\(\int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx\) [155]

Optimal result

Integrand size = 22, antiderivative size = 29 \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3150} \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]

[In]

Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-1),x]

[Out]

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

Rule 3150

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

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]

[In]

Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-1),x]

[Out]

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66

method	result	size
risch	\(\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{a d}\)	\(19\)
derivativedivides	\(\frac {2}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}\)	\(23\)
default	\(\frac {2}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}\)	\(23\)
norman	\(\frac {-\frac {2 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a d}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\)	\(55\)

[In]

int(1/(cos(d*x+c)*a+I*a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {i \, e^{\left (-i \, d x - i \, c\right )}}{a d} \]

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c)),x, algorithm="fricas")

[Out]

I*e^(-I*d*x - I*c)/(a*d)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\begin {cases} \frac {i e^{- i c} e^{- i d x}}{a d} & \text {for}\: a d e^{i c} \neq 0 \\\frac {x e^{- i c}}{a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {2}{{\left (-i \, a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} d} \]

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {2}{a d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}} \]

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c)),x, algorithm="giac")

[Out]

2/(a*d*(tan(1/2*d*x + 1/2*c) - I))

Mupad [B] (verification not implemented)

Time = 22.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx=\frac {2{}\mathrm {i}}{a\,d\,\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )} \]

[In]

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

[Out]

2i/(a*d*(tan(c/2 + (d*x)/2)*1i + 1))