3.27 \(\int \frac{(c+d x)^3}{(a+i a \cot (e+f x))^3} \, dx\)
Optimal. Leaf size=396 \[ -\frac{9 i d^2 (c+d x) e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac{9 i d^2 (c+d x) e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac{i d^2 (c+d x) e^{6 i e+6 i f x}}{288 a^3 f^3}-\frac{9 d (c+d x)^2 e^{2 i e+2 i f x}}{32 a^3 f^2}+\frac{9 d (c+d x)^2 e^{4 i e+4 i f x}}{128 a^3 f^2}-\frac{d (c+d x)^2 e^{6 i e+6 i f x}}{96 a^3 f^2}+\frac{3 i (c+d x)^3 e^{2 i e+2 i f x}}{16 a^3 f}-\frac{3 i (c+d x)^3 e^{4 i e+4 i f x}}{32 a^3 f}+\frac{i (c+d x)^3 e^{6 i e+6 i f x}}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{9 d^3 e^{2 i e+2 i f x}}{64 a^3 f^4}-\frac{9 d^3 e^{4 i e+4 i f x}}{1024 a^3 f^4}+\frac{d^3 e^{6 i e+6 i f x}}{1728 a^3 f^4} \]
[Out]
(9*d^3*E^((2*I)*e + (2*I)*f*x))/(64*a^3*f^4) - (9*d^3*E^((4*I)*e + (4*I)*f*x))/(1024*a^3*f^4) + (d^3*E^((6*I)*
e + (6*I)*f*x))/(1728*a^3*f^4) - (((9*I)/32)*d^2*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(a^3*f^3) + (((9*I)/256)*d
^2*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(a^3*f^3) - ((I/288)*d^2*E^((6*I)*e + (6*I)*f*x)*(c + d*x))/(a^3*f^3) -
(9*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(32*a^3*f^2) + (9*d*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(128*a^3*f^
2) - (d*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^2)/(96*a^3*f^2) + (((3*I)/16)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^3)/(
a^3*f) - (((3*I)/32)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^3)/(a^3*f) + ((I/48)*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^
3)/(a^3*f) + (c + d*x)^4/(32*a^3*d)
________________________________________________________________________________________
Rubi [A] time = 0.381079, antiderivative size = 396, normalized size of antiderivative = 1.,
number of steps used = 14, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used =
{3729, 2176, 2194} \[ -\frac{9 i d^2 (c+d x) e^{2 i e+2 i f x}}{32 a^3 f^3}+\frac{9 i d^2 (c+d x) e^{4 i e+4 i f x}}{256 a^3 f^3}-\frac{i d^2 (c+d x) e^{6 i e+6 i f x}}{288 a^3 f^3}-\frac{9 d (c+d x)^2 e^{2 i e+2 i f x}}{32 a^3 f^2}+\frac{9 d (c+d x)^2 e^{4 i e+4 i f x}}{128 a^3 f^2}-\frac{d (c+d x)^2 e^{6 i e+6 i f x}}{96 a^3 f^2}+\frac{3 i (c+d x)^3 e^{2 i e+2 i f x}}{16 a^3 f}-\frac{3 i (c+d x)^3 e^{4 i e+4 i f x}}{32 a^3 f}+\frac{i (c+d x)^3 e^{6 i e+6 i f x}}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{9 d^3 e^{2 i e+2 i f x}}{64 a^3 f^4}-\frac{9 d^3 e^{4 i e+4 i f x}}{1024 a^3 f^4}+\frac{d^3 e^{6 i e+6 i f x}}{1728 a^3 f^4} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^3/(a + I*a*Cot[e + f*x])^3,x]
[Out]
(9*d^3*E^((2*I)*e + (2*I)*f*x))/(64*a^3*f^4) - (9*d^3*E^((4*I)*e + (4*I)*f*x))/(1024*a^3*f^4) + (d^3*E^((6*I)*
