3.28 \(\int \frac{(c+d x)^2}{(a+i a \cot (e+f x))^3} \, dx\)

Optimal. Leaf size=294 \[ -\frac{3 d (c+d x) e^{2 i e+2 i f x}}{16 a^3 f^2}+\frac{3 d (c+d x) e^{4 i e+4 i f x}}{64 a^3 f^2}-\frac{d (c+d x) e^{6 i e+6 i f x}}{144 a^3 f^2}+\frac{3 i (c+d x)^2 e^{2 i e+2 i f x}}{16 a^3 f}-\frac{3 i (c+d x)^2 e^{4 i e+4 i f x}}{32 a^3 f}+\frac{i (c+d x)^2 e^{6 i e+6 i f x}}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{3 i d^2 e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac{3 i d^2 e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac{i d^2 e^{6 i e+6 i f x}}{864 a^3 f^3} \]

[Out]

(((-3*I)/32)*d^2*E^((2*I)*e + (2*I)*f*x))/(a^3*f^3) + (((3*I)/256)*d^2*E^((4*I)*e + (4*I)*f*x))/(a^3*f^3) - ((
I/864)*d^2*E^((6*I)*e + (6*I)*f*x))/(a^3*f^3) - (3*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(16*a^3*f^2) + (3*d*E^
((4*I)*e + (4*I)*f*x)*(c + d*x))/(64*a^3*f^2) - (d*E^((6*I)*e + (6*I)*f*x)*(c + d*x))/(144*a^3*f^2) + (((3*I)/
16)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(a^3*f) - (((3*I)/32)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(a^3*f) +
((I/48)*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^2)/(a^3*f) + (c + d*x)^3/(24*a^3*d)

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Rubi [A]  time = 0.268709, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3729, 2176, 2194} \[ -\frac{3 d (c+d x) e^{2 i e+2 i f x}}{16 a^3 f^2}+\frac{3 d (c+d x) e^{4 i e+4 i f x}}{64 a^3 f^2}-\frac{d (c+d x) e^{6 i e+6 i f x}}{144 a^3 f^2}+\frac{3 i (c+d x)^2 e^{2 i e+2 i f x}}{16 a^3 f}-\frac{3 i (c+d x)^2 e^{4 i e+4 i f x}}{32 a^3 f}+\frac{i (c+d x)^2 e^{6 i e+6 i f x}}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{3 i d^2 e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac{3 i d^2 e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac{i d^2 e^{6 i e+6 i f x}}{864 a^3 f^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + I*a*Cot[e + f*x])^3,x]

[Out]

(((-3*I)/32)*d^2*E^((2*I)*e + (2*I)*f*x))/(a^3*f^3) + (((3*I)/256)*d^2*E^((4*I)*e + (4*I)*f*x))/(a^3*f^3) - ((
I/864)*d^2*E^((6*I)*e + (6*I)*f*x))/(a^3*f^3) - (3*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(16*a^3*f^2) + (3*d*E^
((4*I)*e + (4*I)*f*x)*(c + d*x))/(64*a^3*f^2) - (d*E^((6*I)*e + (6*I)*f*x)*(c + d*x))/(144*a^3*f^2) + (((3*I)/
16)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(a^3*f) - (((3*I)/32)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(a^3*f) +
((I/48)*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^2)/(a^3*f) + (c + d*x)^3/(24*a^3*d)

