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 = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.45, size = 6, normalized size = 0.43 \[ i \, e^{\left (-i \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*e^(-I*x)

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giac [A]  time = 1.15, size = 10, normalized size = 0.71 \[ \frac {2}{\tan \left (\frac {1}{2} \, x\right ) - i} \]

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)

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maple [A]  time = 0.14, size = 12, normalized size = 0.86 \[ \frac {2}{\tan \left (\frac {x}{2}\right )-i} \]

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 = 0.71, size = 15, normalized size = 1.07 \[ \frac {2}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} - i} \]

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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mupad [B]  time = 0.20, size = 14, normalized size = 1.00 \[ \frac {2{}\mathrm {i}}{1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)*(cot(x) + 1i)),x)

[Out]

2i/(tan(x/2)*1i + 1)

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sympy [A]  time = 0.34, size = 8, normalized size = 0.57 \[ \frac {i \csc {\relax (x )}}{\cot {\relax (x )} + i} \]

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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