e + (6*I)*f*x))/(1728*a^3*f^4) - (((9*I)/32)*d^2*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(a^3*f^3) + (((9*I)/256)*d
^2*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(a^3*f^3) - ((I/288)*d^2*E^((6*I)*e + (6*I)*f*x)*(c + d*x))/(a^3*f^3) -
(9*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(32*a^3*f^2) + (9*d*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(128*a^3*f^
2) - (d*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^2)/(96*a^3*f^2) + (((3*I)/16)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^3)/(
a^3*f) - (((3*I)/32)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^3)/(a^3*f) + ((I/48)*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^
3)/(a^3*f) + (c + d*x)^4/(32*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)^3}{(a+i a \cot (e+f x))^3} \, dx &=\int \left (\frac{(c+d x)^3}{8 a^3}-\frac{3 e^{2 i e+2 i f x} (c+d x)^3}{8 a^3}+\frac{3 e^{4 i e+4 i f x} (c+d x)^3}{8 a^3}-\frac{e^{6 i e+6 i f x} (c+d x)^3}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^4}{32 a^3 d}-\frac{\int e^{6 i e+6 i f x} (c+d x)^3 \, dx}{8 a^3}-\frac{3 \int e^{2 i e+2 i f x} (c+d x)^3 \, dx}{8 a^3}+\frac{3 \int e^{4 i e+4 i f x} (c+d x)^3 \, dx}{8 a^3}\\ &=\frac{3 i e^{2 i e+2 i f x} (c+d x)^3}{16 a^3 f}-\frac{3 i e^{4 i e+4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{6 i e+6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}-\frac{(i d) \int e^{6 i e+6 i f x} (c+d x)^2 \, dx}{16 a^3 f}+\frac{(9 i d) \int e^{4 i e+4 i f x} (c+d x)^2 \, dx}{32 a^3 f}-\frac{(9 i d) \int e^{2 i e+2 i f x} (c+d x)^2 \, dx}{16 a^3 f}\\ &=-\frac{9 d e^{2 i e+2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac{9 d e^{4 i e+4 i f x} (c+d x)^2}{128 a^3 f^2}-\frac{d e^{6 i e+6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac{3 i e^{2 i e+2 i f x} (c+d x)^3}{16 a^3 f}-\frac{3 i e^{4 i e+4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{6 i e+6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{d^2 \int e^{6 i e+6 i f x} (c+d x) \, dx}{48 a^3 f^2}-\frac{\left (9 d^2\right ) \int e^{4 i e+4 i f x} (c+d x) \, dx}{64 a^3 f^2}+\frac{\left (9 d^2\right ) \int e^{2 i e+2 i f x} (c+d x) \, dx}{16 a^3 f^2}\\ &=-\frac{9 i d^2 e^{2 i e+2 i f x} (c+d x)}{32 a^3 f^3}+\frac{9 i d^2 e^{4 i e+4 i f x} (c+d x)}{256 a^3 f^3}-\frac{i d^2 e^{6 i e+6 i f x} (c+d x)}{288 a^3 f^3}-\frac{9 d e^{2 i e+2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac{9 d e^{4 i e+4 i f x} (c+d x)^2}{128 a^3 f^2}-\frac{d e^{6 i e+6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac{3 i e^{2 i e+2 i f x} (c+d x)^3}{16 a^3 f}-\frac{3 i e^{4 i e+4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{6 i e+6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{\left (i d^3\right ) \int e^{6 i e+6 i f x} \, dx}{288 a^3 f^3}-\frac{\left (9 i d^3\right ) \int