3.21 \(\int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx\)

Optimal. Leaf size=227 \[ \frac {i f^2 \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))} \]

[Out]

1/2*I*f/a/d^2/(d*x+c)+f^2*Ci(2*c*f/d+2*f*x)*cos(-2*e+2*c*f/d)/a/d^3-1/2/d/(d*x+c)^2/(a+I*a*cot(f*x+e))-I*f/d^2
/(d*x+c)/(a+I*a*cot(f*x+e))+I*f^2*cos(-2*e+2*c*f/d)*Si(2*c*f/d+2*f*x)/a/d^3-I*f^2*Ci(2*c*f/d+2*f*x)*sin(-2*e+2
*c*f/d)/a/d^3+f^2*Si(2*c*f/d+2*f*x)*sin(-2*e+2*c*f/d)/a/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.31, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3725, 3724, 3303, 3299, 3302} \[ \frac {i f^2 \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^3*(a + I*a*Cot[e + f*x])),x]

[Out]

((I/2)*f)/(a*d^2*(c + d*x)) + (f^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c +
d*x)^2*(a + I*a*Cot[e + f*x])) - (I*f)/(d^2*(c + d*x)*(a + I*a*Cot[e + f*x])) + (I*f^2*CosIntegral[(2*c*f)/d +
2*f*x]*Sin[2*e - (2*c*f)/d])/(a*d^3) + (I*f^2*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) -
(f^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^3)

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3724

Int[1/(((c_.) + (d_.)*(x_))^2*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> -Simp[(d*(c + d*x)*(a + b*
Tan[e + f*x]))^(-1), x] + (-Dist[f/(a*d), Int[Sin[2*e + 2*f*x]/(c + d*x), x], x] + Dist[f/(b*d), Int[Cos[2*e +
2*f*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]

Rule 3725

Int[((c_.) + (d_.)*(x_))^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(f*(c + d*x)^(m + 2))/
(b*d^2*(m + 1)*(m + 2)), x] + (Dist[(2*b*f)/(a*d*(m + 1)), Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x], x]
+ Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^
2, 0] && LtQ[m, -1] && NeQ[m, -2]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx &=\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}+\frac {(i f) \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))} \, dx}{d}\\ &=\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}-\frac {\left (i f^2\right ) \int \frac {\sin \left (2 \left (e+\frac {\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{a d^2}-\frac {f^2 \int \frac {\cos \left (2 \left (e+\frac {\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{a d^2}\\ &=\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {\left (i f^2 \cos \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}+\frac {\left (f^2 \cos \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}+\frac {\left (i f^2 \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}-\frac {\left (f^2 \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d^2}\\ &=\frac {i f}{2 a d^2 (c+d x)}+\frac {f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {i f^2 \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.59, size = 283, normalized size = 1.25 \[ \frac {\left (\cos \left (f \left (x-\frac {c}{d}\right )+e\right )+i \sin \left (f \left (x-\frac {c}{d}\right )+e\right )\right ) \left (4 f^2 (c+d x)^2 \text {Ci}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {f (c+d x)}{d}\right )+i \sin \left (e-\frac {f (c+d x)}{d}\right )\right )+i \left (4 f^2 (c+d x)^2 \text {Si}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {f (c+d x)}{d}\right )+i \sin \left (e-\frac {f (c+d x)}{d}\right )\right )+d \left (d \sin \left (f \left (x-\frac {c}{d}\right )+e\right )+d \sin \left (f \left (\frac {c}{d}+x\right )+e\right )+2 i c f \sin \left (f \left (\frac {c}{d}+x\right )+e\right )+2 i d f x \sin \left (f \left (\frac {c}{d}+x\right )+e\right )+i d \cos \left (f \left (x-\frac {c}{d}\right )+e\right )+(2 c f+2 d f x-i d) \cos \left (f \left (\frac {c}{d}+x\right )+e\right )\right )\right )\right )}{4 a d^3 (c+d x)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((c + d*x)^3*(a + I*a*Cot[e + f*x])),x]

[Out]

((Cos[e + f*(-(c/d) + x)] + I*Sin[e + f*(-(c/d) + x)])*(4*f^2*(c + d*x)^2*CosIntegral[(2*f*(c + d*x))/d]*(Cos[
e - (f*(c + d*x))/d] + I*Sin[e - (f*(c + d*x))/d]) + I*(d*(I*d*Cos[e + f*(-(c/d) + x)] + ((-I)*d + 2*c*f + 2*d
*f*x)*Cos[e + f*(c/d + x)] + d*Sin[e + f*(-(c/d) + x)] + d*Sin[e + f*(c/d + x)] + (2*I)*c*f*Sin[e + f*(c/d + x
)] + (2*I)*d*f*x*Sin[e + f*(c/d + x)]) + 4*f^2*(c + d*x)^2*(Cos[e - (f*(c + d*x))/d] + I*Sin[e - (f*(c + d*x))
/d])*SinIntegral[(2*f*(c + d*x))/d])))/(4*a*d^3*(c + d*x)^2)

