3.22 \(\int \frac{(c+d x)^3}{(a+i a \cot (e+f x))^2} \, dx\)
Optimal. Leaf size=270 \[ -\frac{3 i d^2 (c+d x) e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac{3 i d^2 (c+d x) e^{4 i e+4 i f x}}{128 a^2 f^3}-\frac{3 d (c+d x)^2 e^{2 i e+2 i f x}}{8 a^2 f^2}+\frac{3 d (c+d x)^2 e^{4 i e+4 i f x}}{64 a^2 f^2}+\frac{i (c+d x)^3 e^{2 i e+2 i f x}}{4 a^2 f}-\frac{i (c+d x)^3 e^{4 i e+4 i f x}}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac{3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4} \]
[Out]
(3*d^3*E^((2*I)*e + (2*I)*f*x))/(16*a^2*f^4) - (3*d^3*E^((4*I)*e + (4*I)*f*x))/(512*a^2*f^4) - (((3*I)/8)*d^2*
E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(a^2*f^3) + (((3*I)/128)*d^2*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(a^2*f^3) -
(3*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(8*a^2*f^2) + (3*d*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(64*a^2*f^2
) + ((I/4)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^3)/(a^2*f) - ((I/16)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^3)/(a^2*f)
+ (c + d*x)^4/(16*a^2*d)
________________________________________________________________________________________
Rubi [A] time = 0.27597, antiderivative size = 270, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used =
{3729, 2176, 2194} \[ -\frac{3 i d^2 (c+d x) e^{2 i e+2 i f x}}{8 a^2 f^3}+\frac{3 i d^2 (c+d x) e^{4 i e+4 i f x}}{128 a^2 f^3}-\frac{3 d (c+d x)^2 e^{2 i e+2 i f x}}{8 a^2 f^2}+\frac{3 d (c+d x)^2 e^{4 i e+4 i f x}}{64 a^2 f^2}+\frac{i (c+d x)^3 e^{2 i e+2 i f x}}{4 a^2 f}-\frac{i (c+d x)^3 e^{4 i e+4 i f x}}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac{3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^3/(a + I*a*Cot[e + f*x])^2,x]
[Out]
(3*d^3*E^((2*I)*e + (2*I)*f*x))/(16*a^2*f^4) - (3*d^3*E^((4*I)*e + (4*I)*f*x))/(512*a^2*f^4) - (((3*I)/8)*d^2*
E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(a^2*f^3) + (((3*I)/128)*d^2*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(a^2*f^3) -
(3*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(8*a^2*f^2) + (3*d*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(64*a^2*f^2
) + ((I/4)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^3)/(a^2*f) - ((I/16)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^3)/(a^2*f)
+ (c + d*x)^4/(16*a^2*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))^2} \, dx &=\int \left (\frac{(c+d x)^3}{4 a^2}-\frac{e^{2 i e+2 i f x} (c+d x)^3}{2 a^2}+\frac{e^{4 i e+4 i f x} (c+d x)^3}{4 a^2}\right ) \, dx\\ &=\frac{(c+d x)^4}{16 a^2 d}+\frac{\int e^{4 i e+4 i f x} (c+d x)^3 \, dx}{4 a^2}-\frac{\int e^{2 i e+2 i f x} (c+d x)^3 \, dx}{2 a^2}\\ &=\frac{i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac{i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{(3 i d) \int e^{4 i e+4 i f x} (c+d x)^2 \, dx}{16 a^2 f}-\frac{(3 i d) \int e^{2 i e+2 i f x} (c+d x)^2 \, dx}{4 a^2 f}\\ &=-\frac{3 d e^{2 i e+2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac{3 d e^{4 i e+4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac{i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac{i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{\left (3 d^2\right ) \int e^{4 i e+4 i f x} (c+d x) \, dx}{32 a^2 f^2}+\frac{\left (3 d^2\right ) \int e^{2 i e+2 i f x} (c+d x) \, dx}{4 a^2 f^2}\\ &=-\frac{3 i d^2 e^{2 i e+2 i f x} (c+d x)}{8 a^2 f^3}+\frac{3 i d^2 e^{4 i e+4 i f x} (c+d x)}{128 a^2 f^3}-\frac{3 d e^{2 i e+2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac{3 d e^{4 i e+4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac{i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac{i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{\left (3 i d^3\right ) \int e^{4 i e+4 i f x} \, dx}{128 a^2 f^3}+\frac{\left (3 i d^3\right ) \int e^{2 i e+2 i f x} \, dx}{8 a^2 f^3}\\ &=\frac{3 d^3 e^{2 i e+2 i f x}}{16 a^2 f^4}-\frac{3 d^3 e^{4 i e+4 i f x}}{512 a^2 f^4}-\frac{3 i d^2 e^{2 i e+2 i f x} (c+d x)}{8 a^2 f^3}+\frac{3 i d^2 e^{4 i e+4 i f x} (c+d x)}{128 a^2 f^3}-\frac{3 d e^{2 i e+2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac{3 d e^{4 i e+4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac{i e^{2 i e+2 i f x} (c+d x)^3}{4 a^2 f}-\frac{i e^{4 i e+4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.53663, size = 362, normalized size = 1.34 \[ \frac{(\cos (2 (e+f x))+i \sin (2 (e+f x))) \left (\left (24 c^2 d f^2 \left (8 f^2 x^2-4 i f x+1\right )+32 c^3 f^3 (4 f x-i)+4 c d^2 f \left (32 f^3 x^3-24 i f^2 x^2+12 f x+3 i\right )+d^3 \left (32 f^4 x^4-32 i f^3 x^3+24 f^2 x^2+12 i f x-3\right )\right ) \cos (2 (e+f x))-i \left (\left (24 c^2 d f^2 \left (8 f^2 x^2+4 i f x-1\right )+32 c^3 f^3 (4 f x+i)+4 c d^2 f \left (32 f^3 x^3+24 i f^2 x^2-12 f x-3 i\right )+d^3 \left (32 f^4 x^4+32 i f^3 x^3-24 f^2 x^2-12 i f x+3\right )\right ) \sin (2 (e+f x))-32 \left (6 c^2 d f^2 (2 f x+i)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 i f x-1\right )+d^3 \left (4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right )\right )\right )\right )}{512 a^2 f^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^3/(a + I*a*Cot[e + f*x])^2,x]
[Out]
((Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*((32*c^3*f^3*(-I + 4*f*x) + 24*c^2*d*f^2*(1 - (4*I)*f*x + 8*f^2*x^2)
+ 4*c*d^2*f*(3*I + 12*f*x - (24*I)*f^2*x^2 + 32*f^3*x^3) + d^3*(-3 + (12*I)*f*x + 24*f^2*x^2 - (32*I)*f^3*x^3
+ 32*f^4*x^4))*Cos[2*(e + f*x)] - I*(-32*(4*c^3*f^3 + 6*c^2*d*f^2*(I + 2*f*x) + 6*c*d^2*f*(-1 + (2*I)*f*x + 2*
f^2*x^2) + d^3*(-3*I - 6*f*x + (6*I)*f^2*x^2 + 4*f^3*x^3)) + (32*c^3*f^3*(I + 4*f*x) + 24*c^2*d*f^2*(-1 + (4*I
)*f*x + 8*f^2*x^2) + 4*c*d^2*f*(-3*I - 12*f*x + (24*I)*f^2*x^2 + 32*f^3*x^3) + d^3*(3 - (12*I)*f*x - 24*f^2*x^
2 + (32*I)*f^3*x^3 + 32*f^4*x^4))*Sin[2*(e + f*x)])))/(512*a^2*f^4)
________________________________________________________________________________________
Maple [B] time = 0.201, size = 2261, normalized size = 8.4 \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)^3/(a+I*a*cot(f*x+e))^2,x)
[Out]
-1/a^2/f*(-12*I/f^2*c*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)+2/f^3*d^3*((f*x+e)^3*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-3/16*(f*x+e)^2*cos(f*x+e)^2+3/8*(f*x+e)