Rule 3729

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+i a \cot (e+f x))^3} \, dx &=\int \left (\frac{(c+d x)^2}{8 a^3}-\frac{3 e^{2 i e+2 i f x} (c+d x)^2}{8 a^3}+\frac{3 e^{4 i e+4 i f x} (c+d x)^2}{8 a^3}-\frac{e^{6 i e+6 i f x} (c+d x)^2}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^3}{24 a^3 d}-\frac{\int e^{6 i e+6 i f x} (c+d x)^2 \, dx}{8 a^3}-\frac{3 \int e^{2 i e+2 i f x} (c+d x)^2 \, dx}{8 a^3}+\frac{3 \int e^{4 i e+4 i f x} (c+d x)^2 \, dx}{8 a^3}\\ &=\frac{3 i e^{2 i e+2 i f x} (c+d x)^2}{16 a^3 f}-\frac{3 i e^{4 i e+4 i f x} (c+d x)^2}{32 a^3 f}+\frac{i e^{6 i e+6 i f x} (c+d x)^2}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{(i d) \int e^{6 i e+6 i f x} (c+d x) \, dx}{24 a^3 f}+\frac{(3 i d) \int e^{4 i e+4 i f x} (c+d x) \, dx}{16 a^3 f}-\frac{(3 i d) \int e^{2 i e+2 i f x} (c+d x) \, dx}{8 a^3 f}\\ &=-\frac{3 d e^{2 i e+2 i f x} (c+d x)}{16 a^3 f^2}+\frac{3 d e^{4 i e+4 i f x} (c+d x)}{64 a^3 f^2}-\frac{d e^{6 i e+6 i f x} (c+d x)}{144 a^3 f^2}+\frac{3 i e^{2 i e+2 i f x} (c+d x)^2}{16 a^3 f}-\frac{3 i e^{4 i e+4 i f x} (c+d x)^2}{32 a^3 f}+\frac{i e^{6 i e+6 i f x} (c+d x)^2}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}+\frac{d^2 \int e^{6 i e+6 i f x} \, dx}{144 a^3 f^2}-\frac{\left (3 d^2\right ) \int e^{4 i e+4 i f x} \, dx}{64 a^3 f^2}+\frac{\left (3 d^2\right ) \int e^{2 i e+2 i f x} \, dx}{16 a^3 f^2}\\ &=-\frac{3 i d^2 e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac{3 i d^2 e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac{i d^2 e^{6 i e+6 i f x}}{864 a^3 f^3}-\frac{3 d e^{2 i e+2 i f x} (c+d x)}{16 a^3 f^2}+\frac{3 d e^{4 i e+4 i f x} (c+d x)}{64 a^3 f^2}-\frac{d e^{6 i e+6 i f x} (c+d x)}{144 a^3 f^2}+\frac{3 i e^{2 i e+2 i f x} (c+d x)^2}{16 a^3 f}-\frac{3 i e^{4 i e+4 i f x} (c+d x)^2}{32 a^3 f}+\frac{i e^{6 i e+6 i f x} (c+d x)^2}{48 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.755708, size = 369, normalized size = 1.26 \[ \frac{288 f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )+648 (\cos (2 e)+i \sin (2 e)) \cos (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-81 (\cos (4 e)+i \sin (4 e)) \cos (4 f x) ((2+2 i) c f+d (-1+(2+2 i) f x)) ((2+2 i) c f+d ((2+2 i) f x+i))+8 (\cos (6 e)+i \sin (6 e)) \cos (6 f x) ((3+3 i) c f+d (-1+(3+3 i) f x)) ((3+3 i) c f+d ((3+3 i) f x+i))+648 i (\cos (2 e)+i \sin (2 e)) \sin (2 f x) ((1+i) c f+d (-1+(1+i) f x)) ((1+i) c f+d ((1+i) f x+i))-81 (\cos (4 e)+i \sin (4 e)) \sin (4 f x) (-(2+2 i) c f+(-2-2 i) d f x+d) ((2-2 i) c f+(2-2 i) d f x+d)+8 i (\cos (6 e)+i \sin (6 e)) \sin (6 f x) ((3+3 i) c f+d (-1+(3+3 i) f x)) ((3+3 i) c f+d ((3+3 i) f x+i))}{6912 a^3 f^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + I*a*Cot[e + f*x])^3,x]

[Out]