e^{4 i e+4 i f x} \, dx}{256 a^3 f^3}+\frac{\left (9 i d^3\right ) \int e^{2 i e+2 i f x} \, dx}{32 a^3 f^3}\\ &=\frac{9 d^3 e^{2 i e+2 i f x}}{64 a^3 f^4}-\frac{9 d^3 e^{4 i e+4 i f x}}{1024 a^3 f^4}+\frac{d^3 e^{6 i e+6 i f x}}{1728 a^3 f^4}-\frac{9 i d^2 e^{2 i e+2 i f x} (c+d x)}{32 a^3 f^3}+\frac{9 i d^2 e^{4 i e+4 i f x} (c+d x)}{256 a^3 f^3}-\frac{i d^2 e^{6 i e+6 i f x} (c+d x)}{288 a^3 f^3}-\frac{9 d e^{2 i e+2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac{9 d e^{4 i e+4 i f x} (c+d x)^2}{128 a^3 f^2}-\frac{d e^{6 i e+6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac{3 i e^{2 i e+2 i f x} (c+d x)^3}{16 a^3 f}-\frac{3 i e^{4 i e+4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{6 i e+6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}\\ \end{align*}
Mathematica [A] time = 2.41125, size = 664, normalized size = 1.68 \[ \frac{(\cos (3 (e+f x))+i \sin (3 (e+f x))) \left (81 i \left (24 c^2 d f^2 (4 f x+3 i)+32 c^3 f^3+12 c d^2 f \left (8 f^2 x^2+12 i f x-7\right )+d^3 \left (32 f^3 x^3+72 i f^2 x^2-84 f x-45 i\right )\right ) \cos (e+f x)+16 \left (18 c^2 d f^2 \left (18 f^2 x^2+6 i f x-1\right )+36 c^3 f^3 (6 f x+i)+6 c d^2 f \left (36 f^3 x^3+18 i f^2 x^2-6 f x-i\right )+d^3 \left (54 f^4 x^4+36 i f^3 x^3-18 f^2 x^2-6 i f x+1\right )\right ) \cos (3 (e+f x))-5184 i c^2 d f^4 x^2 \sin (3 (e+f x))+23328 c^2 d f^3 x \sin (e+f x)-1728 c^2 d f^3 x \sin (3 (e+f x))+9720 i c^2 d f^2 \sin (e+f x)-288 i c^2 d f^2 \sin (3 (e+f x))-3456 i c^3 f^4 x \sin (3 (e+f x))+7776 c^3 f^3 \sin (e+f x)-576 c^3 f^3 \sin (3 (e+f x))-3456 i c d^2 f^4 x^3 \sin (3 (e+f x))+23328 c d^2 f^3 x^2 \sin (e+f x)-1728 c d^2 f^3 x^2 \sin (3 (e+f x))+19440 i c d^2 f^2 x \sin (e+f x)-576 i c d^2 f^2 x \sin (3 (e+f x))-8748 c d^2 f \sin (e+f x)+96 c d^2 f \sin (3 (e+f x))-864 i d^3 f^4 x^4 \sin (3 (e+f x))+7776 d^3 f^3 x^3 \sin (e+f x)-576 d^3 f^3 x^3 \sin (3 (e+f x))+9720 i d^3 f^2 x^2 \sin (e+f x)-288 i d^3 f^2 x^2 \sin (3 (e+f x))-8748 d^3 f x \sin (e+f x)+96 d^3 f x \sin (3 (e+f x))-4131 i d^3 \sin (e+f x)+16 i d^3 \sin (3 (e+f x))\right )}{27648 a^3 f^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^3/(a + I*a*Cot[e + f*x])^3,x]
[Out]
((Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*((81*I)*(32*c^3*f^3 + 24*c^2*d*f^2*(3*I + 4*f*x) + 12*c*d^2*f*(-7 + (
12*I)*f*x + 8*f^2*x^2) + d^3*(-45*I - 84*f*x + (72*I)*f^2*x^2 + 32*f^3*x^3))*Cos[e + f*x] + 16*(36*c^3*f^3*(I
+ 6*f*x) + 18*c^2*d*f^2*(-1 + (6*I)*f*x + 18*f^2*x^2) + 6*c*d^2*f*(-I - 6*f*x + (18*I)*f^2*x^2 + 36*f^3*x^3) +
d^3*(1 - (6*I)*f*x - 18*f^2*x^2 + (36*I)*f^3*x^3 + 54*f^4*x^4))*Cos[3*(e + f*x)] - (4131*I)*d^3*Sin[e + f*x]
- 8748*c*d^2*f*Sin[e + f*x] + (9720*I)*c^2*d*f^2*Sin[e + f*x] + 7776*c^3*f^3*Sin[e + f*x] - 8748*d^3*f*x*Sin[e
