3.5 \(\int \frac{\csc (x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=14 \[ \frac{i \csc (x)}{\cot (x)+i} \]

[Out]

(I*Csc[x])/(I + Cot[x])

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Rubi [A]  time = 0.0187808, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3488} \[ \frac{i \csc (x)}{\cot (x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]/(I + Cot[x]),x]

[Out]

(I*Csc[x])/(I + Cot[x])

Rule 3488

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*} \int \frac{\csc (x)}{i+\cot (x)} \, dx &=\frac{i \csc (x)}{i+\cot (x)}\\ \end{align*}

Mathematica [A]  time = 0.0035542, size = 9, normalized size = 0.64 \[ \sin (x)+i \cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]/(I + Cot[x]),x]

[Out]

I*Cos[x] + Sin[x]

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Maple [A]  time = 0.035, size = 12, normalized size = 0.9 \begin{align*} 2\, \left ( \tan \left ( x/2 \right ) -i \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(tan(1/2*x)-I)

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Maxima [A]  time = 1.17676, size = 20, normalized size = 1.43 \begin{align*} \frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sin(x)/(cos(x) + 1) - I)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{\left (3 i \, x\right )} - e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{e^{\left (2 i \, x\right )} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e^(3*I*x) - e^(I*x))*e^(-2*I*x)/(e^(2*I*x) - 1), x)

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Sympy [A]  time = 1.0056, size = 8, normalized size = 0.57 \begin{align*} \frac{i \csc{\left (x \right )}}{\cot{\left (x \right )} + i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*csc(x)/(cot(x) + I)

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Giac [A]  time = 1.23846, size = 14, normalized size = 1. \begin{align*} \frac{2}{\tan \left (\frac{1}{2} \, x\right ) - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(tan(1/2*x) - I)