(288*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 648*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))*((1 + I)*c*f + d*(I + (1 + I
)*f*x))*Cos[2*f*x]*(Cos[2*e] + I*Sin[2*e]) - 81*((2 + 2*I)*c*f + d*(-1 + (2 + 2*I)*f*x))*((2 + 2*I)*c*f + d*(I
+ (2 + 2*I)*f*x))*Cos[4*f*x]*(Cos[4*e] + I*Sin[4*e]) + 8*((3 + 3*I)*c*f + d*(-1 + (3 + 3*I)*f*x))*((3 + 3*I)*
c*f + d*(I + (3 + 3*I)*f*x))*Cos[6*f*x]*(Cos[6*e] + I*Sin[6*e]) + (648*I)*((1 + I)*c*f + d*(-1 + (1 + I)*f*x))
*((1 + I)*c*f + d*(I + (1 + I)*f*x))*(Cos[2*e] + I*Sin[2*e])*Sin[2*f*x] - 81*(d - (2 + 2*I)*c*f - (2 + 2*I)*d*
f*x)*(d + (2 - 2*I)*c*f + (2 - 2*I)*d*f*x)*(Cos[4*e] + I*Sin[4*e])*Sin[4*f*x] + (8*I)*((3 + 3*I)*c*f + d*(-1 +
(3 + 3*I)*f*x))*((3 + 3*I)*c*f + d*(I + (3 + 3*I)*f*x))*(Cos[6*e] + I*Sin[6*e])*Sin[6*f*x])/(6912*a^3*f^3)

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Maple [B]  time = 0.078, size = 1843, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a+I*a*cot(f*x+e))^3,x)

[Out]

