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]

3/16*d^3*exp(2*I*e+2*I*f*x)/a^2/f^4-3/512*d^3*exp(4*I*e+4*I*f*x)/a^2/f^4-3/8*I*d^2*exp(2*I*e+2*I*f*x)*(d*x+c)/
a^2/f^3+3/128*I*d^2*exp(4*I*e+4*I*f*x)*(d*x+c)/a^2/f^3-3/8*d*exp(2*I*e+2*I*f*x)*(d*x+c)^2/a^2/f^2+3/64*d*exp(4
*I*e+4*I*f*x)*(d*x+c)^2/a^2/f^2+1/4*I*exp(2*I*e+2*I*f*x)*(d*x+c)^3/a^2/f-1/16*I*exp(4*I*e+4*I*f*x)*(d*x+c)^3/a
^2/f+1/16*(d*x+c)^4/a^2/d

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Rubi [A]  time = 0.28, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 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))^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*}

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Mathematica [A]  time = 1.61, size = 362, normalized size = 1.34 \[ \frac {(\cos (2 (e+f x))+i \sin (2 (e+f x))) \left (\left (32 c^3 f^3 (4 f x-i)+24 c^2 d f^2 \left (8 f^2 x^2-4 i f x+1\right )+4 c d^2 f \left (32 f^3 x^3-24 i f^2 x^2+12 f x+3 i\right )+d^3 \left (32 f^4 x^4-32 i f^3 x^3+24 f^2 x^2+12 i f x-3\right )\right ) \cos (2 (e+f x))-i \left (\left (32 c^3 f^3 (4 f x+i)+24 c^2 d f^2 \left (8 f^2 x^2+4 i f x-1\right )+4 c d^2 f \left (32 f^3 x^3+24 i f^2 x^2-12 f x-3 i\right )+d^3 \left (32 f^4 x^4+32 i f^3 x^3-24 f^2 x^2-12 i f x+3\right )\right ) \sin (2 (e+f x))-32 \left (4 c^3 f^3+6 c^2 d f^2 (2 f x+i)+6 c d^2 f \left (2 f^2 x^2+2 i f x-1\right )+d^3 \left (4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right )\right )\right )\right )}{512 a^2 f^4} \]

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)

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fricas [A]  time = 1.90, size = 253, normalized size = 0.94 \[ \frac {32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} + 192 \, c^{2} d f^{4} x^{2} + 128 \, c^{3} f^{4} x + {\left (-32 i \, d^{3} f^{3} x^{3} - 32 i \, c^{3} f^{3} + 24 \, c^{2} d f^{2} + 12 i \, c d^{2} f - 3 \, d^{3} + {\left (-96 i \, c d^{2} f^{3} + 24 \, d^{3} f^{2}\right )} x^{2} + {\left (-96 i \, c^{2} d f^{3} + 48 \, c d^{2} f^{2} + 12 i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (128 i \, d^{3} f^{3} x^{3} + 128 i \, c^{3} f^{3} - 192 \, c^{2} d f^{2} - 192 i \, c d^{2} f + 96 \, d^{3} + {\left (384 i \, c d^{2} f^{3} - 192 \, d^{3} f^{2}\right )} x^{2} + {\left (384 i \, c^{2} d f^{3} - 384 \, c d^{2} f^{2} - 192 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} f^{4}} \]

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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giac [B]  time = 0.51, size = 433, normalized size = 1.60 \[ \frac {32 \, d^{3} f^{4} x^{4} + 128 \, c d^{2} f^{4} x^{3} - 32 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 192 \, c^{2} d f^{4} x^{2} - 96 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 128 \, c^{3} f^{4} x - 96 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 24 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} - 192 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 32 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 48 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 384 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} - 192 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 192 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 12 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 192 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 3 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )}}{512 \, a^{2} f^{4}} \]

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)

