\(\int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx\) [19]

Optimal result

Integrand size = 23, antiderivative size = 161 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=-\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}+\frac {\log (c+d x)}{2 a d}-\frac {i \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}+\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3807, 3384, 3380, 3383} \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=-\frac {i \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a d}-\frac {\operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{2 a d}+\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{2 a d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{2 a d}+\frac {\log (c+d x)}{2 a d} \]

[In]

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

[Out]

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

Rule 3380

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 3383

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 3384

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 3807

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

Rubi steps \begin{align*} \text {integral}& = \frac {\log (c+d x)}{2 a d}+\frac {i \int \frac {\sin \left (2 \left (e+\frac {\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{2 a}+\frac {\int \frac {\cos \left (2 \left (e+\frac {\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{2 a} \\ & = \frac {\log (c+d x)}{2 a d}-\frac {\left (i \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}{2 a}-\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}-\frac {\left (i \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}{2 a}+\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a} \\ & = -\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}+\frac {\log (c+d x)}{2 a d}-\frac {i \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}+\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=\frac {\log (c+d x)-\left (\cos \left (2 e-\frac {2 c f}{d}\right )+i \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 a d} \]

[In]

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

[Out]

(Log[c + d*x] - (Cos[2*e - (2*c*f)/d] + I*Sin[2*e - (2*c*f)/d])*(CosIntegral[(2*f*(c + d*x))/d] + I*SinIntegra
l[(2*f*(c + d*x))/d]))/(2*a*d)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.42

method	result	size
risch	\(\frac {\ln \left (d x +c \right )}{2 a d}+\frac {{\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{2 a d}\)	\(67\)
derivativedivides	\(\frac {-\frac {i \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{4}+\frac {\ln \left (c f -d e +d \left (f x +e \right )\right )}{2 d}-\frac {\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{2 d}-\frac {\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{2 d}}{a}\)	\(195\)
default	\(\frac {-\frac {i \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{4}+\frac {\ln \left (c f -d e +d \left (f x +e \right )\right )}{2 d}-\frac {\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{2 d}-\frac {\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{2 d}}{a}\)	\(195\)

[In]

int(1/(d*x+c)/(a+I*a*cot(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(d*x+c)/a/d+1/2/a/d*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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.33 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=-\frac {{\rm Ei}\left (-\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, d e + i \, c f\right )}}{d}\right )} - \log \left (\frac {d x + c}{d}\right )}{2 \, a d} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=- \frac {i \int \frac {1}{c \cot {\left (e + f x \right )} - i c + d x \cot {\left (e + f x \right )} - i d x}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=\frac {f \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - i \, f E_{1}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + f \log \left ({\left (f x + e\right )} d - d e + c f\right )}{2 \, a d f} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (147) = 294\).

Time = 0.28 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=-\frac {\cos \left (e\right )^{2} \cos \left (\frac {2 \, c f}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 i \, \cos \left (e\right ) \cos \left (\frac {2 \, c f}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (e\right ) - \cos \left (\frac {2 \, c f}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (e\right )^{2} - i \, \cos \left (e\right )^{2} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {2 \, c f}{d}\right ) + 2 \, \cos \left (e\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (e\right ) \sin \left (\frac {2 \, c f}{d}\right ) + i \, \operatorname {Ci}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (e\right )^{2} \sin \left (\frac {2 \, c f}{d}\right ) + i \, \cos \left (e\right )^{2} \cos \left (\frac {2 \, c f}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - 2 \, \cos \left (e\right ) \cos \left (\frac {2 \, c f}{d}\right ) \sin \left (e\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - i \, \cos \left (\frac {2 \, c f}{d}\right ) \sin \left (e\right )^{2} \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + \cos \left (e\right )^{2} \sin \left (\frac {2 \, c f}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 i \, \cos \left (e\right ) \sin \left (e\right ) \sin \left (\frac {2 \, c f}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - \sin \left (e\right )^{2} \sin \left (\frac {2 \, c f}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - \log \left (d x + c\right )}{2 \, a d} \]

[In]

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

[Out]

-1/2*(cos(e)^2*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d) + 2*I*cos(e)*cos(2*c*f/d)*cos_integral(2*(d*f*x +
c*f)/d)*sin(e) - cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e)^2 - I*cos(e)^2*cos_integral(2*(d*f*x + c*
f)/d)*sin(2*c*f/d) + 2*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(e)*sin(2*c*f/d) + I*cos_integral(2*(d*f*x +
c*f)/d)*sin(e)^2*sin(2*c*f/d) + I*cos(e)^2*cos(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - 2*cos(e)*cos(2*c*f/d
)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) - I*cos(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x + c*f)/d) + cos(e)^2*
sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 2*I*cos(e)*sin(e)*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d)
- sin(e)^2*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - log(d*x + c))/(a*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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