*(1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-21/128*(f*x+e)^2-3/128*sin(f*x+e)^2-3/32*(f*x+e)^4-(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)+3/128*sin(f*x+e)^4)-1/f^3*d^3*((f*x+e)^3*(-1/2*cos(f*x+e)*sin(
f*x+e)+1/2*f*x+1/2*e)-3/4*(f*x+e)^2*cos(f*x+e)^2+3/2*(f*x+e)*(1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-3/8*(f*
x+e)^2-3/8*sin(f*x+e)^2-3/8*(f*x+e)^4)+1/2*I*c^3*sin(f*x+e)^4+3/2*I/f^2*c*d^2*e^2*sin(f*x+e)^4-6*I/f^3*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)+6*I/f^3*d^3*e^2*(1/4*(f*x+e)*sin(f*x+e)^4+1/16*(sin(f*x+e)^3+3/2*si
n(f*x+e))*cos(f*x+e)-3/32*f*x-3/32*e)-3/2*I/f*c^2*d*e*sin(f*x+e)^4-3/f^3*d^3*e^2*((f*x+e)*(-1/2*cos(f*x+e)*sin
(f*x+e)+1/2*f*x+1/2*e)-1/4*(f*x+e)^2+1/4*sin(f*x+e)^2)+6/f^2*c*d^2*((f*x+e)^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*
f*x+1/2*e)-1/8*(f*x+e)*cos(f*x+e)^2+1/16*cos(f*x+e)*sin(f*x+e)+7/64*f*x+7/64*e-1/12*(f*x+e)^3-(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))+6/f*c^2*d*((f*x+e)*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/16*(f*x+e)^2+1/16*sin(f*x+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)-1/16*sin(f*x+e)^4)-2/f^3*d^3*e^3*(-1/
4*sin(f*x+e)*cos(f*x+e)^3+1/8*cos(f*x+e)*sin(f*x+e)+1/8*f*x+1/8*e)-c^3*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2
*e)+2*c^3*(-1/4*sin(f*x+e)*cos(f*x+e)^3+1/8*cos(f*x+e)*sin(f*x+e)+1/8*f*x+1/8*e)-6/f^3*d^3*e*((f*x+e)^2*(-1/2*
cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/8*(f*x+e)*cos(f*x+e)^2+1/16*cos(f*x+e)*sin(f*x+e)+7/64*f*x+7/64*e-1/12*
(f*x+e)^3-(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)^4-1/3
2*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e))+2*I/f^3*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*(sin(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)+3/f^3*d^
3*e*((f*x+e)^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/2*(f*x+e)*cos(f*x+e)^2+1/4*cos(f*x+e)*sin(f*x+e)+1
/4*f*x+1/4*e-1/3*(f*x+e)^3)+6/f^3*d^3*e^2*((f*x+e)*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/16*(f*x+e)^2+1
/16*sin(f*x+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)-1/16*sin(f*x+e)^4)+1/f^
3*d^3*e^3*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-3/f^2*c*d^2*((f*x+e)^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*
x+1/2*e)-1/2*(f*x+e)*cos(f*x+e)^2+1/4*cos(f*x+e)*sin(f*x+e)+1/4*f*x+1/4*e-1/3*(f*x+e)^3)-3/f*c^2*d*((f*x+e)*(-
1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/4*(f*x+e)^2+1/4*sin(f*x+e)^2)+6*I/f^2*c*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)-1/2*I/f^3*d^3*e^3*sin(f*x+e)^4-3/f^2*c*d^2*e^2*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)
+6/f^2*c*d^2*e*((f*x+e)*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/4*(f*x+e)^2+1/4*sin(f*x+e)^2)-6/f*c^2*d*e
*(-1/4*sin(f*x+e)*cos(f*x+e)^3+1/8*cos(f*x+e)*sin(f*x+e)+1/8*f*x+1/8*e)+3/f*c^2*d*e*(-1/2*cos(f*x+e)*sin(f*x+e
)+1/2*f*x+1/2*e)+6*I/f*c^2*d*(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)+6/f^2*c*d^2*e^2*(-1/4*sin(f*x+e)*cos(f*x+e)^3+1/8*cos(f*x+e)*sin(f*x+e)+1/8*f*x+1/8*e)-12/f^2*c*d^2*e*
((f*x+e)*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/16*(f*x+e)^2+1/16*sin(f*x+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)-1/16*sin(f*x+e)^4))