+ f*x] + (19440*I)*c*d^2*f^2*x*Sin[e + f*x] + 23328*c^2*d*f^3*x*Sin[e + f*x] + (9720*I)*d^3*f^2*x^2*Sin[e + f
*x] + 23328*c*d^2*f^3*x^2*Sin[e + f*x] + 7776*d^3*f^3*x^3*Sin[e + f*x] + (16*I)*d^3*Sin[3*(e + f*x)] + 96*c*d^
2*f*Sin[3*(e + f*x)] - (288*I)*c^2*d*f^2*Sin[3*(e + f*x)] - 576*c^3*f^3*Sin[3*(e + f*x)] + 96*d^3*f*x*Sin[3*(e
+ f*x)] - (576*I)*c*d^2*f^2*x*Sin[3*(e + f*x)] - 1728*c^2*d*f^3*x*Sin[3*(e + f*x)] - (3456*I)*c^3*f^4*x*Sin[3
*(e + f*x)] - (288*I)*d^3*f^2*x^2*Sin[3*(e + f*x)] - 1728*c*d^2*f^3*x^2*Sin[3*(e + f*x)] - (5184*I)*c^2*d*f^4*
x^2*Sin[3*(e + f*x)] - 576*d^3*f^3*x^3*Sin[3*(e + f*x)] - (3456*I)*c*d^2*f^4*x^3*Sin[3*(e + f*x)] - (864*I)*d^
3*f^4*x^4*Sin[3*(e + f*x)]))/(27648*a^3*f^4)
________________________________________________________________________________________
Maple [B] time = 0.133, size = 3997, normalized size = 10.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^3/(a+I*a*cot(f*x+e))^3,x)
[Out]
1/f^4/a^3*(-12*c*d^2*e^2*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)-6*c*d^2*e*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)+3*c*d^2*e^2*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)-3*c^2*d*e*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)-9*I*c*d^2*f*(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)^2-1/3
2*sin(f*x+e)^4-3/32*sin(f*x+e)^2)+24*c*d^2*e*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*s
in(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)-1/36*sin(f*x+e)^6)+12*I*c*d^2*f*(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)+12*I*c^2*d*f^2*(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))+9/4*I*c^2*d*e*f^2*sin(f*x+e)^4-9/4*I*c*d^2*e^2*f*sin(f*x+e)^4-12*I*c^2*d*e*f^2*(-1/6*sin(f*x+e)^2*cos(
f*x+e)^4-1/12*cos(f*x+e)^4)-4*d^3*((f*x+e)^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+1/3
2*(f*x+e)^2*sin(f*x+e)^4-1/16*(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)-47/768*(f*
x+e)^2+11/2304*sin(f*x+e)^4-25/768*sin(f*x+e)^2-3/32*(f*x+e)^2*cos(f*x+e)^2+3/16*(f*x+e)*(1/2*cos(f*x+e)*sin(f
*x+e)+1/2*f*x+1/2*e)-3/64*(f*x+e)^4-(f*x+e)^3*(-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)-1/12*(f*x+e)^2*sin(f*x+e)^6+1/6*(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)+1/216*sin(f*x+e)^6)+d^3*((f*x+e)^3*(-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*sin(f*x+e)^4-3/8*(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)-27/128*(f*x+e)^2-3/128*sin(f*x+e)^4-45/128*sin(f*x+e)^2-9/16*(f*x+e)^2*cos(f*x+e)^2+9/8*(f*x+e)*(1/2*
cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-9/32*(f*x+e)^4)-4*I*d^3*e^3*(-1/6*sin(f*x+e)^2*cos(f*x+e)^4-1/12*cos(f*x+
e)^4)+9*I*d^3*e*(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)-9*I*d^3*e^2*(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)-12*c^2*d*f^2*((f*x+e)*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*c
