\(\int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx\) [154]

Optimal result

Integrand size = 29, antiderivative size = 46 \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {x}{2 a}+\frac {i \cos (c+d x)}{2 d (a \cos (c+d x)+i a \sin (c+d x))} \]

[Out]

1/2*x/a+1/2*I*cos(d*x+c)/d/(a*cos(d*x+c)+I*a*sin(d*x+c))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {3161, 8} \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {x}{2 a}+\frac {i \cos (c+d x)}{2 d (a \cos (c+d x)+i a \sin (c+d x))} \]

[In]

Int[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]

[Out]

x/(2*a) + ((I/2)*Cos[c + d*x])/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3161

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> Simp[(-b)*((a*Cos[c + d*x] + b*Sin[c + d*x])^n/(2*a*d*n*Cos[c + d*x]^n)), x] + Dist[1/(2*a), Int[(a*Cos
[c + d*x] + b*Sin[c + d*x])^(n + 1)/Cos[c + d*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] &&
EqQ[a^2 + b^2, 0] && LtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {i \cos (c+d x)}{2 d (a \cos (c+d x)+i a \sin (c+d x))}+\frac {\int 1 \, dx}{2 a} \\ & = \frac {x}{2 a}+\frac {i \cos (c+d x)}{2 d (a \cos (c+d x)+i a \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {2 (c+d x)+i \cos (2 (c+d x))+\sin (2 (c+d x))}{4 a d} \]

[In]

Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]

[Out]

(2*(c + d*x) + I*Cos[2*(c + d*x)] + Sin[2*(c + d*x)])/(4*a*d)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {x}{2 a}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a d}\)	\(26\)
derivativedivides	\(\frac {\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{4}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{4}+\frac {1}{2 \tan \left (d x +c \right )-2 i}}{d a}\)	\(48\)
default	\(\frac {\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{4}-\frac {i \ln \left (\tan \left (d x +c \right )-i\right )}{4}+\frac {1}{2 \tan \left (d x +c \right )-2 i}}{d a}\)	\(48\)

[In]

int(cos(d*x+c)/(cos(d*x+c)*a+I*a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/2*x/a+1/4*I/a/d*exp(-2*I*(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {{\left (2 \, d x e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \]

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\begin {cases} \frac {i e^{- 2 i c} e^{- 2 i d x}}{4 a d} & \text {for}\: a d e^{2 i c} \neq 0 \\x \left (\frac {\left (e^{2 i c} + 1\right ) e^{- 2 i c}}{2 a} - \frac {1}{2 a}\right ) & \text {otherwise} \end {cases} + \frac {x}{2 a} \]

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c)),x)

[Out]

Piecewise((I*exp(-2*I*c)*exp(-2*I*d*x)/(4*a*d), Ne(a*d*exp(2*I*c), 0)), (x*((exp(2*I*c) + 1)*exp(-2*I*c)/(2*a)
- 1/(2*a)), True)) + x/(2*a)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=-\frac {-\frac {i \, \log \left (\tan \left (d x + c\right ) + i\right )}{a} + \frac {i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a} + \frac {-i \, \tan \left (d x + c\right ) - 3}{a {\left (\tan \left (d x + c\right ) - i\right )}}}{4 \, d} \]

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/4*(-I*log(tan(d*x + c) + I)/a + I*log(tan(d*x + c) - I)/a + (-I*tan(d*x + c) - 3)/(a*(tan(d*x + c) - I)))/d

Mupad [B] (verification not implemented)

Time = 22.66 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {x}{2\,a}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d\,{\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^2} \]

[In]

int(cos(c + d*x)/(a*cos(c + d*x) + a*sin(c + d*x)*1i),x)

[Out]

x/(2*a) + tan(c/2 + (d*x)/2)/(a*d*(tan(c/2 + (d*x)/2)*1i + 1)^2)