3.29 \(\int \frac {(c+d x)^3}{(a+i a \tan (e+f x))^3} \, dx\)

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

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Rubi [A]  time = 0.40, 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*Tan[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)/2
56)*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)/(1
28*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 \tan (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 = 2.74, size = 667, normalized size = 1.68 \[ \frac {i \sec ^3(e+f x) \left (3456 i c^3 f^4 x \sin (3 (e+f x))-2592 c^3 f^3 \sin (e+f x)+576 c^3 f^3 \sin (3 (e+f x))+5184 i c^2 d f^4 x^2 \sin (3 (e+f x))-7776 c^2 d f^3 x \sin (e+f x)+1728 c^2 d f^3 x \sin (3 (e+f x))+5832 i c^2 d f^2 \sin (e+f x)-288 i c^2 d f^2 \sin (3 (e+f x))+243 \left (32 i c^3 f^3+8 c^2 d f^2 (5+12 i f x)+4 c d^2 f \left (24 i f^2 x^2+20 f x-9 i\right )+d^3 \left (32 i f^3 x^3+40 f^2 x^2-36 i f x-17\right )\right ) \cos (e+f x)+16 \left (36 c^3 f^3 (6 f x+i)+18 c^2 d f^2 \left (18 f^2 x^2+6 i f x+1\right )+6 c d^2 f \left (36 f^3 x^3+18 i f^2 x^2+6 f x-i\right )+d^3 \left (54 f^4 x^4+36 i f^3 x^3+18 f^2 x^2-6 i f x-1\right )\right ) \cos (3 (e+f x))+3456 i c d^2 f^4 x^3 \sin (3 (e+f x))-7776 c d^2 f^3 x^2 \sin (e+f x)+1728 c d^2 f^3 x^2 \sin (3 (e+f x))+11664 i c d^2 f^2 x \sin (e+f x)-576 i c d^2 f^2 x \sin (3 (e+f x))+6804 c d^2 f \sin (e+f x)-96 c d^2 f \sin (3 (e+f x))+864 i d^3 f^4 x^4 \sin (3 (e+f x))-2592 d^3 f^3 x^3 \sin (e+f x)+576 d^3 f^3 x^3 \sin (3 (e+f x))+5832 i d^3 f^2 x^2 \sin (e+f x)-288 i d^3 f^2 x^2 \sin (3 (e+f x))+6804 d^3 f x \sin (e+f x)-96 d^3 f x \sin (3 (e+f x))-3645 i d^3 \sin (e+f x)+16 i d^3 \sin (3 (e+f x))\right )}{27648 a^3 f^4 (\tan (e+f x)-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/27648)*Sec[e + f*x]^3*(243*((32*I)*c^3*f^3 + 8*c^2*d*f^2*(5 + (12*I)*f*x) + 4*c*d^2*f*(-9*I + 20*f*x + (24
*I)*f^2*x^2) + d^3*(-17 - (36*I)*f*x + 40*f^2*x^2 + (32*I)*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)] - (3645*I)*d^3*Sin[e + f*x] + 6804*c
*d^2*f*Sin[e + f*x] + (5832*I)*c^2*d*f^2*Sin[e + f*x] - 2592*c^3*f^3*Sin[e + f*x] + 6804*d^3*f*x*Sin[e + f*x]
+ (11664*I)*c*d^2*f^2*x*Sin[e + f*x] - 7776*c^2*d*f^3*x*Sin[e + f*x] + (5832*I)*d^3*f^2*x^2*Sin[e + f*x] - 777
6*c*d^2*f^3*x^2*Sin[e + f*x] - 2592*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)]))/(a^3*f^4*(-I + Tan[e + f*x])^3)

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fricas [A]  time = 0.61, size = 362, normalized size = 0.91 \[ \frac {{\left (576 i \, d^{3} f^{3} x^{3} + 576 i \, c^{3} f^{3} + 288 \, c^{2} d f^{2} - 96 i \, c d^{2} f - 16 \, d^{3} + {\left (1728 i \, c d^{2} f^{3} + 288 \, d^{3} f^{2}\right )} x^{2} + {\left (1728 i \, c^{2} d f^{3} + 576 \, c d^{2} f^{2} - 96 i \, d^{3} f\right )} x + 864 \, {\left (d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} + 6 \, c^{2} d f^{4} x^{2} + 4 \, c^{3} f^{4} x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (5184 i \, d^{3} f^{3} x^{3} + 5184 i \, c^{3} f^{3} + 7776 \, c^{2} d f^{2} - 7776 i \, c d^{2} f - 3888 \, d^{3} + {\left (15552 i \, c d^{2} f^{3} + 7776 \, d^{3} f^{2}\right )} x^{2} + {\left (15552 i \, c^{2} d f^{3} + 15552 \, c d^{2} f^{2} - 7776 i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (2592 i \, d^{3} f^{3} x^{3} + 2592 i \, c^{3} f^{3} + 1944 \, c^{2} d f^{2} - 972 i \, c d^{2} f - 243 \, d^{3} + {\left (7776 i \, c d^{2} f^{3} + 1944 \, d^{3} f^{2}\right )} x^{2} + {\left (7776 i \, c^{2} d f^{3} + 3888 \, c d^{2} f^{2} - 972 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{27648 \, a^{3} f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27648*(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 + 864*(d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 6*c^2*d
*f^4*x^2 + 4*c^3*f^4*x)*e^(6*I*f*x + 6*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 - 3888*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^(4*I*f*x + 4*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^(2*I*f*x + 2*I
*e))*e^(-6*I*f*x - 6*I*e)/(a^3*f^4)

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giac [A]  time = 2.33, size = 573, normalized size = 1.45 \[ \frac {{\left (864 \, d^{3} f^{4} x^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 3456 \, c d^{2} f^{4} x^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 5184 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 576 i \, d^{3} f^{3} x^{3} + 3456 \, c^{3} f^{4} x e^{\left (6 i \, f x + 6 i \, e\right )} + 15552 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 1728 i \, c d^{2} f^{3} x^{2} + 15552 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} + 1944 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 1728 i \, c^{2} d f^{3} x + 288 \, d^{3} f^{2} x^{2} + 5184 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 15552 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3888 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 576 i \, c^{3} f^{3} + 576 \, c d^{2} f^{2} x + 7776 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 7776 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 1944 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 972 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 288 \, c^{2} d f^{2} - 96 i \, d^{3} f x - 7776 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 972 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 96 i \, c d^{2} f - 3888 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 243 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 16 \, d^{3}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{27648 \, a^{3} f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27648*(864*d^3*f^4*x^4*e^(6*I*f*x + 6*I*e) + 3456*c*d^2*f^4*x^3*e^(6*I*f*x + 6*I*e) + 5184*c^2*d*f^4*x^2*e^(
6*I*f*x + 6*I*e) + 5184*I*d^3*f^3*x^3*e^(4*I*f*x + 4*I*e) + 2592*I*d^3*f^3*x^3*e^(2*I*f*x + 2*I*e) + 576*I*d^3
*f^3*x^3 + 3456*c^3*f^4*x*e^(6*I*f*x + 6*I*e) + 15552*I*c*d^2*f^3*x^2*e^(4*I*f*x + 4*I*e) + 7776*I*c*d^2*f^3*x
^2*e^(2*I*f*x + 2*I*e) + 1728*I*c*d^2*f^3*x^2 + 15552*I*c^2*d*f^3*x*e^(4*I*f*x + 4*I*e) + 7776*d^3*f^2*x^2*e^(
4*I*f*x + 4*I*e) + 7776*I*c^2*d*f^3*x*e^(2*I*f*x + 2*I*e) + 1944*d^3*f^2*x^2*e^(2*I*f*x + 2*I*e) + 1728*I*c^2*