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maple [B]  time = 2.16, size = 2261, normalized size = 8.37 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^2/f*(1/2*I*c^3*sin(f*x+e)^4+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)+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)-12/f^2*c*d^2*e*((f*x+e)*(-1/
2*sin(f*x+e)*cos(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)-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*sin(f*x+e))*cos(f*x+e)-3/32*f*x-3/32*e)-1/f^3*
d^3*((f*x+e)^3*(-1/2*sin(f*x+e)*cos(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*sin(f*x+
e)*cos(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)+2/f^3*d^3*((f*x+e)^3*(-1/2*sin(f*x+
e)*cos(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*sin(f*x+e)*cos(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)+3/f*c^2*d*e*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+6/f^2*c*d^2*e*((f*x+e)*(-1/2*
sin(f*x+e)*cos(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*e^2*(-1/4*sin(f*x+e)*cos(f*x+
e)^3+1/8*sin(f*x+e)*cos(f*x+e)+1/8*f*x+1/8*e)-3/f^2*c*d^2*e^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-6/f*c
^2*d*e*(-1/4*sin(f*x+e)*cos(f*x+e)^3+1/8*sin(f*x+e)*cos(f*x+e)+1/8*f*x+1/8*e)-3/f*c^2*d*((f*x+e)*(-1/2*sin(f*x
+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/4*(f*x+e)^2+1/4*sin(f*x+e)^2)+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*sin(f*x+e)*cos(f*x+e)+3/16*(
f*x+e)^3)-2/f^3*d^3*e^3*(-1/4*sin(f*x+e)*cos(f*x+e)^3+1/8*sin(f*x+e)*cos(f*x+e)+1/8*f*x+1/8*e)-3/f^3*d^3*e^2*(
(f*x+e)*(-1/2*sin(f*x+e)*cos(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*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/8*(f*x+e)*cos(f*x+e)^2+1/16*sin(f*x+e)*cos(f*x+e)+7/64*f*x+7/64*e-1/1
2*(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))+3/f^3*d^3*e*((f*x+e)^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e
)-1/2*(f*x+e)*cos(f*x+e)^2+1/4*sin(f*x+e)*cos(f*x+e)+1/4*f*x+1/4*e-1/3*(f*x+e)^3)-3/f^2*c*d^2*((f*x+e)^2*(-1/2
*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/2*(f*x+e)*cos(f*x+e)^2+1/4*sin(f*x+e)*cos(f*x+e)+1/4*f*x+1/4*e-1/3*(f*
x+e)^3)-6/f^3*d^3*e*((f*x+e)^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/8*(f*x+e)*cos(f*x+e)^2+1/16*sin(f*
x+e)*cos(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^3*d^3*e^2*((f*x+e)*(-1/2*
sin(f*x+e)*cos(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)+6/f*c^2*d*((f*x+e)*(-1/2*sin(f*x+e)*cos(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*s
in(f*x+e)^4)+1/f^3*d^3*e^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/2*I/f^3*d^3*e^3*sin(f*x+e)^4-3/2*I/f*c
^2*d*e*sin(f*x+e)^4+3/2*I/f^2*c*d^2*e^2*sin(f*x+e)^4-12*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)-c^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+2*c^3*(-1/4*si
n(f*x+e)*cos(f*x+e)^3+1/8*sin(f*x+e)*cos(f*x+e)+1/8*f*x+1/8*e))

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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))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 1.12, size = 294, normalized size = 1.09 \[ {\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (-\frac {\left (-4\,c^3\,f^3-c^2\,d\,f^2\,6{}\mathrm {i}+6\,c\,d^2\,f+d^3\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^2\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{4\,a^2\,f}+\frac {d\,x\,\left (2\,c^2\,f^2+c\,d\,f\,2{}\mathrm {i}-d^2\right )\,3{}\mathrm {i}}{8\,a^2\,f^3}+\frac {d^2\,x^2\,\left (2\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,a^2\,f^2}\right )-{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (-\frac {\left (-32\,c^3\,f^3-c^2\,d\,f^2\,24{}\mathrm {i}+12\,c\,d^2\,f+d^3\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{512\,a^2\,f^4}+\frac {d^3\,x^3\,1{}\mathrm {i}}{16\,a^2\,f}+\frac {d\,x\,\left (8\,c^2\,f^2+c\,d\,f\,4{}\mathrm {i}-d^2\right )\,3{}\mathrm {i}}{128\,a^2\,f^3}+\frac {d^2\,x^2\,\left (4\,c\,f+d\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,a^2\,f^2}\right )+\frac {c^3\,x}{4\,a^2}+\frac {d^3\,x^4}{16\,a^2}+\frac {3\,c^2\,d\,x^2}{8\,a^2}+\frac {c\,d^2\,x^3}{4\,a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + a*cot(e + f*x)*1i)^2,x)

[Out]

exp(e*2i + f*x*2i)*((d^3*x^3*1i)/(4*a^2*f) - ((d^3*3i - 4*c^3*f^3 - c^2*d*f^2*6i + 6*c*d^2*f)*1i)/(16*a^2*f^4)
+ (d*x*(2*c^2*f^2 - d^2 + c*d*f*2i)*3i)/(8*a^2*f^3) + (d^2*x^2*(d*1i + 2*c*f)*3i)/(8*a^2*f^2)) - exp(e*4i + f
*x*4i)*((d^3*x^3*1i)/(16*a^2*f) - ((d^3*3i - 32*c^3*f^3 - c^2*d*f^2*24i + 12*c*d^2*f)*1i)/(512*a^2*f^4) + (d*x
*(8*c^2*f^2 - d^2 + c*d*f*4i)*3i)/(128*a^2*f^3) + (d^2*x^2*(d*1i + 4*c*f)*3i)/(64*a^2*f^2)) + (c^3*x)/(4*a^2)
+ (d^3*x^4)/(16*a^2) + (3*c^2*d*x^2)/(8*a^2) + (c*d^2*x^3)/(4*a^2)