os(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*si
n(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)+3/4*I*d^3*e^3*sin(f*x+e)^4+18*I*c*d
^2*e*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)+12*I*c*d^2*e^2
*f*(-1/6*sin(f*x+e)^2*cos(f*x+e)^4-1/12*cos(f*x+e)^4)-d^3*e^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3
/8*f*x+3/8*e)+c^3*f^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+4*I*d^3*(1/4*(f*x+e)^3*sin
(f*x+e)^4-3/4*(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)-1/24*(f*x+e)*sin(f*x+e)^
4-1/96*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)-1/18*f*x-1/18*e+1/8*(f*x+e)*cos(f*x+e)^2-1/16*cos(f*x+e)*sin(f
*x+e)+1/12*(f*x+e)^3-1/6*(f*x+e)^3*sin(f*x+e)^6+1/2*(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)+1/36*(f*x+e)*sin(f*x+e)^6+1/216*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+
e))*cos(f*x+e))+3*d^3*e^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)-3/16*(f*x+e)^
2+1/16*sin(f*x+e)^4+3/16*sin(f*x+e)^2)-4*c^3*f^3*(-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^3*e^3*(-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)-3*d^3*e*((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)-12*d^3*e^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)-1/32*(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)+12*d^3*e*((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)*cos(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))-3*I*d^3*(1/4*(f*x+e)^3*sin(f*x+e)^4-3
/4*(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)-3/32*(f*x+e)*sin(f*x+e)^4-3/128*(si
n(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)-27/256*f*x-27/256*e+9/32*(f*x+e)*cos(f*x+e)^2-9/64*cos(f*x+e)*sin(f*x+e)
+3/16*(f*x+e)^3)-24*I*c*d^2*e*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))+12*c^2*d*e
*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)-9
*I*c^2*d*f^2*(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^
3*f^3*sin(f*x+e)^4+12*I*d^3*e^2*(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))-12*c*d^2*f
*((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)*cos(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))+4*I*c^3*f^3*(-1/6*s
in(f*x+e)^2*cos(f*x+e)^4-1/12*cos(f*x+e)^4)+3*c*d^2*f*((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)-12*I*d^3*e*(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*c^2*d*f^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)-3/16*(f*x+e)^2+1/16*sin(f*x+e)^4+3/16*sin(f*x+e)^2))
________________________________________________________________________________________
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)^3/(a+I*a*cot(f*x+e))^3,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