d*f^3*x + 288*d^3*f^2*x^2 + 5184*I*c^3*f^3*e^(4*I*f*x + 4*I*e) + 15552*c*d^2*f^2*x*e^(4*I*f*x + 4*I*e) + 2592*
I*c^3*f^3*e^(2*I*f*x + 2*I*e) + 3888*c*d^2*f^2*x*e^(2*I*f*x + 2*I*e) + 576*I*c^3*f^3 + 576*c*d^2*f^2*x + 7776*
c^2*d*f^2*e^(4*I*f*x + 4*I*e) - 7776*I*d^3*f*x*e^(4*I*f*x + 4*I*e) + 1944*c^2*d*f^2*e^(2*I*f*x + 2*I*e) - 972*
I*d^3*f*x*e^(2*I*f*x + 2*I*e) + 288*c^2*d*f^2 - 96*I*d^3*f*x - 7776*I*c*d^2*f*e^(4*I*f*x + 4*I*e) - 972*I*c*d^
2*f*e^(2*I*f*x + 2*I*e) - 96*I*c*d^2*f - 3888*d^3*e^(4*I*f*x + 4*I*e) - 243*d^3*e^(2*I*f*x + 2*I*e) - 16*d^3)*
e^(-6*I*f*x - 6*I*e)/(a^3*f^4)

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maple [B]  time = 0.93, size = 3404, normalized size = 8.60 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3/f*(2/3*I*c^3*cos(f*x+e)^6-3/4*I/f^2*c*d^2*e^2*cos(f*x+e)^4-2*I/f*c^2*d*e*cos(f*x+e)^6+2*I/f^2*c*d^2*e^2*
cos(f*x+e)^6+4/f^3*d^3*((f*x+e)^3*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/1
6*e)+1/12*(f*x+e)^2*cos(f*x+e)^6-1/6*(f*x+e)*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5
/16*f*x+5/16*e)+245/768*(f*x+e)^2-1/216*cos(f*x+e)^6-65/2304*cos(f*x+e)^4-65/768*cos(f*x+e)^2+5/32*(f*x+e)^2*c
os(f*x+e)^4-5/16*(f*x+e)*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)+15/32*(f*x+e)^2*cos(f*x+
e)^2-15/16*(f*x+e)*(1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+15/64*sin(f*x+e)^2-15/64*(f*x+e)^4)-3/f^3*d^3*((f
*x+e)^3*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)+3/16*(f*x+e)^2*cos(f*x+e)^4-3/8*(f*x+e)*(
1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)+45/128*(f*x+e)^2-3/128*cos(f*x+e)^4-9/128*cos(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*sin(f*x+e)^2-9/32*
(f*x+e)^4)-1/4*I*c^3*cos(f*x+e)^4-3*c^3*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)+4*c^3*(1/
6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)-9/f*c^2*d*((f*x+e)*(1/4*(cos(f*x
+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)-3/16*(f*x+e)^2+1/16*cos(f*x+e)^4+3/16*cos(f*x+e)^2)+9/f^3*d^3*
e*((f*x+e)^2*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)+1/8*(f*x+e)*cos(f*x+e)^4-1/32*(cos(f
*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)-15/64*f*x-15/64*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)-9/f^3*d^3*e^2*((f*x+e)*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)-3/16*(f*x+e)^2+1
/16*cos(f*x+e)^4+3/16*cos(f*x+e)^2)-9/f^2*c*d^2*((f*x+e)^2*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f
*x+3/8*e)+1/8*(f*x+e)*cos(f*x+e)^4-1/32*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)-15/64*f*x-15/64*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/f^3*d^3*e^3*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f
*x+e)+3/8*f*x+3/8*e)+I/f^3*d^3*(-1/4*(f*x+e)^3*cos(f*x+e)^4+3/4*(f*x+e)^2*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*s
in(f*x+e)+3/8*f*x+3/8*e)+3/32*(f*x+e)*cos(f*x+e)^4-3/128*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)-45/256*f*x-4
5/256*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)-4*I/f^3*d^3*(-1/6*(f*x+e)^3*cos(f