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sympy [A]  time = 0.64, size = 653, normalized size = 2.42 \[ \begin {cases} \frac {\left (2048 i a^{2} c^{3} f^{7} e^{2 i e} + 6144 i a^{2} c^{2} d f^{7} x e^{2 i e} - 3072 a^{2} c^{2} d f^{6} e^{2 i e} + 6144 i a^{2} c d^{2} f^{7} x^{2} e^{2 i e} - 6144 a^{2} c d^{2} f^{6} x e^{2 i e} - 3072 i a^{2} c d^{2} f^{5} e^{2 i e} + 2048 i a^{2} d^{3} f^{7} x^{3} e^{2 i e} - 3072 a^{2} d^{3} f^{6} x^{2} e^{2 i e} - 3072 i a^{2} d^{3} f^{5} x e^{2 i e} + 1536 a^{2} d^{3} f^{4} e^{2 i e}\right ) e^{2 i f x} + \left (- 512 i a^{2} c^{3} f^{7} e^{4 i e} - 1536 i a^{2} c^{2} d f^{7} x e^{4 i e} + 384 a^{2} c^{2} d f^{6} e^{4 i e} - 1536 i a^{2} c d^{2} f^{7} x^{2} e^{4 i e} + 768 a^{2} c d^{2} f^{6} x e^{4 i e} + 192 i a^{2} c d^{2} f^{5} e^{4 i e} - 512 i a^{2} d^{3} f^{7} x^{3} e^{4 i e} + 384 a^{2} d^{3} f^{6} x^{2} e^{4 i e} + 192 i a^{2} d^{3} f^{5} x e^{4 i e} - 48 a^{2} d^{3} f^{4} e^{4 i e}\right ) e^{4 i f x}}{8192 a^{4} f^{8}} & \text {for}\: 8192 a^{4} f^{8} \neq 0 \\\frac {x^{4} \left (d^{3} e^{4 i e} - 2 d^{3} e^{2 i e}\right )}{16 a^{2}} + \frac {x^{3} \left (c d^{2} e^{4 i e} - 2 c d^{2} e^{2 i e}\right )}{4 a^{2}} + \frac {x^{2} \left (3 c^{2} d e^{4 i e} - 6 c^{2} d e^{2 i e}\right )}{8 a^{2}} + \frac {x \left (c^{3} e^{4 i e} - 2 c^{3} e^{2 i e}\right )}{4 a^{2}} & \text {otherwise} \end {cases} + \frac {c^{3} x}{4 a^{2}} + \frac {3 c^{2} d x^{2}}{8 a^{2}} + \frac {c d^{2} x^{3}}{4 a^{2}} + \frac {d^{3} x^{4}}{16 a^{2}} \]

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**2*c**3*f**7*exp(2*I*e) + 6144*I*a**2*c**2*d*f**7*x*exp(2*I*e) - 3072*a**2*c**2*d*f**6*e
xp(2*I*e) + 6144*I*a**2*c*d**2*f**7*x**2*exp(2*I*e) - 6144*a**2*c*d**2*f**6*x*exp(2*I*e) - 3072*I*a**2*c*d**2*
f**5*exp(2*I*e) + 2048*I*a**2*d**3*f**7*x**3*exp(2*I*e) - 3072*a**2*d**3*f**6*x**2*exp(2*I*e) - 3072*I*a**2*d*
*3*f**5*x*exp(2*I*e) + 1536*a**2*d**3*f**4*exp(2*I*e))*exp(2*I*f*x) + (-512*I*a**2*c**3*f**7*exp(4*I*e) - 1536
*I*a**2*c**2*d*f**7*x*exp(4*I*e) + 384*a**2*c**2*d*f**6*exp(4*I*e) - 1536*I*a**2*c*d**2*f**7*x**2*exp(4*I*e) +
768*a**2*c*d**2*f**6*x*exp(4*I*e) + 192*I*a**2*c*d**2*f**5*exp(4*I*e) - 512*I*a**2*d**3*f**7*x**3*exp(4*I*e)
+ 384*a**2*d**3*f**6*x**2*exp(4*I*e) + 192*I*a**2*d**3*f**5*x*exp(4*I*e) - 48*a**2*d**3*f**4*exp(4*I*e))*exp(4
*I*f*x))/(8192*a**4*f**8), Ne(8192*a**4*f**8, 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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