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 118, normalized size = 0.52 \[ \frac {4 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 i \, d f x + 2 i \, c f}{d}\right ) e^{\left (\frac {2 i \, d e - 2 i \, c f}{d}\right )} - d^{2} + {\left (2 i \, d^{2} f x + 2 i \, c d f + d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{4 \, {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+I*a*cot(f*x+e)),x, algorithm="fricas")

[Out]

1/4*(4*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei((2*I*d*f*x + 2*I*c*f)/d)*e^((2*I*d*e - 2*I*c*f)/d) - d^2 + (2*
I*d^2*f*x + 2*I*c*d*f + d^2)*e^(2*I*f*x + 2*I*e))/(a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)

________________________________________________________________________________________

giac [B]  time = 0.62, size = 1630, normalized size = 7.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+I*a*cot(f*x+e)),x, algorithm="giac")

[Out]

1/4*(4*d^2*f^2*x^2*cos(2*c*f/d)*cos(e)^2*cos_integral(2*(d*f*x + c*f)/d) - 4*I*d^2*f^2*x^2*cos(e)^2*cos_integr
al(2*(d*f*x + c*f)/d)*sin(2*c*f/d) + 8*I*d^2*f^2*x^2*cos(2*c*f/d)*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(e
) + 8*d^2*f^2*x^2*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(e) - 4*d^2*f^2*x^2*cos(2*c*f/d)*cos_
integral(2*(d*f*x + c*f)/d)*sin(e)^2 + 4*I*d^2*f^2*x^2*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(e)^2 +
4*I*d^2*f^2*x^2*cos(2*c*f/d)*cos(e)^2*sin_integral(2*(d*f*x + c*f)/d) + 4*d^2*f^2*x^2*cos(e)^2*sin(2*c*f/d)*s
in_integral(2*(d*f*x + c*f)/d) - 8*d^2*f^2*x^2*cos(2*c*f/d)*cos(e)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) + 8*
I*d^2*f^2*x^2*cos(e)*sin(2*c*f/d)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) - 4*I*d^2*f^2*x^2*cos(2*c*f/d)*sin(e)
^2*sin_integral(2*(d*f*x + c*f)/d) - 4*d^2*f^2*x^2*sin(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x + c*f)/d) + 8*c
*d*f^2*x*cos(2*c*f/d)*cos(e)^2*cos_integral(2*(d*f*x + c*f)/d) - 8*I*c*d*f^2*x*cos(e)^2*cos_integral(2*(d*f*x
+ c*f)/d)*sin(2*c*f/d) + 16*I*c*d*f^2*x*cos(2*c*f/d)*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(e) + 16*c*d*f^
2*x*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(e) - 8*c*d*f^2*x*cos(2*c*f/d)*cos_integral(2*(d*f*
x + c*f)/d)*sin(e)^2 + 8*I*c*d*f^2*x*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(e)^2 + 8*I*c*d*f^2*x*cos
(2*c*f/d)*cos(e)^2*sin_integral(2*(d*f*x + c*f)/d) + 8*c*d*f^2*x*cos(e)^2*sin(2*c*f/d)*sin_integral(2*(d*f*x +
c*f)/d) - 16*c*d*f^2*x*cos(2*c*f/d)*cos(e)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) + 16*I*c*d*f^2*x*cos(e)*sin
(2*c*f/d)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) - 8*I*c*d*f^2*x*cos(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x +
c*f)/d) - 8*c*d*f^2*x*sin(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x + c*f)/d) + 4*c^2*f^2*cos(2*c*f/d)*cos(e)^2
*cos_integral(2*(d*f*x + c*f)/d) - 4*I*c^2*f^2*cos(e)^2*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d) + 8*I*c^2
*f^2*cos(2*c*f/d)*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(e) + 8*c^2*f^2*cos(e)*cos_integral(2*(d*f*x + c*f
)/d)*sin(2*c*f/d)*sin(e) - 4*c^2*f^2*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e)^2 + 4*I*c^2*f^2*cos_i
ntegral(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(e)^2 + 4*I*c^2*f^2*cos(2*c*f/d)*cos(e)^2*sin_integral(2*(d*f*x + c
*f)/d) + 4*c^2*f^2*cos(e)^2*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - 8*c^2*f^2*cos(2*c*f/d)*cos(e)*sin(e
)*sin_integral(2*(d*f*x + c*f)/d) + 8*I*c^2*f^2*cos(e)*sin(2*c*f/d)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) - 4
*I*c^2*f^2*cos(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x + c*f)/d) - 4*c^2*f^2*sin(2*c*f/d)*sin(e)^2*sin_integra
l(2*(d*f*x + c*f)/d) + 2*I*d^2*f*x*cos(2*f*x)*cos(e)^2 - 2*d^2*f*x*cos(e)^2*sin(2*f*x) - 4*d^2*f*x*cos(2*f*x)*
cos(e)*sin(e) - 4*I*d^2*f*x*cos(e)*sin(2*f*x)*sin(e) - 2*I*d^2*f*x*cos(2*f*x)*sin(e)^2 + 2*d^2*f*x*sin(2*f*x)*
sin(e)^2 + 2*I*c*d*f*cos(2*f*x)*cos(e)^2 - 2*c*d*f*cos(e)^2*sin(2*f*x) - 4*c*d*f*cos(2*f*x)*cos(e)*sin(e) - 4*
I*c*d*f*cos(e)*sin(2*f*x)*sin(e) - 2*I*c*d*f*cos(2*f*x)*sin(e)^2 + 2*c*d*f*sin(2*f*x)*sin(e)^2 + d^2*cos(2*f*x
)*cos(e)^2 + I*d^2*cos(e)^2*sin(2*f*x) + 2*I*d^2*cos(2*f*x)*cos(e)*sin(e) - 2*d^2*cos(e)*sin(2*f*x)*sin(e) - d
^2*cos(2*f*x)*sin(e)^2 - I*d^2*sin(2*f*x)*sin(e)^2 - d^2)/(a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)