*x+e)^6+1/2*(f*x+e)^2*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)+1/36*(f
*x+e)*cos(f*x+e)^6-1/216*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)-245/2304*f*x-245/2304*e+5/
96*(f*x+e)*cos(f*x+e)^4-5/384*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+5/32*(f*x+e)*cos(f*x+e)^2-5/64*sin(f*x+
e)*cos(f*x+e)-5/48*(f*x+e)^3)-4/f^3*d^3*e^3*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/
16*f*x+5/16*e)+12/f^2*c*d^2*((f*x+e)^2*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*
x+5/16*e)+1/18*(f*x+e)*cos(f*x+e)^6-1/108*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)-245/1152*
f*x-245/1152*e+5/48*(f*x+e)*cos(f*x+e)^4-5/192*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+5/16*(f*x+e)*cos(f*x+e
)^2-5/32*sin(f*x+e)*cos(f*x+e)-5/24*(f*x+e)^3)+12/f*c^2*d*((f*x+e)*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*co
s(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)-5/32*(f*x+e)^2+1/36*cos(f*x+e)^6+5/96*cos(f*x+e)^4+5/32*cos(f*x+e)^2)-12
/f^3*d^3*e*((f*x+e)^2*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)+1/18*(f
*x+e)*cos(f*x+e)^6-1/108*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)-245/1152*f*x-245/1152*e+5/
48*(f*x+e)*cos(f*x+e)^4-5/192*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+5/16*(f*x+e)*cos(f*x+e)^2-5/32*sin(f*x+
e)*cos(f*x+e)-5/24*(f*x+e)^3)+12/f^3*d^3*e^2*((f*x+e)*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin
(f*x+e)+5/16*f*x+5/16*e)-5/32*(f*x+e)^2+1/36*cos(f*x+e)^6+5/96*cos(f*x+e)^4+5/32*cos(f*x+e)^2)+3*I/f^3*d^3*e^2
*(-1/4*(f*x+e)*cos(f*x+e)^4+1/16*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/32*f*x+3/32*e)+12*I/f^3*d^3*e*(-1/
6*(f*x+e)^2*cos(f*x+e)^6+1/3*(f*x+e)*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+
5/16*e)-5/96*(f*x+e)^2+1/108*cos(f*x+e)^6+5/288*cos(f*x+e)^4+5/96*cos(f*x+e)^2)-12*I/f^3*d^3*e^2*(-1/6*(f*x+e)
*cos(f*x+e)^6+1/36*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/96*f*x+5/96*e)+3*I/f^2*c*d^2*(
-1/4*(f*x+e)^2*cos(f*x+e)^4+1/2*(f*x+e)*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)-3/32*(f*x
+e)^2+1/32*cos(f*x+e)^4+3/32*cos(f*x+e)^2)+3*I/f*c^2*d*(-1/4*(f*x+e)*cos(f*x+e)^4+1/16*(cos(f*x+e)^3+3/2*cos(f
*x+e))*sin(f*x+e)+3/32*f*x+3/32*e)-12*I/f^2*c*d^2*(-1/6*(f*x+e)^2*cos(f*x+e)^6+1/3*(f*x+e)*(1/6*(cos(f*x+e)^5+
5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)-5/96*(f*x+e)^2+1/108*cos(f*x+e)^6+5/288*cos(f*x+
e)^4+5/96*cos(f*x+e)^2)-12*I/f*c^2*d*(-1/6*(f*x+e)*cos(f*x+e)^6+1/36*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f
*x+e))*sin(f*x+e)+5/96*f*x+5/96*e)+1/4*I/f^3*d^3*e^3*cos(f*x+e)^4-2/3*I/f^3*d^3*e^3*cos(f*x+e)^6+12/f^2*c*d^2*
e^2*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)-9/f^2*c*d^2*e^2*(1/4*(cos
(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)+9/f*c^2*d*e*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+
3/8*f*x+3/8*e)+18/f^2*c*d^2*e*((f*x+e)*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)-3/16*(f*x+
e)^2+1/16*cos(f*x+e)^4+3/16*cos(f*x+e)^2)-24/f^2*c*d^2*e*((f*x+e)*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos
(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)-5/32*(f*x+e)^2+1/36*cos(f*x+e)^6+5/96*cos(f*x+e)^4+5/32*cos(f*x+e)^2)-12/
f*c^2*d*e*(1/6*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/16*f*x+5/16*e)-3*I/f^3*d^3*e*(-1/4
*(f*x+e)^2*cos(f*x+e)^4+1/2*(f*x+e)*(1/4*(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/8*f*x+3/8*e)-3/32*(f*x+e)^
2+1/32*cos(f*x+e)^4+3/32*cos(f*x+e)^2)-6*I/f^2*c*d^2*e*(-1/4*(f*x+e)*cos(f*x+e)^4+1/16*(cos(f*x+e)^3+3/2*cos(f
*x+e))*sin(f*x+e)+3/32*f*x+3/32*e)+3/4*I/f*c^2*d*e*cos(f*x+e)^4+24*I/f^2*c*d^2*e*(-1/6*(f*x+e)*cos(f*x+e)^6+1/
36*(cos(f*x+e)^5+5/4*cos(f*x+e)^3+15/8*cos(f*x+e))*sin(f*x+e)+5/96*f*x+5/96*e))

________________________________________________________________________________________

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*tan(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 = 3.85, size = 411, normalized size = 1.04 \[ {\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,\left (\frac {\left (12\,c^3\,f^3-c^2\,d\,f^2\,18{}\mathrm {i}-18\,c\,d^2\,f+d^3\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^3\,f^4}+\frac {d^3\,x^3\,3{}\mathrm {i}}{16\,a^3\,f}-\frac {d\,x\,\left (-2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}+d^2\right )\,9{}\mathrm {i}}{32\,a^3\,f^3}-\frac {d^2\,x^2\,\left (-2\,c\,f+d\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{32\,a^3\,f^2}\right )+{\mathrm {e}}^{-e\,4{}\mathrm {i}-f\,x\,4{}\mathrm {i}}\,\left (\frac {\left (96\,c^3\,f^3-c^2\,d\,f^2\,72{}\mathrm {i}-36\,c\,d^2\,f+d^3\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{1024\,a^3\,f^4}+\frac {d^3\,x^3\,3{}\mathrm {i}}{32\,a^3\,f}-\frac {d\,x\,\left (-8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}+d^2\right )\,9{}\mathrm {i}}{256\,a^3\,f^3}-\frac {d^2\,x^2\,\left (-4\,c\,f+d\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{128\,a^3\,f^2}\right )+{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,\left (\frac {\left (36\,c^3\,f^3-c^2\,d\,f^2\,18{}\mathrm {i}-6\,c\,d^2\,f+d^3\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{1728\,a^3\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{48\,a^3\,f}-\frac {d\,x\,\left (-18\,c^2\,f^2+c\,d\,f\,6{}\mathrm {i}+d^2\right )\,1{}\mathrm {i}}{288\,a^3\,f^3}-\frac {d^2\,x^2\,\left (-6\,c\,f+d\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{96\,a^3\,f^2}\right )+\frac {c^3\,x}{8\,a^3}+\frac {d^3\,x^4}{32\,a^3}+\frac {3\,c^2\,d\,x^2}{16\,a^3}+\frac {c\,d^2\,x^3}{8\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

exp(- e*2i - f*x*2i)*(((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^3*x^3*3i)/(16*
a^3*f) - (d*x*(d^2 - 2*c^2*f^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*9i + 96*c^3*f^3 - c^2*d*f^2*72i - 36*c*d^2*f)*1i)/(1024*a^3*f^4) + (d^3*x^3*3i)/(32*a^
3*f) - (d*x*(d^2 - 8*c^2*f^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*1i + 36*c^3*f^3 - c^2*d*f^2*18i - 6*c*d^2*f)*1i)/(1728*a^3*f^4) + (d^3*x^3*1i)/(48*a^3