1/f^3/a^3*(-6*I*c*d*f*(1/4*(f*x+e)*sin(f*x+e)^4+1/16*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)-3/32*f*x-3/32*e)
-8*I*c*d*e*f*(-1/6*sin(f*x+e)^2*cos(f*x+e)^4-1/12*cos(f*x+e)^4)+3/2*I*c*d*e*f*sin(f*x+e)^4-3*I*d^2*(1/4*(f*x+e
)^2*sin(f*x+e)^4-1/2*(f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+3/32*(f*x+e)^2-1/32
*sin(f*x+e)^4-3/32*sin(f*x+e)^2)+4*I*d^2*(1/4*(f*x+e)^2*sin(f*x+e)^4-1/2*(f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f
*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+1/24*(f*x+e)^2-1/72*sin(f*x+e)^4-1/24*sin(f*x+e)^2-1/6*(f*x+e)^2*sin(f*x+e)^6
+1/3*(f*x+e)*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)+1/108*sin(f*x+e
)^6)-3/4*I*d^2*e^2*sin(f*x+e)^4-4*d^2*((f*x+e)^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)
+1/48*(f*x+e)*sin(f*x+e)^4+1/192*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+47/1152*f*x+47/1152*e-1/16*(f*x+e)*c
os(f*x+e)^2+1/32*cos(f*x+e)*sin(f*x+e)-1/24*(f*x+e)^3-(f*x+e)^2*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(
f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)-1/18*(f*x+e)*sin(f*x+e)^6-1/108*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*
x+e))*cos(f*x+e))-8*c*d*f*((f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-1/32*(f*x+e)^
2+1/96*sin(f*x+e)^4+1/32*sin(f*x+e)^2-(f*x+e)*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)
+5/16*f*x+5/16*e)-1/36*sin(f*x+e)^6)+8*d^2*e*((f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3
/8*e)-1/32*(f*x+e)^2+1/96*sin(f*x+e)^4+1/32*sin(f*x+e)^2-(f*x+e)*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin
(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)-1/36*sin(f*x+e)^6)-4*c^2*f^2*(-1/6*sin(f*x+e)^3*cos(f*x+e)^3-1/8*sin(f*x+
e)*cos(f*x+e)^3+1/16*cos(f*x+e)*sin(f*x+e)+1/16*f*x+1/16*e)+8*c*d*e*f*(-1/6*sin(f*x+e)^3*cos(f*x+e)^3-1/8*sin(
f*x+e)*cos(f*x+e)^3+1/16*cos(f*x+e)*sin(f*x+e)+1/16*f*x+1/16*e)-4*d^2*e^2*(-1/6*sin(f*x+e)^3*cos(f*x+e)^3-1/8*
sin(f*x+e)*cos(f*x+e)^3+1/16*cos(f*x+e)*sin(f*x+e)+1/16*f*x+1/16*e)+8*I*c*d*f*(1/4*(f*x+e)*sin(f*x+e)^4+1/16*(
sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)-1/24*f*x-1/24*e-1/6*(f*x+e)*sin(f*x+e)^6-1/36*(sin(f*x+e)^5+5/4*sin(f*
x+e)^3+15/8*sin(f*x+e))*cos(f*x+e))+4*I*c^2*f^2*(-1/6*sin(f*x+e)^2*cos(f*x+e)^4-1/12*cos(f*x+e)^4)-8*I*d^2*e*(
1/4*(f*x+e)*sin(f*x+e)^4+1/16*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)-1/24*f*x-1/24*e-1/6*(f*x+e)*sin(f*x+e)^
6-1/36*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e))+4*I*d^2*e^2*(-1/6*sin(f*x+e)^2*cos(f*x+e)^4
-1/12*cos(f*x+e)^4)+6*I*d^2*e*(1/4*(f*x+e)*sin(f*x+e)^4+1/16*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)-3/32*f*x
-3/32*e)-3/4*I*c^2*f^2*sin(f*x+e)^4+d^2*((f*x+e)^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*
e)+1/8*(f*x+e)*sin(f*x+e)^4+1/32*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+9/64*f*x+9/64*e-3/8*(f*x+e)*cos(f*x+
e)^2+3/16*cos(f*x+e)*sin(f*x+e)-1/4*(f*x+e)^3)+2*c*d*f*((f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)
+3/8*f*x+3/8*e)-3/16*(f*x+e)^2+1/16*sin(f*x+e)^4+3/16*sin(f*x+e)^2)-2*d^2*e*((f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*s
in(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-3/16*(f*x+e)^2+1/16*sin(f*x+e)^4+3/16*sin(f*x+e)^2)+c^2*f^2*(-1/4*(sin(f*
x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-2*c*d*e*f*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*
f*x+3/8*e)+d^2*e^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e))

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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((d*x+c)^2/(a+I*a*cot(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.61122, size = 560, normalized size = 1.9 \begin{align*} \frac{288 \, d^{2} f^{3} x^{3} + 864 \, c d f^{3} x^{2} + 864 \, c^{2} f^{3} x +{\left (144 i \, d^{2} f^{2} x^{2} + 144 i \, c^{2} f^{2} - 48 \, c d f - 8 i \, d^{2} +{\left (288 i \, c d f^{2} - 48 \, d^{2} f\right )} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-648 i \, d^{2} f^{2} x^{2} - 648 i \, c^{2} f^{2} + 324 \, c d f + 81 i \, d^{2} +{\left (-1296 i \, c d f^{2} + 324 \, d^{2} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (1296 i \, d^{2} f^{2} x^{2} + 1296 i \, c^{2} f^{2} - 1296 \, c d f - 648 i \, d^{2} +{\left (2592 i \, c d f^{2} - 1296 \, d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{6912 \, a^{3} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+I*a*cot(f*x+e))^3,x, algorithm="fricas")

[Out]

