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]
9/64*d^3*exp(2*I*e+2*I*f*x)/a^3/f^4-9/1024*d^3*exp(4*I*e+4*I*f*x)/a^3/f^4+1/1728*d^3*exp(6*I*e+6*I*f*x)/a^3/f^
4-9/32*I*d^2*exp(2*I*e+2*I*f*x)*(d*x+c)/a^3/f^3+9/256*I*d^2*exp(4*I*e+4*I*f*x)*(d*x+c)/a^3/f^3-1/288*I*d^2*exp
(6*I*e+6*I*f*x)*(d*x+c)/a^3/f^3-9/32*d*exp(2*I*e+2*I*f*x)*(d*x+c)^2/a^3/f^2+9/128*d*exp(4*I*e+4*I*f*x)*(d*x+c)
^2/a^3/f^2-1/96*d*exp(6*I*e+6*I*f*x)*(d*x+c)^2/a^3/f^2+3/16*I*exp(2*I*e+2*I*f*x)*(d*x+c)^3/a^3/f-3/32*I*exp(4*
I*e+4*I*f*x)*(d*x+c)^3/a^3/f+1/48*I*exp(6*I*e+6*I*f*x)*(d*x+c)^3/a^3/f+1/32*(d*x+c)^4/a^3/d
________________________________________________________________________________________
Rubi [A] time = 0.38, antiderivative size = 396, normalized size of antiderivative = 1.00,
number of steps used = 14, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 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]
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]
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*}
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Mathematica [A] time = 2.85, size = 664, normalized size = 1.68 \[ \frac {(\cos (3 (e+f x))+i \sin (3 (e+f x))) \left (-3456 i c^3 f^4 x \sin (3 (e+f x))+7776 c^3 f^3 \sin (e+f x)-576 c^3 f^3 \sin (3 (e+f x))-5184 i c^2 d f^4 x^2 \sin (3 (e+f x))+23328 c^2 d f^3 x \sin (e+f x)-1728 c^2 d f^3 x \sin (3 (e+f x))+9720 i c^2 d f^2 \sin (e+f x)-288 i c^2 d f^2 \sin (3 (e+f x))+81 i \left (32 c^3 f^3+24 c^2 d f^2 (4 f x+3 i)+12 c d^2 f \left (8 f^2 x^2+12 i f x-7\right )+d^3 \left (32 f^3 x^3+72 i f^2 x^2-84 f x-45 i\right )\right ) \cos (e+f x)+16 \left (36 c^3 f^3 (6 f x+i)+18 c^2 d f^2 \left (18 f^2 x^2+6 i f x-1\right )+6 c d^2 f \left (36 f^3 x^3+18 i f^2 x^2-6 f x-i\right )+d^3 \left (54 f^4 x^4+36 i f^3 x^3-18 f^2 x^2-6 i f x+1\right )\right ) \cos (3 (e+f x))-3456 i c d^2 f^4 x^3 \sin (3 (e+f x))+23328 c d^2 f^3 x^2 \sin (e+f x)-1728 c d^2 f^3 x^2 \sin (3 (e+f x))+19440 i c d^2 f^2 x \sin (e+f x)-576 i c d^2 f^2 x \sin (3 (e+f x))-8748 c d^2 f \sin (e+f x)+96 c d^2 f \sin (3 (e+f x))-864 i d^3 f^4 x^4 \sin (3 (e+f x))+7776 d^3 f^3 x^3 \sin (e+f x)-576 d^3 f^3 x^3 \sin (3 (e+f x))+9720 i d^3 f^2 x^2 \sin (e+f x)-288 i d^3 f^2 x^2 \sin (3 (e+f x))-8748 d^3 f x \sin (e+f x)+96 d^3 f x \sin (3 (e+f x))-4131 i d^3 \sin (e+f x)+16 i d^3 \sin (3 (e+f x))\right )}{27648 a^3 f^4} \]
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)
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fricas [A] time = 0.95, size = 353, normalized size = 0.89 \[ \frac {864 \, d^{3} f^{4} x^{4} + 3456 \, c d^{2} f^{4} x^{3} + 5184 \, c^{2} d f^{4} x^{2} + 3456 \, c^{3} f^{4} x + {\left (576 i \, d^{3} f^{3} x^{3} + 576 i \, c^{3} f^{3} - 288 \, c^{2} d f^{2} - 96 i \, c d^{2} f + 16 \, d^{3} + {\left (1728 i \, c d^{2} f^{3} - 288 \, d^{3} f^{2}\right )} x^{2} + {\left (1728 i \, c^{2} d f^{3} - 576 \, c d^{2} f^{2} - 96 i \, d^{3} f\right )} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-2592 i \, d^{3} f^{3} x^{3} - 2592 i \, c^{3} f^{3} + 1944 \, c^{2} d f^{2} + 972 i \, c d^{2} f - 243 \, d^{3} + {\left (-7776 i \, c d^{2} f^{3} + 1944 \, d^{3} f^{2}\right )} x^{2} + {\left (-7776 i \, c^{2} d f^{3} + 3888 \, c d^{2} f^{2} + 972 i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (5184 i \, d^{3} f^{3} x^{3} + 5184 i \, c^{3} f^{3} - 7776 \, c^{2} d f^{2} - 7776 i \, c d^{2} f + 3888 \, d^{3} + {\left (15552 i \, c d^{2} f^{3} - 7776 \, d^{3} f^{2}\right )} x^{2} + {\left (15552 i \, c^{2} d f^{3} - 15552 \, c d^{2} f^{2} - 7776 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{27648 \, a^{3} f^{4}} \]
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)
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giac [B] time = 0.61, size = 623, normalized size = 1.57 \[ \frac {864 \, d^{3} f^{4} x^{4} + 3456 \, c d^{2} f^{4} x^{3} + 576 i \, d^{3} f^{3} x^{3} e^{\left (6 i \, f x + 6 i \, e\right )} - 2592 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 5184 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} + 1728 i \, c d^{2} f^{3} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 7776 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 15552 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 3456 \, c^{3} f^{4} x + 1728 i \, c^{2} d f^{3} x e^{\left (6 i \, f x + 6 i \, e\right )} - 288 \, d^{3} f^{2} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 7776 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 1944 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 15552 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} - 7776 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 576 i \, c^{3} f^{3} e^{\left (6 i \, f x + 6 i \, e\right )} - 576 \, c d^{2} f^{2} x e^{\left (6 i \, f x + 6 i \, e\right )} - 2592 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 3888 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 5184 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 15552 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} - 288 \, c^{2} d f^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 96 i \, d^{3} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 1944 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 972 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} - 7776 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 7776 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} - 96 i \, c d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + 972 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 7776 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + 16 \, d^{3} e^{\left (6 i \, f x + 6 i \, e\right )} - 243 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 3888 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )}}{27648 \, a^{3} f^{4}} \]
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)
________________________________________________________________________________________
maple [B] time = 1.72, size = 3997, normalized size = 10.09 \[ \text {output too large to display} \]
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*(-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*sin(f*x+e)
*cos(f*x+e)-1/4*(f*x+e)^3)+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)+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)+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*sin(f*x+e)*cos
(f*x+e)+1/16*f*x+1/16*e)+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
)-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^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)-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*sin(f*x+e)*cos(f*x+e)+1/16*f*x+1/16*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)+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*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)-3/4*I*c^3*f^3*sin(f*x+e)^4+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)-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*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*si
n(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)-1/36*sin(f*x+e)^6)+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)-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)-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*(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*sin(f*x
+e)*cos(f*x+e)+3/16*(f*x+e)^3)+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*sin(f*x+e)*cos(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*si
n(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*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*sin(f*x+e)*cos(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+1/21
6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e))+4*d^3*e^3*(-1/6*sin(f*x+e)^3*cos(f*x+e)^3-1/8*si
n(f*x+e)*cos(f*x+e)^3+1/16*sin(f*x+e)*cos(f*x+e)+1/16*f*x+1/16*e)-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*sin(f*x+e)*cos(f*x+e)+1/16*f*x+1/16*e)+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*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)-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*si
n(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*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)+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*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)-1/36*sin(f*x+e)^6)+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)+3/3
2*(f*x+e)^2-1/32*sin(f*x+e)^4-3/32*sin(f*x+e)^2)-12*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)-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*I*d^3*e^3*(-1/6*sin(f*x+e)^2*cos(f*
x+e)^4-1/12*cos(f*x+e)^4)-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)*co
s(f*x+e)^2+1/32*sin(f*x+e)*cos(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))+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))-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))+9/4*I*c^2*d*e*f^2*sin(f*x+e)^4+3/4*I*d^
3*e^3*sin(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*s
in(f*x+e)*cos(f*x+e)-1/4*(f*x+e)^3)-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)+1/32*(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*sin(f*x+
e)*cos(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))*co
s(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*si
n(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))*c
os(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*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-9/32*(f*x+e)^4))
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.