*f) - (d*x*(d^2 - 18*c^2*f^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 = 0.92, size = 949, normalized size = 2.40 \[ \begin {cases} - \frac {\left (\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} + \left (- 10616832 i a^{6} c^{3} f^{11} e^{8 i e} - 31850496 i a^{6} c^{2} d f^{11} x e^{8 i e} - 7962624 a^{6} c^{2} d f^{10} e^{8 i e} - 31850496 i a^{6} c d^{2} f^{11} x^{2} e^{8 i e} - 15925248 a^{6} c d^{2} f^{10} x e^{8 i e} + 3981312 i a^{6} c d^{2} f^{9} e^{8 i e} - 10616832 i a^{6} d^{3} f^{11} x^{3} e^{8 i e} - 7962624 a^{6} d^{3} f^{10} x^{2} e^{8 i e} + 3981312 i a^{6} d^{3} f^{9} x e^{8 i e} + 995328 a^{6} d^{3} f^{8} e^{8 i e}\right ) e^{- 4 i f x} + \left (- 21233664 i a^{6} c^{3} f^{11} e^{10 i e} - 63700992 i a^{6} c^{2} d f^{11} x e^{10 i e} - 31850496 a^{6} c^{2} d f^{10} e^{10 i e} - 63700992 i a^{6} c d^{2} f^{11} x^{2} e^{10 i e} - 63700992 a^{6} c d^{2} f^{10} x e^{10 i e} + 31850496 i a^{6} c d^{2} f^{9} e^{10 i e} - 21233664 i a^{6} d^{3} f^{11} x^{3} e^{10 i e} - 31850496 a^{6} d^{3} f^{10} x^{2} e^{10 i e} + 31850496 i a^{6} d^{3} f^{9} x e^{10 i e} + 15925248 a^{6} d^{3} f^{8} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{113246208 a^{9} f^{12}} & \text {for}\: 113246208 a^{9} f^{12} e^{12 i e} \neq 0 \\\frac {x^{4} \left (3 d^{3} e^{4 i e} + 3 d^{3} e^{2 i e} + d^{3}\right ) e^{- 6 i e}}{32 a^{3}} + \frac {x^{3} \left (3 c d^{2} e^{4 i e} + 3 c d^{2} e^{2 i e} + c d^{2}\right ) e^{- 6 i e}}{8 a^{3}} + \frac {x^{2} \left (9 c^{2} d e^{4 i e} + 9 c^{2} d e^{2 i e} + 3 c^{2} d\right ) e^{- 6 i e}}{16 a^{3}} + \frac {x \left (3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3}\right ) e^{- 6 i e}}{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*tan(f*x+e))**3,x)

[Out]

Piecewise((-((-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*exp(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) + (-10
616832*I*a**6*c**3*f**11*exp(8*I*e) - 31850496*I*a**6*c**2*d*f**11*x*exp(8*I*e) - 7962624*a**6*c**2*d*f**10*ex
p(8*I*e) - 31850496*I*a**6*c*d**2*f**11*x**2*exp(8*I*e) - 15925248*a**6*c*d**2*f**10*x*exp(8*I*e) + 3981312*I*
a**6*c*d**2*f**9*exp(8*I*e) - 10616832*I*a**6*d**3*f**11*x**3*exp(8*I*e) - 7962624*a**6*d**3*f**10*x**2*exp(8*
I*e) + 3981312*I*a**6*d**3*f**9*x*exp(8*I*e) + 995328*a**6*d**3*f**8*exp(8*I*e))*exp(-4*I*f*x) + (-21233664*I*
a**6*c**3*f**11*exp(10*I*e) - 63700992*I*a**6*c**2*d*f**11*x*exp(10*I*e) - 31850496*a**6*c**2*d*f**10*exp(10*I
*e) - 63700992*I*a**6*c*d**2*f**11*x**2*exp(10*I*e) - 63700992*a**6*c*d**2*f**10*x*exp(10*I*e) + 31850496*I*a*
*6*c*d**2*f**9*exp(10*I*e) - 21233664*I*a**6*d**3*f**11*x**3*exp(10*I*e) - 31850496*a**6*d**3*f**10*x**2*exp(1
0*I*e) + 31850496*I*a**6*d**3*f**9*x*exp(10*I*e) + 15925248*a**6*d**3*f**8*exp(10*I*e))*exp(-2*I*f*x))*exp(-12
*I*e)/(113246208*a**9*f**12), Ne(113246208*a**9*f**12*exp(12*I*e), 0)), (x**4*(3*d**3*exp(4*I*e) + 3*d**3*exp(
2*I*e) + d**3)*exp(-6*I*e)/(32*a**3) + x**3*(3*c*d**2*exp(4*I*e) + 3*c*d**2*exp(2*I*e) + c*d**2)*exp(-6*I*e)/(
8*a**3) + x**2*(9*c**2*d*exp(4*I*e) + 9*c**2*d*exp(2*I*e) + 3*c**2*d)*exp(-6*I*e)/(16*a**3) + x*(3*c**3*exp(4*
I*e) + 3*c**3*exp(2*I*e) + c**3)*exp(-6*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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