3.20 \(\int \frac{1}{(c+d x)^2 (a+i a \cot (e+f x))} \, dx\)

Optimal. Leaf size=166 \[ \frac{f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{a d^2}-\frac{i f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{a d^2}+\frac{i f \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}+\frac{f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}-\frac{1}{d (c+d x) (a+i a \cot (e+f x))} \]

[Out]

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

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Rubi [A]  time = 0.227157, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3724, 3303, 3299, 3302} \[ \frac{f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{a d^2}-\frac{i f \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{a d^2}+\frac{i f \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}+\frac{f \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}-\frac{1}{d (c+d x) (a+i a \cot (e+f x))} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^2*(a + I*a*Cot[e + f*x])),x]

[Out]

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

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 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 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]

Rubi steps

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

Mathematica [A]  time = 1.19851, size = 215, normalized size = 1.3 \[ \frac{\left (\cos \left (f \left (x-\frac{c}{d}\right )+e\right )+i \sin \left (f \left (x-\frac{c}{d}\right )+e\right )\right ) \left (2 f (c+d x) \text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right ) \left (\sin \left (e-\frac{f (c+d x)}{d}\right )-i \cos \left (e-\frac{f (c+d x)}{d}\right )\right )+2 f (c+d x) \text{Si}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac{f (c+d x)}{d}\right )+i \sin \left (e-\frac{f (c+d x)}{d}\right )\right )+d \left (i \left (\sin \left (f \left (x-\frac{c}{d}\right )+e\right )+\sin \left (f \left (\frac{c}{d}+x\right )+e\right )\right )-\cos \left (f \left (x-\frac{c}{d}\right )+e\right )+\cos \left (f \left (\frac{c}{d}+x\right )+e\right )\right )\right )}{2 a d^2 (c+d x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((c + d*x)^2*(a + I*a*Cot[e + f*x])),x]

[Out]

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

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Maple [A]  time = 0.158, size = 105, normalized size = 0.6 \begin{align*} -{\frac{1}{2\, \left ( dx+c \right ) ad}}+{\frac{{\frac{i}{2}}f{{\rm e}^{2\,i \left ( fx+e \right ) }}}{a{d}^{2}} \left ( ifx+{\frac{icf}{d}} \right ) ^{-1}}+{\frac{if}{a{d}^{2}}{{\rm e}^{{\frac{-2\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,-2\,ifx-2\,ie-2\,{\frac{icf-ide}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.45037, size = 163, normalized size = 0.98 \begin{align*} \frac{f^{2} \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) E_{2}\left (-\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) - i \, f^{2} E_{2}\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^{2}}{2 \,{\left ({\left (f x + e\right )} a d^{2} - a d^{2} e + a c d f\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+I*a*cot(f*x+e)),x, algorithm="maxima")

[Out]

1/2*(f^2*cos(-2*(d*e - c*f)/d)*exp_integral_e(2, -(2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d) - I*f^2*exp_integra
l_e(2, -(2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d)*sin(-2*(d*e - c*f)/d) - f^2)/(((f*x + e)*a*d^2 - a*d^2*e + a*
c*d*f)*f)

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Fricas [A]  time = 1.6447, size = 178, normalized size = 1.07 \begin{align*} \frac{{\left (-2 i \, d f x - 2 i \, c f\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 e^{\left (2 i \, f x + 2 i \, e\right )} - d}{2 \,{\left (a d^{3} x + a c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+I*a*cot(f*x+e)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.54179, size = 797, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+I*a*cot(f*x+e)),x, algorithm="giac")

[Out]

-1/2*(2*I*d*f*x*cos(2*c*f/d)*cos(2*e)*cos_integral(2*(d*f*x + c*f)/d) + 2*d*f*x*cos(2*e)*cos_integral(2*(d*f*x
+ c*f)/d)*sin(2*c*f/d) - 2*d*f*x*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(2*e) + 2*I*d*f*x*cos_integr
al(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(2*e) - 2*d*f*x*cos(2*c*f/d)*cos(2*e)*sin_integral(2*(d*f*x + c*f)/d) +
2*I*d*f*x*cos(2*e)*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - 2*I*d*f*x*cos(2*c*f/d)*sin(2*e)*sin_integral
(2*(d*f*x + c*f)/d) - 2*d*f*x*sin(2*c*f/d)*sin(2*e)*sin_integral(2*(d*f*x + c*f)/d) + 2*I*c*f*cos(2*c*f/d)*cos
(2*e)*cos_integral(2*(d*f*x + c*f)/d) + 2*c*f*cos(2*e)*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d) - 2*c*f*co
s(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(2*e) + 2*I*c*f*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin
(2*e) - 2*c*f*cos(2*c*f/d)*cos(2*e)*sin_integral(2*(d*f*x + c*f)/d) + 2*I*c*f*cos(2*e)*sin(2*c*f/d)*sin_integr
al(2*(d*f*x + c*f)/d) - 2*I*c*f*cos(2*c*f/d)*sin(2*e)*sin_integral(2*(d*f*x + c*f)/d) - 2*c*f*sin(2*c*f/d)*sin
(2*e)*sin_integral(2*(d*f*x + c*f)/d) - d*cos(2*f*x)*cos(2*e) - I*d*cos(2*e)*sin(2*f*x) - I*d*cos(2*f*x)*sin(2
*e) + d*sin(2*f*x)*sin(2*e))/((d^3*x + c*d^2)*a) - 1/2/((d*x + c)*a*d)