\(\int \frac {\csc (x)}{i+\cot (x)} \, dx\) [5]

Optimal result

Integrand size = 11, antiderivative size = 14 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {i \csc (x)}{i+\cot (x)} \]

[Out]

I*csc(x)/(I+cot(x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3569} \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {i \csc (x)}{\cot (x)+i} \]

[In]

Int[Csc[x]/(I + Cot[x]),x]

[Out]

(I*Csc[x])/(I + Cot[x])

Rule 3569

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(d*
Sec[e + f*x])^m*((a + b*Tan[e + f*x])^n/(a*f*m)), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rubi steps \begin{align*} \text {integral}& = \frac {i \csc (x)}{i+\cot (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=i \cos (x)+\sin (x) \]

[In]

Integrate[Csc[x]/(I + Cot[x]),x]

[Out]

I*Cos[x] + Sin[x]

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64

method	result	size
risch	\(i {\mathrm e}^{-i x}\)	\(9\)
default	\(\frac {2}{-i+\tan \left (\frac {x}{2}\right )}\)	\(12\)

[In]

int(csc(x)/(I+cot(x)),x,method=_RETURNVERBOSE)

[Out]

I*exp(-I*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.43 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=i \, e^{\left (-i \, x\right )} \]

[In]

integrate(csc(x)/(I+cot(x)),x, algorithm="fricas")

[Out]

I*e^(-I*x)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {i \csc {\left (x \right )}}{\cot {\left (x \right )} + i} \]

[In]

integrate(csc(x)/(I+cot(x)),x)

[Out]

I*csc(x)/(cot(x) + I)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {2}{\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - i} \]

[In]

integrate(csc(x)/(I+cot(x)),x, algorithm="maxima")

[Out]

2/(sin(x)/(cos(x) + 1) - I)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {2}{\tan \left (\frac {1}{2} \, x\right ) - i} \]

[In]

integrate(csc(x)/(I+cot(x)),x, algorithm="giac")

[Out]

2/(tan(1/2*x) - I)

Mupad [B] (verification not implemented)

Time = 12.48 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (x)}{i+\cot (x)} \, dx=\frac {2{}\mathrm {i}}{1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}} \]

[In]

int(1/(sin(x)*(cot(x) + 1i)),x)

[Out]

2i/(tan(x/2)*1i + 1)