1/6912*(288*d^2*f^3*x^3 + 864*c*d*f^3*x^2 + 864*c^2*f^3*x + (144*I*d^2*f^2*x^2 + 144*I*c^2*f^2 - 48*c*d*f - 8*
I*d^2 + (288*I*c*d*f^2 - 48*d^2*f)*x)*e^(6*I*f*x + 6*I*e) + (-648*I*d^2*f^2*x^2 - 648*I*c^2*f^2 + 324*c*d*f +
81*I*d^2 + (-1296*I*c*d*f^2 + 324*d^2*f)*x)*e^(4*I*f*x + 4*I*e) + (1296*I*d^2*f^2*x^2 + 1296*I*c^2*f^2 - 1296*
c*d*f - 648*I*d^2 + (2592*I*c*d*f^2 - 1296*d^2*f)*x)*e^(2*I*f*x + 2*I*e))/(a^3*f^3)

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Sympy [A]  time = 1.48968, size = 576, normalized size = 1.96 \begin{align*} \begin{cases} \frac{\left (1327104 i a^{24} c^{2} f^{17} e^{2 i e} + 2654208 i a^{24} c d f^{17} x e^{2 i e} - 1327104 a^{24} c d f^{16} e^{2 i e} + 1327104 i a^{24} d^{2} f^{17} x^{2} e^{2 i e} - 1327104 a^{24} d^{2} f^{16} x e^{2 i e} - 663552 i a^{24} d^{2} f^{15} e^{2 i e}\right ) e^{2 i f x} + \left (- 663552 i a^{24} c^{2} f^{17} e^{4 i e} - 1327104 i a^{24} c d f^{17} x e^{4 i e} + 331776 a^{24} c d f^{16} e^{4 i e} - 663552 i a^{24} d^{2} f^{17} x^{2} e^{4 i e} + 331776 a^{24} d^{2} f^{16} x e^{4 i e} + 82944 i a^{24} d^{2} f^{15} e^{4 i e}\right ) e^{4 i f x} + \left (147456 i a^{24} c^{2} f^{17} e^{6 i e} + 294912 i a^{24} c d f^{17} x e^{6 i e} - 49152 a^{24} c d f^{16} e^{6 i e} + 147456 i a^{24} d^{2} f^{17} x^{2} e^{6 i e} - 49152 a^{24} d^{2} f^{16} x e^{6 i e} - 8192 i a^{24} d^{2} f^{15} e^{6 i e}\right ) e^{6 i f x}}{7077888 a^{27} f^{18}} & \text{for}\: 7077888 a^{27} f^{18} \neq 0 \\\frac{x^{3} \left (- d^{2} e^{6 i e} + 3 d^{2} e^{4 i e} - 3 d^{2} e^{2 i e}\right )}{24 a^{3}} + \frac{x^{2} \left (- c d e^{6 i e} + 3 c d e^{4 i e} - 3 c d e^{2 i e}\right )}{8 a^{3}} + \frac{x \left (- c^{2} e^{6 i e} + 3 c^{2} e^{4 i e} - 3 c^{2} e^{2 i e}\right )}{8 a^{3}} & \text{otherwise} \end{cases} + \frac{c^{2} x}{8 a^{3}} + \frac{c d x^{2}}{8 a^{3}} + \frac{d^{2} x^{3}}{24 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a+I*a*cot(f*x+e))**3,x)

[Out]

