3.4 \(\int \frac {\sin (x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (\cot (x)+i)} \]

[Out]

2/3*I*cos(x)+1/3*I*sin(x)/(I+cot(x))

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Rubi [A]  time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3502, 2638} \[ \frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (\cot (x)+i)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(I + Cot[x]),x]

[Out]

((2*I)/3)*Cos[x] + ((I/3)*Sin[x])/(I + Cot[x])

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3502

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(b*f*(m + 2*n)), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]

Rubi steps

\begin {align*} \int \frac {\sin (x)}{i+\cot (x)} \, dx &=\frac {i \sin (x)}{3 (i+\cot (x))}-\frac {2}{3} i \int \sin (x) \, dx\\ &=\frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (i+\cot (x))}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 27, normalized size = 1.08 \[ \frac {1}{6} (\sin (x)+i \cos (x)) (2 i \sin (2 x)+\cos (2 x)+3) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(I + Cot[x]),x]

[Out]

((I*Cos[x] + Sin[x])*(3 + Cos[2*x] + (2*I)*Sin[2*x]))/6

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fricas [A]  time = 0.52, size = 20, normalized size = 0.80 \[ \frac {1}{12} \, {\left (3 i \, e^{\left (4 i \, x\right )} + 6 i \, e^{\left (2 i \, x\right )} - i\right )} e^{\left (-3 i \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*I*e^(4*I*x) + 6*I*e^(2*I*x) - I)*e^(-3*I*x)

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giac [B]  time = 0.57, size = 37, normalized size = 1.48 \[ -\frac {1}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}} + \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 12 i \, \tan \left (\frac {1}{2} \, x\right ) - 5}{6 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(tan(1/2*x) + I) + 1/6*(3*tan(1/2*x)^2 - 12*I*tan(1/2*x) - 5)/(tan(1/2*x) - I)^3

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maple [B]  time = 0.31, size = 47, normalized size = 1.88 \[ -\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )+i\right )}-\frac {i}{\left (\tan \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {2}{3 \left (\tan \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {1}{2 \tan \left (\frac {x}{2}\right )-2 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(tan(1/2*x)+I)-I/(tan(1/2*x)-I)^2+2/3/(tan(1/2*x)-I)^3+1/2/(tan(1/2*x)-I)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.30, size = 31, normalized size = 1.24 \[ -\frac {\frac {4}{3}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,8{}\mathrm {i}}{3}}{{\left (1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cot(x) + 1i),x)

[Out]

-((tan(x/2)*8i)/3 + 4/3)/((tan(x/2)*1i + 1)^3*(tan(x/2) + 1i))

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sympy [A]  time = 0.14, size = 26, normalized size = 1.04 \[ \frac {i e^{i x}}{4} + \frac {i e^{- i x}}{2} - \frac {i e^{- 3 i x}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*exp(I*x)/4 + I*exp(-I*x)/2 - I*exp(-3*I*x)/12

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