________________________________________________________________________________________
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))^2,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
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Fricas [A] time = 1.6487, size = 622, normalized size = 2.3 \begin{align*} \frac{32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} + 192 \, c^{2} d f^{4} x^{2} + 128 \, c^{3} f^{4} x +{\left (-32 i \, d^{3} f^{3} x^{3} - 32 i \, c^{3} f^{3} + 24 \, c^{2} d f^{2} + 12 i \, c d^{2} f - 3 \, d^{3} +{\left (-96 i \, c d^{2} f^{3} + 24 \, d^{3} f^{2}\right )} x^{2} +{\left (-96 i \, c^{2} d f^{3} + 48 \, c d^{2} f^{2} + 12 i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (128 i \, d^{3} f^{3} x^{3} + 128 i \, c^{3} f^{3} - 192 \, c^{2} d f^{2} - 192 i \, c d^{2} f + 96 \, d^{3} +{\left (384 i \, c d^{2} f^{3} - 192 \, d^{3} f^{2}\right )} x^{2} +{\left (384 i \, c^{2} d f^{3} - 384 \, c d^{2} f^{2} - 192 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} 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))^2,x, algorithm="fricas")
[Out]
1/512*(32*d^3*f^4*x^4 + 128*c*d^2*f^4*x^3 + 192*c^2*d*f^4*x^2 + 128*c^3*f^4*x + (-32*I*d^3*f^3*x^3 - 32*I*c^3*
f^3 + 24*c^2*d*f^2 + 12*I*c*d^2*f - 3*d^3 + (-96*I*c*d^2*f^3 + 24*d^3*f^2)*x^2 + (-96*I*c^2*d*f^3 + 48*c*d^2*f
^2 + 12*I*d^3*f)*x)*e^(4*I*f*x + 4*I*e) + (128*I*d^3*f^3*x^3 + 128*I*c^3*f^3 - 192*c^2*d*f^2 - 192*I*c*d^2*f +
96*d^3 + (384*I*c*d^2*f^3 - 192*d^3*f^2)*x^2 + (384*I*c^2*d*f^3 - 384*c*d^2*f^2 - 192*I*d^3*f)*x)*e^(2*I*f*x
+ 2*I*e))/(a^2*f^4)
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Sympy [A] time = 1.42495, size = 653, normalized size = 2.42 \begin{align*} \begin{cases} \frac{\left (2048 i a^{14} c^{3} f^{19} e^{2 i e} + 6144 i a^{14} c^{2} d f^{19} x e^{2 i e} - 3072 a^{14} c^{2} d f^{18} e^{2 i e} + 6144 i a^{14} c d^{2} f^{19} x^{2} e^{2 i e} - 6144 a^{14} c d^{2} f^{18} x e^{2 i e} - 3072 i a^{14} c d^{2} f^{17} e^{2 i e} + 2048 i a^{14} d^{3} f^{19} x^{3} e^{2 i e} - 3072 a^{14} d^{3} f^{18} x^{2} e^{2 i e} - 3072 i a^{14} d^{3} f^{17} x e^{2 i e} + 1536 a^{14} d^{3} f^{16} e^{2 i e}\right ) e^{2 i f x} + \left (- 512 i a^{14} c^{3} f^{19} e^{4 i e} - 1536 i a^{14} c^{2} d f^{19} x e^{4 i e} + 384 a^{14} c^{2} d f^{18} e^{4 i e} - 1536 i a^{14} c d^{2} f^{19} x^{2} e^{4 i e} + 768 a^{14} c d^{2} f^{18} x e^{4 i e} + 192 i a^{14} c d^{2} f^{17} e^{4 i e} - 512 i a^{14} d^{3} f^{19} x^{3} e^{4 i e} + 384 a^{14} d^{3} f^{18} x^{2} e^{4 i e} + 192 i a^{14} d^{3} f^{17} x e^{4 i e} - 48 a^{14} d^{3} f^{16} e^{4 i e}\right ) e^{4 i f x}}{8192 a^{16} f^{20}} & \text{for}\: 8192 a^{16} f^{20} \neq 0 \\\frac{x^{4} \left (d^{3} e^{4 i e} - 2 d^{3} e^{2 i e}\right )}{16 a^{2}} + \frac{x^{3} \left (c d^{2} e^{4 i e} - 2 c d^{2} e^{2 i e}\right )}{4 a^{2}} + \frac{x^{2} \left (3 c^{2} d e^{4 i e} - 6 c^{2} d e^{2 i e}\right )}{8 a^{2}} + \frac{x \left (c^{3} e^{4 i e} - 2 c^{3} e^{2 i e}\right )}{4 a^{2}} & \text{otherwise} \end{cases} + \frac{c^{3} x}{4 a^{2}} + \frac{3 c^{2} d x^{2}}{8 a^{2}} + \frac{c d^{2} x^{3}}{4 a^{2}} + \frac{d^{3} x^{4}}{16 a^{2}} \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))**2,x)