________________________________________________________________________________________
mupad [B] time = 1.62, size = 418, normalized size = 1.06 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (-\frac {\left (-12\,c^3\,f^3-c^2\,d\,f^2\,18{}\mathrm {i}+18\,c\,d^2\,f+d^3\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^3\,f^4}+\frac {d^3\,x^3\,3{}\mathrm {i}}{16\,a^3\,f}+\frac {d\,x\,\left (2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}-d^2\right )\,9{}\mathrm {i}}{32\,a^3\,f^3}+\frac {d^2\,x^2\,\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{32\,a^3\,f^2}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (-\frac {\left (-96\,c^3\,f^3-c^2\,d\,f^2\,72{}\mathrm {i}+36\,c\,d^2\,f+d^3\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{1024\,a^3\,f^4}+\frac {d^3\,x^3\,3{}\mathrm {i}}{32\,a^3\,f}+\frac {d\,x\,\left (8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}-d^2\right )\,9{}\mathrm {i}}{256\,a^3\,f^3}+\frac {d^2\,x^2\,\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{128\,a^3\,f^2}\right )+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (-\frac {\left (-36\,c^3\,f^3-c^2\,d\,f^2\,18{}\mathrm {i}+6\,c\,d^2\,f+d^3\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{1728\,a^3\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{48\,a^3\,f}+\frac {d\,x\,\left (18\,c^2\,f^2+c\,d\,f\,6{}\mathrm {i}-d^2\right )\,1{}\mathrm {i}}{288\,a^3\,f^3}+\frac {d^2\,x^2\,\left (6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{96\,a^3\,f^2}\right )+\frac {c^3\,x}{8\,a^3}+\frac {d^3\,x^4}{32\,a^3}+\frac {3\,c^2\,d\,x^2}{16\,a^3}+\frac {c\,d^2\,x^3}{8\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^3/(a + a*cot(e + f*x)*1i)^3,x)
[Out]
exp(e*2i + f*x*2i)*((d^3*x^3*3i)/(16*a^3*f) - ((d^3*9i - 12*c^3*f^3 - c^2*d*f^2*18i + 18*c*d^2*f)*1i)/(64*a^3*
f^4) + (d*x*(2*c^2*f^2 - d^2 + c*d*f*2i)*9i)/(32*a^3*f^3) + (d^2*x^2*(d*1i + 2*c*f)*9i)/(32*a^3*f^2)) - exp(e*
4i + f*x*4i)*((d^3*x^3*3i)/(32*a^3*f) - ((d^3*9i - 96*c^3*f^3 - c^2*d*f^2*72i + 36*c*d^2*f)*1i)/(1024*a^3*f^4)
+ (d*x*(8*c^2*f^2 - d^2 + c*d*f*4i)*9i)/(256*a^3*f^3) + (d^2*x^2*(d*1i + 4*c*f)*9i)/(128*a^3*f^2)) + exp(e*6i
+ f*x*6i)*((d^3*x^3*1i)/(48*a^3*f) - ((d^3*1i - 36*c^3*f^3 - c^2*d*f^2*18i + 6*c*d^2*f)*1i)/(1728*a^3*f^4) +
(d*x*(18*c^2*f^2 - d^2 + c*d*f*6i)*1i)/(288*a^3*f^3) + (d^2*x^2*(d*1i + 6*c*f)*1i)/(96*a^3*f^2)) + (c^3*x)/(8*
a^3) + (d^3*x^4)/(32*a^3) + (3*c^2*d*x^2)/(16*a^3) + (c*d^2*x^3)/(8*a^3)
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sympy [A] time = 0.84, size = 935, normalized size = 2.36 \[ \begin {cases} - \frac {\left (- 21233664 i a^{6} c^{3} f^{11} e^{2 i e} - 63700992 i a^{6} c^{2} d f^{11} x e^{2 i e} + 31850496 a^{6} c^{2} d f^{10} e^{2 i e} - 63700992 i a^{6} c d^{2} f^{11} x^{2} e^{2 i e} + 63700992 a^{6} c d^{2} f^{10} x e^{2 i e} + 31850496 i a^{6} c d^{2} f^{9} e^{2 i e} - 21233664 i a^{6} d^{3} f^{11} x^{3} e^{2 i e} + 31850496 a^{6} d^{3} f^{10} x^{2} e^{2 i e} + 31850496 i a^{6} d^{3} f^{9} x e^{2 i e} - 15925248 a^{6} d^{3} f^{8} e^{2 i e}\right ) e^{2 i f x} + \left (10616832 i a^{6} c^{3} f^{11} e^{4 i e} + 31850496 i a^{6} c^{2} d f^{11} x e^{4 i e} - 7962624 a^{6} c^{2} d f^{10} e^{4 i e} + 31850496 i a^{6} c d^{2} f^{11} x^{2} e^{4 i e} - 15925248 a^{6} c d^{2} f^{10} x e^{4 i e} - 3981312 i a^{6} c d^{2} f^{9} e^{4 i e} + 10616832 i a^{6} d^{3} f^{11} x^{3} e^{4 i e} - 7962624 a^{6} d^{3} f^{10} x^{2} e^{4 i e} - 3981312 i a^{6} d^{3} f^{9} x e^{4 i e} + 995328 a^{6} d^{3} f^{8} e^{4 i e}\right ) e^{4 i f x} + \left (- 2359296 i a^{6} c^{3} f^{11} e^{6 i e} - 7077888 i a^{6} c^{2} d f^{11} x e^{6 i e} + 1179648 a^{6} c^{2} d f^{10} e^{6 i e} - 7077888 i a^{6} c d^{2} f^{11} x^{2} e^{6 i e} + 2359296 a^{6} c d^{2} f^{10} x e^{6 i e} + 393216 i a^{6} c d^{2} f^{9} e^{6 i e} - 2359296 i a^{6} d^{3} f^{11} x^{3} e^{6 i e} + 1179648 a^{6} d^{3} f^{10} x^{2} e^{6 i e} + 393216 i a^{6} d^{3} f^{9} x e^{6 i e} - 65536 a^{6} d^{3} f^{8} e^{6 i e}\right ) e^{6 i f x}}{113246208 a^{9} f^{12}} & \text {for}\: 113246208 a^{9} f^{12} \neq 0 \\\frac {x^{4} \left (- d^{3} e^{6 i e} + 3 d^{3} e^{4 i e} - 3 d^{3} e^{2 i e}\right )}{32 a^{3}} + \frac {x^{3} \left (- c d^{2} e^{6 i e} + 3 c d^{2} e^{4 i e} - 3 c d^{2} e^{2 i e}\right )}{8 a^{3}} + \frac {x^{2} \left (- 3 c^{2} d e^{6 i e} + 9 c^{2} d e^{4 i e} - 9 c^{2} d e^{2 i e}\right )}{16 a^{3}} + \frac {x \left (- c^{3} e^{6 i e} + 3 c^{3} e^{4 i e} - 3 c^{3} e^{2 i e}\right )}{8 a^{3}} & \text {otherwise} \end {cases} + \frac {c^{3} x}{8 a^{3}} + \frac {3 c^{2} d x^{2}}{16 a^{3}} + \frac {c d^{2} x^{3}}{8 a^{3}} + \frac {d^{3} x^{4}}{32 a^{3}} \]
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**6*c**3*f**11*exp(2*I*e) - 63700992*I*a**6*c**2*d*f**11*x*exp(2*I*e) + 31850496*a*
*6*c**2*d*f**10*exp(2*I*e) - 63700992*I*a**6*c*d**2*f**11*x**2*exp(2*I*e) + 63700992*a**6*c*d**2*f**10*x*exp(2
*I*e) + 31850496*I*a**6*c*d**2*f**9*exp(2*I*e) - 21233664*I*a**6*d**3*f**11*x**3*exp(2*I*e) + 31850496*a**6*d*
*3*f**10*x**2*exp(2*I*e) + 31850496*I*a**6*d**3*f**9*x*exp(2*I*e) - 15925248*a**6*d**3*f**8*exp(2*I*e))*exp(2*
I*f*x) + (10616832*I*a**6*c**3*f**11*exp(4*I*e) + 31850496*I*a**6*c**2*d*f**11*x*exp(4*I*e) - 7962624*a**6*c**
2*d*f**10*exp(4*I*e) + 31850496*I*a**6*c*d**2*f**11*x**2*exp(4*I*e) - 15925248*a**6*c*d**2*f**10*x*exp(4*I*e)
- 3981312*I*a**6*c*d**2*f**9*exp(4*I*e) + 10616832*I*a**6*d**3*f**11*x**3*exp(4*I*e) - 7962624*a**6*d**3*f**10
*x**2*exp(4*I*e) - 3981312*I*a**6*d**3*f**9*x*exp(4*I*e) + 995328*a**6*d**3*f**8*exp(4*I*e))*exp(4*I*f*x) + (-
2359296*I*a**6*c**3*f**11*exp(6*I*e) - 7077888*I*a**6*c**2*d*f**11*x*exp(6*I*e) + 1179648*a**6*c**2*d*f**10*ex
p(6*I*e) - 7077888*I*a**6*c*d**2*f**11*x**2*exp(6*I*e) + 2359296*a**6*c*d**2*f**10*x*exp(6*I*e) + 393216*I*a**
6*c*d**2*f**9*exp(6*I*e) - 2359296*I*a**6*d**3*f**11*x**3*exp(6*I*e) + 1179648*a**6*d**3*f**10*x**2*exp(6*I*e)
+ 393216*I*a**6*d**3*f**9*x*exp(6*I*e) - 65536*a**6*d**3*f**8*exp(6*I*e))*exp(6*I*f*x))/(113246208*a**9*f**12
), Ne(113246208*a**9*f**12, 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
)
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