________________________________________________________________________________________
Fricas [A] time = 1.70601, size = 929, normalized size = 2.35 \begin{align*} \frac{864 \, d^{3} f^{4} x^{4} + 3456 \, c d^{2} f^{4} x^{3} + 5184 \, c^{2} d f^{4} x^{2} + 3456 \, c^{3} f^{4} x +{\left (576 i \, d^{3} f^{3} x^{3} + 576 i \, c^{3} f^{3} - 288 \, c^{2} d f^{2} - 96 i \, c d^{2} f + 16 \, d^{3} +{\left (1728 i \, c d^{2} f^{3} - 288 \, d^{3} f^{2}\right )} x^{2} +{\left (1728 i \, c^{2} d f^{3} - 576 \, c d^{2} f^{2} - 96 i \, d^{3} f\right )} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-2592 i \, d^{3} f^{3} x^{3} - 2592 i \, c^{3} f^{3} + 1944 \, c^{2} d f^{2} + 972 i \, c d^{2} f - 243 \, d^{3} +{\left (-7776 i \, c d^{2} f^{3} + 1944 \, d^{3} f^{2}\right )} x^{2} +{\left (-7776 i \, c^{2} d f^{3} + 3888 \, c d^{2} f^{2} + 972 i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (5184 i \, d^{3} f^{3} x^{3} + 5184 i \, c^{3} f^{3} - 7776 \, c^{2} d f^{2} - 7776 i \, c d^{2} f + 3888 \, d^{3} +{\left (15552 i \, c d^{2} f^{3} - 7776 \, d^{3} f^{2}\right )} x^{2} +{\left (15552 i \, c^{2} d f^{3} - 15552 \, c d^{2} f^{2} - 7776 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{27648 \, a^{3} f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3/(a+I*a*cot(f*x+e))^3,x, algorithm="fricas")
[Out]
1/27648*(864*d^3*f^4*x^4 + 3456*c*d^2*f^4*x^3 + 5184*c^2*d*f^4*x^2 + 3456*c^3*f^4*x + (576*I*d^3*f^3*x^3 + 576
*I*c^3*f^3 - 288*c^2*d*f^2 - 96*I*c*d^2*f + 16*d^3 + (1728*I*c*d^2*f^3 - 288*d^3*f^2)*x^2 + (1728*I*c^2*d*f^3
- 576*c*d^2*f^2 - 96*I*d^3*f)*x)*e^(6*I*f*x + 6*I*e) + (-2592*I*d^3*f^3*x^3 - 2592*I*c^3*f^3 + 1944*c^2*d*f^2
+ 972*I*c*d^2*f - 243*d^3 + (-7776*I*c*d^2*f^3 + 1944*d^3*f^2)*x^2 + (-7776*I*c^2*d*f^3 + 3888*c*d^2*f^2 + 972
*I*d^3*f)*x)*e^(4*I*f*x + 4*I*e) + (5184*I*d^3*f^3*x^3 + 5184*I*c^3*f^3 - 7776*c^2*d*f^2 - 7776*I*c*d^2*f + 38
88*d^3 + (15552*I*c*d^2*f^3 - 7776*d^3*f^2)*x^2 + (15552*I*c^2*d*f^3 - 15552*c*d^2*f^2 - 7776*I*d^3*f)*x)*e^(2
*I*f*x + 2*I*e))/(a^3*f^4)
________________________________________________________________________________________
Sympy [A] time = 2.22282, size = 933, normalized size = 2.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**3/(a+I*a*cot(f*x+e))**3,x)
[Out]
Piecewise((((21233664*I*a**33*c**3*f**29*exp(2*I*e) + 63700992*I*a**33*c**2*d*f**29*x*exp(2*I*e) - 31850496*a*
*33*c**2*d*f**28*exp(2*I*e) + 63700992*I*a**33*c*d**2*f**29*x**2*exp(2*I*e) - 63700992*a**33*c*d**2*f**28*x*ex
p(2*I*e) - 31850496*I*a**33*c*d**2*f**27*exp(2*I*e) + 21233664*I*a**33*d**3*f**29*x**3*exp(2*I*e) - 31850496*a
**33*d**3*f**28*x**2*exp(2*I*e) - 31850496*I*a**33*d**3*f**27*x*exp(2*I*e) + 15925248*a**33*d**3*f**26*exp(2*I
*e))*exp(2*I*f*x) + (-10616832*I*a**33*c**3*f**29*exp(4*I*e) - 31850496*I*a**33*c**2*d*f**29*x*exp(4*I*e) + 79