[Out]
Piecewise((((2048*I*a**14*c**3*f**19*exp(2*I*e) + 6144*I*a**14*c**2*d*f**19*x*exp(2*I*e) - 3072*a**14*c**2*d*f
**18*exp(2*I*e) + 6144*I*a**14*c*d**2*f**19*x**2*exp(2*I*e) - 6144*a**14*c*d**2*f**18*x*exp(2*I*e) - 3072*I*a*
*14*c*d**2*f**17*exp(2*I*e) + 2048*I*a**14*d**3*f**19*x**3*exp(2*I*e) - 3072*a**14*d**3*f**18*x**2*exp(2*I*e)
- 3072*I*a**14*d**3*f**17*x*exp(2*I*e) + 1536*a**14*d**3*f**16*exp(2*I*e))*exp(2*I*f*x) + (-512*I*a**14*c**3*f
**19*exp(4*I*e) - 1536*I*a**14*c**2*d*f**19*x*exp(4*I*e) + 384*a**14*c**2*d*f**18*exp(4*I*e) - 1536*I*a**14*c*
d**2*f**19*x**2*exp(4*I*e) + 768*a**14*c*d**2*f**18*x*exp(4*I*e) + 192*I*a**14*c*d**2*f**17*exp(4*I*e) - 512*I
*a**14*d**3*f**19*x**3*exp(4*I*e) + 384*a**14*d**3*f**18*x**2*exp(4*I*e) + 192*I*a**14*d**3*f**17*x*exp(4*I*e)
- 48*a**14*d**3*f**16*exp(4*I*e))*exp(4*I*f*x))/(8192*a**16*f**20), Ne(8192*a**16*f**20, 0)), (x**4*(d**3*exp
(4*I*e) - 2*d**3*exp(2*I*e))/(16*a**2) + x**3*(c*d**2*exp(4*I*e) - 2*c*d**2*exp(2*I*e))/(4*a**2) + x**2*(3*c**
2*d*exp(4*I*e) - 6*c**2*d*exp(2*I*e))/(8*a**2) + x*(c**3*exp(4*I*e) - 2*c**3*exp(2*I*e))/(4*a**2), True)) + c*
*3*x/(4*a**2) + 3*c**2*d*x**2/(8*a**2) + c*d**2*x**3/(4*a**2) + d**3*x**4/(16*a**2)
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Giac [B] time = 1.2899, size = 585, normalized size = 2.17 \begin{align*} \frac{32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} - 32 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 192 \, c^{2} d f^{4} x^{2} - 96 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 128 \, c^{3} f^{4} x - 96 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 24 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} - 192 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 32 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 48 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 384 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} - 192 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 192 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 12 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 192 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 3 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} 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))^2,x, algorithm="giac")
[Out]
1/512*(32*d^3*f^4*x^4 + 128*c*d^2*f^4*x^3 - 32*I*d^3*f^3*x^3*e^(4*I*f*x + 4*I*e) + 128*I*d^3*f^3*x^3*e^(2*I*f*
x + 2*I*e) + 192*c^2*d*f^4*x^2 - 96*I*c*d^2*f^3*x^2*e^(4*I*f*x + 4*I*e) + 384*I*c*d^2*f^3*x^2*e^(2*I*f*x + 2*I
*e) + 128*c^3*f^4*x - 96*I*c^2*d*f^3*x*e^(4*I*f*x + 4*I*e) + 24*d^3*f^2*x^2*e^(4*I*f*x + 4*I*e) + 384*I*c^2*d*
f^3*x*e^(2*I*f*x + 2*I*e) - 192*d^3*f^2*x^2*e^(2*I*f*x + 2*I*e) - 32*I*c^3*f^3*e^(4*I*f*x + 4*I*e) + 48*c*d^2*
f^2*x*e^(4*I*f*x + 4*I*e) + 128*I*c^3*f^3*e^(2*I*f*x + 2*I*e) - 384*c*d^2*f^2*x*e^(2*I*f*x + 2*I*e) + 24*c^2*d
*f^2*e^(4*I*f*x + 4*I*e) + 12*I*d^3*f*x*e^(4*I*f*x + 4*I*e) - 192*c^2*d*f^2*e^(2*I*f*x + 2*I*e) - 192*I*d^3*f*
x*e^(2*I*f*x + 2*I*e) + 12*I*c*d^2*f*e^(4*I*f*x + 4*I*e) - 192*I*c*d^2*f*e^(2*I*f*x + 2*I*e) - 3*d^3*e^(4*I*f*
x + 4*I*e) + 96*d^3*e^(2*I*f*x + 2*I*e))/(a^2*f^4)