Piecewise((((1327104*I*a**24*c**2*f**17*exp(2*I*e) + 2654208*I*a**24*c*d*f**17*x*exp(2*I*e) - 1327104*a**24*c*
d*f**16*exp(2*I*e) + 1327104*I*a**24*d**2*f**17*x**2*exp(2*I*e) - 1327104*a**24*d**2*f**16*x*exp(2*I*e) - 6635
52*I*a**24*d**2*f**15*exp(2*I*e))*exp(2*I*f*x) + (-663552*I*a**24*c**2*f**17*exp(4*I*e) - 1327104*I*a**24*c*d*
f**17*x*exp(4*I*e) + 331776*a**24*c*d*f**16*exp(4*I*e) - 663552*I*a**24*d**2*f**17*x**2*exp(4*I*e) + 331776*a*
*24*d**2*f**16*x*exp(4*I*e) + 82944*I*a**24*d**2*f**15*exp(4*I*e))*exp(4*I*f*x) + (147456*I*a**24*c**2*f**17*e
xp(6*I*e) + 294912*I*a**24*c*d*f**17*x*exp(6*I*e) - 49152*a**24*c*d*f**16*exp(6*I*e) + 147456*I*a**24*d**2*f**
17*x**2*exp(6*I*e) - 49152*a**24*d**2*f**16*x*exp(6*I*e) - 8192*I*a**24*d**2*f**15*exp(6*I*e))*exp(6*I*f*x))/(
7077888*a**27*f**18), Ne(7077888*a**27*f**18, 0)), (x**3*(-d**2*exp(6*I*e) + 3*d**2*exp(4*I*e) - 3*d**2*exp(2*
I*e))/(24*a**3) + x**2*(-c*d*exp(6*I*e) + 3*c*d*exp(4*I*e) - 3*c*d*exp(2*I*e))/(8*a**3) + x*(-c**2*exp(6*I*e)
+ 3*c**2*exp(4*I*e) - 3*c**2*exp(2*I*e))/(8*a**3), True)) + c**2*x/(8*a**3) + c*d*x**2/(8*a**3) + d**2*x**3/(2
4*a**3)

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Giac [A]  time = 1.26437, size = 474, normalized size = 1.61 \begin{align*} \frac{288 \, d^{2} f^{3} x^{3} + 864 \, c d f^{3} x^{2} + 144 i \, d^{2} f^{2} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 648 i \, d^{2} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 864 \, c^{2} f^{3} x + 288 i \, c d f^{2} x e^{\left (6 i \, f x + 6 i \, e\right )} - 1296 i \, c d f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 144 i \, c^{2} f^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 48 \, d^{2} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 648 i \, c^{2} f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 324 \, d^{2} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 1296 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 1296 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} - 48 \, c d f e^{\left (6 i \, f x + 6 i \, e\right )} + 324 \, c d f e^{\left (4 i \, f x + 4 i \, e\right )} - 1296 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 81 i \, d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 648 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )}}{6912 \, a^{3} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+I*a*cot(f*x+e))^3,x, algorithm="giac")

[Out]

1/6912*(288*d^2*f^3*x^3 + 864*c*d*f^3*x^2 + 144*I*d^2*f^2*x^2*e^(6*I*f*x + 6*I*e) - 648*I*d^2*f^2*x^2*e^(4*I*f
*x + 4*I*e) + 1296*I*d^2*f^2*x^2*e^(2*I*f*x + 2*I*e) + 864*c^2*f^3*x + 288*I*c*d*f^2*x*e^(6*I*f*x + 6*I*e) - 1
296*I*c*d*f^2*x*e^(4*I*f*x + 4*I*e) + 2592*I*c*d*f^2*x*e^(2*I*f*x + 2*I*e) + 144*I*c^2*f^2*e^(6*I*f*x + 6*I*e)
- 48*d^2*f*x*e^(6*I*f*x + 6*I*e) - 648*I*c^2*f^2*e^(4*I*f*x + 4*I*e) + 324*d^2*f*x*e^(4*I*f*x + 4*I*e) + 1296
*I*c^2*f^2*e^(2*I*f*x + 2*I*e) - 1296*d^2*f*x*e^(2*I*f*x + 2*I*e) - 48*c*d*f*e^(6*I*f*x + 6*I*e) + 324*c*d*f*e
^(4*I*f*x + 4*I*e) - 1296*c*d*f*e^(2*I*f*x + 2*I*e) - 8*I*d^2*e^(6*I*f*x + 6*I*e) + 81*I*d^2*e^(4*I*f*x + 4*I*
e) - 648*I*d^2*e^(2*I*f*x + 2*I*e))/(a^3*f^3)