2} + 9 i \, B \tan \left (d x + c\right )^{2} + 6 i \, A \tan \left (d x + c\right ) - 2 \, B \tan \left (d x + c\right ) - A\right )}}{a^{3} \tan \left (d x + c\right )^{2}} + \frac{1221 i \, A \tan \left (d x + c\right )^{3} - 539 \, B \tan \left (d x + c\right )^{3} + 4035 \, A \tan \left (d x + c\right )^{2} + 1821 i \, B \tan \left (d x + c\right )^{2} - 4491 i \, A \tan \left (d x + c\right ) + 2085 \, B \tan \left (d x + c\right ) - 1693 \, A - 819 i \, B}{a^{3}{\left (i \, \tan \left (d x + c\right ) + 1\right )}^{3}}}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/96*(6*(111*A + 49*I*B)*log(I*tan(d*x + c) + 1)/a^3 + 6*(A - I*B)*log(I*tan(d*x + c) - 1)/a^3 - 96*(7*A + 3*I
*B)*log(abs(tan(d*x + c)))/a^3 + 48*(21*A*tan(d*x + c)^2 + 9*I*B*tan(d*x + c)^2 + 6*I*A*tan(d*x + c) - 2*B*tan
(d*x + c) - A)/(a^3*tan(d*x + c)^2) + (1221*I*A*tan(d*x + c)^3 - 539*B*tan(d*x + c)^3 + 4035*A*tan(d*x + c)^2
+ 1821*I*B*tan(d*x + c)^2 - 4491*I*A*tan(d*x + c) + 2085*B*tan(d*x + c) - 1693*A - 819*I*B)/(a^3*(I*tan(d*x +
c) + 1)^3))/d