62624*a**33*c**2*d*f**28*exp(4*I*e) - 31850496*I*a**33*c*d**2*f**29*x**2*exp(4*I*e) + 15925248*a**33*c*d**2*f*
*28*x*exp(4*I*e) + 3981312*I*a**33*c*d**2*f**27*exp(4*I*e) - 10616832*I*a**33*d**3*f**29*x**3*exp(4*I*e) + 796
2624*a**33*d**3*f**28*x**2*exp(4*I*e) + 3981312*I*a**33*d**3*f**27*x*exp(4*I*e) - 995328*a**33*d**3*f**26*exp(
4*I*e))*exp(4*I*f*x) + (2359296*I*a**33*c**3*f**29*exp(6*I*e) + 7077888*I*a**33*c**2*d*f**29*x*exp(6*I*e) - 11
79648*a**33*c**2*d*f**28*exp(6*I*e) + 7077888*I*a**33*c*d**2*f**29*x**2*exp(6*I*e) - 2359296*a**33*c*d**2*f**2
8*x*exp(6*I*e) - 393216*I*a**33*c*d**2*f**27*exp(6*I*e) + 2359296*I*a**33*d**3*f**29*x**3*exp(6*I*e) - 1179648
*a**33*d**3*f**28*x**2*exp(6*I*e) - 393216*I*a**33*d**3*f**27*x*exp(6*I*e) + 65536*a**33*d**3*f**26*exp(6*I*e)
)*exp(6*I*f*x))/(113246208*a**36*f**30), Ne(113246208*a**36*f**30, 0)), (x**4*(-d**3*exp(6*I*e) + 3*d**3*exp(4
*I*e) - 3*d**3*exp(2*I*e))/(32*a**3) + x**3*(-c*d**2*exp(6*I*e) + 3*c*d**2*exp(4*I*e) - 3*c*d**2*exp(2*I*e))/(
8*a**3) + x**2*(-3*c**2*d*exp(6*I*e) + 9*c**2*d*exp(4*I*e) - 9*c**2*d*exp(2*I*e))/(16*a**3) + x*(-c**3*exp(6*I
*e) + 3*c**3*exp(4*I*e) - 3*c**3*exp(2*I*e))/(8*a**3), True)) + c**3*x/(8*a**3) + 3*c**2*d*x**2/(16*a**3) + c*
d**2*x**3/(8*a**3) + d**3*x**4/(32*a**3)
________________________________________________________________________________________
Giac [B] time = 1.33025, size = 841, normalized size = 2.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3/(a+I*a*cot(f*x+e))^3,x, algorithm="giac")
[Out]
1/27648*(864*d^3*f^4*x^4 + 3456*c*d^2*f^4*x^3 + 576*I*d^3*f^3*x^3*e^(6*I*f*x + 6*I*e) - 2592*I*d^3*f^3*x^3*e^(
4*I*f*x + 4*I*e) + 5184*I*d^3*f^3*x^3*e^(2*I*f*x + 2*I*e) + 5184*c^2*d*f^4*x^2 + 1728*I*c*d^2*f^3*x^2*e^(6*I*f
*x + 6*I*e) - 7776*I*c*d^2*f^3*x^2*e^(4*I*f*x + 4*I*e) + 15552*I*c*d^2*f^3*x^2*e^(2*I*f*x + 2*I*e) + 3456*c^3*
f^4*x + 1728*I*c^2*d*f^3*x*e^(6*I*f*x + 6*I*e) - 288*d^3*f^2*x^2*e^(6*I*f*x + 6*I*e) - 7776*I*c^2*d*f^3*x*e^(4
*I*f*x + 4*I*e) + 1944*d^3*f^2*x^2*e^(4*I*f*x + 4*I*e) + 15552*I*c^2*d*f^3*x*e^(2*I*f*x + 2*I*e) - 7776*d^3*f^
2*x^2*e^(2*I*f*x + 2*I*e) + 576*I*c^3*f^3*e^(6*I*f*x + 6*I*e) - 576*c*d^2*f^2*x*e^(6*I*f*x + 6*I*e) - 2592*I*c
^3*f^3*e^(4*I*f*x + 4*I*e) + 3888*c*d^2*f^2*x*e^(4*I*f*x + 4*I*e) + 5184*I*c^3*f^3*e^(2*I*f*x + 2*I*e) - 15552
*c*d^2*f^2*x*e^(2*I*f*x + 2*I*e) - 288*c^2*d*f^2*e^(6*I*f*x + 6*I*e) - 96*I*d^3*f*x*e^(6*I*f*x + 6*I*e) + 1944
*c^2*d*f^2*e^(4*I*f*x + 4*I*e) + 972*I*d^3*f*x*e^(4*I*f*x + 4*I*e) - 7776*c^2*d*f^2*e^(2*I*f*x + 2*I*e) - 7776
*I*d^3*f*x*e^(2*I*f*x + 2*I*e) - 96*I*c*d^2*f*e^(6*I*f*x + 6*I*e) + 972*I*c*d^2*f*e^(4*I*f*x + 4*I*e) - 7776*I
*c*d^2*f*e^(2*I*f*x + 2*I*e) + 16*d^3*e^(6*I*f*x + 6*I*e) - 243*d^3*e^(4*I*f*x + 4*I*e) + 3888*d^3*e^(2*I*f*x
+ 2*I*e))/(a^3*f^4)