________________________________________________________________________________________

maple [A]  time = 1.64, size = 143, normalized size = 0.63 \[ -\frac {1}{4 d \left (d x +c \right )^{2} a}-\frac {f^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{4 a \,d^{3} \left (i f x +\frac {i c f}{d}\right )^{2}}-\frac {f^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{2 a \,d^{3} \left (i f x +\frac {i c f}{d}\right )}-\frac {f^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \Ei \left (1, -2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{a \,d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d/(d*x+c)^2/a-1/4*f^2/a/d^3*exp(2*I*(f*x+e))/(I*f*x+I/d*c*f)^2-1/2*f^2/a/d^3*exp(2*I*(f*x+e))/(I*f*x+I/d*
c*f)-f^2/a/d^3*exp(-2*I*(c*f-d*e)/d)*Ei(1,-2*I*f*x-2*I*e-2*(I*c*f-I*d*e)/d)

________________________________________________________________________________________

maxima [A]  time = 0.91, size = 160, normalized size = 0.70 \[ \frac {2 \, f^{3} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{3}\left (-\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) - 2 i \, f^{3} E_{3}\left (-\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - f^{3}}{4 \, {\left ({\left (f x + e\right )}^{2} a d^{3} + a d^{3} e^{2} - 2 \, a c d^{2} e f + a c^{2} d f^{2} - 2 \, {\left (a d^{3} e - a c d^{2} f\right )} {\left (f x + e\right )}\right )} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+I*a*cot(f*x+e)),x, algorithm="maxima")

[Out]

1/4*(2*f^3*cos(-2*(d*e - c*f)/d)*exp_integral_e(3, -(2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d) - 2*I*f^3*exp_int
egral_e(3, -(2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d)*sin(-2*(d*e - c*f)/d) - f^3)/(((f*x + e)^2*a*d^3 + a*d^3*
e^2 - 2*a*c*d^2*e*f + a*c^2*d*f^2 - 2*(a*d^3*e - a*c*d^2*f)*(f*x + e))*f)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {1}{c^{3} \cot {\left (e + f x \right )} - i c^{3} + 3 c^{2} d x \cot {\left (e + f x \right )} - 3 i c^{2} d x + 3 c d^{2} x^{2} \cot {\left (e + f x \right )} - 3 i c d^{2} x^{2} + d^{3} x^{3} \cot {\left (e + f x \right )} - i d^{3} x^{3}}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**3/(a+I*a*cot(f*x+e)),x)

[Out]

-I*Integral(1/(c**3*cot(e + f*x) - I*c**3 + 3*c**2*d*x*cot(e + f*x) - 3*I*c**2*d*x + 3*c*d**2*x**2*cot(e + f*x
) - 3*I*c*d**2*x**2 + d**3*x**3*cot(e + f*x) - I*d**3*x**3), x)/a

________________________________________________________________________________________