3.2 \(\int \frac{\sin ^3(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=37 \[ -\frac{4}{15} i \cos ^3(x)+\frac{4}{5} i \cos (x)+\frac{i \sin ^3(x)}{5 (\cot (x)+i)} \]

[Out]

((4*I)/5)*Cos[x] - ((4*I)/15)*Cos[x]^3 + ((I/5)*Sin[x]^3)/(I + Cot[x])

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Rubi [A]  time = 0.0399543, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3502, 2633} \[ -\frac{4}{15} i \cos ^3(x)+\frac{4}{5} i \cos (x)+\frac{i \sin ^3(x)}{5 (\cot (x)+i)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(I + Cot[x]),x]

[Out]

((4*I)/5)*Cos[x] - ((4*I)/15)*Cos[x]^3 + ((I/5)*Sin[x]^3)/(I + Cot[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]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\sin ^3(x)}{i+\cot (x)} \, dx &=\frac{i \sin ^3(x)}{5 (i+\cot (x))}-\frac{4}{5} i \int \sin ^3(x) \, dx\\ &=\frac{i \sin ^3(x)}{5 (i+\cot (x))}+\frac{4}{5} i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=\frac{4}{5} i \cos (x)-\frac{4}{15} i \cos ^3(x)+\frac{i \sin ^3(x)}{5 (i+\cot (x))}\\ \end{align*}

Mathematica [A]  time = 0.0926275, size = 46, normalized size = 1.24 \[ \frac{\csc (x) (-40 \sin (2 x)+4 \sin (4 x)+20 i \cos (2 x)-i \cos (4 x)+45 i)}{120 (\cot (x)+i)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(I + Cot[x]),x]

[Out]

(Csc[x]*(45*I + (20*I)*Cos[2*x] - I*Cos[4*x] - 40*Sin[2*x] + 4*Sin[4*x]))/(120*(I + Cot[x]))

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Maple [B]  time = 0.064, size = 93, normalized size = 2.5 \begin{align*}{-i \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) ^{-4}}-{{\frac{i}{2}} \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}+{\frac{2}{5} \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) ^{-5}}-{\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}}+{\frac{3}{8} \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}-{{\frac{i}{4}} \left ( \tan \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}}-{\frac{1}{6} \left ( \tan \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}-{\frac{3}{8} \left ( \tan \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(I+cot(x)),x)

[Out]

-I/(tan(1/2*x)-I)^4-1/2*I/(tan(1/2*x)-I)^2+2/5/(tan(1/2*x)-I)^5-1/3/(tan(1/2*x)-I)^3+3/8/(tan(1/2*x)-I)-1/4*I/
(tan(1/2*x)+I)^2-1/6/(tan(1/2*x)+I)^3-3/8/(tan(1/2*x)+I)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(I+cot(x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 1.55839, size = 194, normalized size = 5.24 \begin{align*} \frac{1}{240} \,{\left ({\left (-5 i \, e^{\left (6 i \, x\right )} + 45 i \, e^{\left (4 i \, x\right )} + 45 i \, e^{\left (2 i \, x\right )} - 5 i\right )} e^{\left (2 i \, x\right )} + 15 i \, e^{\left (6 i \, x\right )} + 45 i \, e^{\left (4 i \, x\right )} - 15 i \, e^{\left (2 i \, x\right )} + 3 i\right )} e^{\left (-5 i \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(I+cot(x)),x, algorithm="fricas")

[Out]

1/240*((-5*I*e^(6*I*x) + 45*I*e^(4*I*x) + 45*I*e^(2*I*x) - 5*I)*e^(2*I*x) + 15*I*e^(6*I*x) + 45*I*e^(4*I*x) -
15*I*e^(2*I*x) + 3*I)*e^(-5*I*x)

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Sympy [A]  time = 0.297597, size = 48, normalized size = 1.3 \begin{align*} - \frac{i e^{3 i x}}{48} + \frac{i e^{i x}}{4} + \frac{3 i e^{- i x}}{8} - \frac{i e^{- 3 i x}}{12} + \frac{i e^{- 5 i x}}{80} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3/(I+cot(x)),x)

[Out]

-I*exp(3*I*x)/48 + I*exp(I*x)/4 + 3*I*exp(-I*x)/8 - I*exp(-3*I*x)/12 + I*exp(-5*I*x)/80

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Giac [B]  time = 1.31075, size = 93, normalized size = 2.51 \begin{align*} -\frac{9 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 24 i \, \tan \left (\frac{1}{2} \, x\right ) - 11}{24 \,{\left (\tan \left (\frac{1}{2} \, x\right ) + i\right )}^{3}} + \frac{45 \, \tan \left (\frac{1}{2} \, x\right )^{4} - 240 i \, \tan \left (\frac{1}{2} \, x\right )^{3} - 490 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 320 i \, \tan \left (\frac{1}{2} \, x\right ) + 73}{120 \,{\left (\tan \left (\frac{1}{2} \, x\right ) - i\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(I+cot(x)),x, algorithm="giac")

[Out]

-1/24*(9*tan(1/2*x)^2 + 24*I*tan(1/2*x) - 11)/(tan(1/2*x) + I)^3 + 1/120*(45*tan(1/2*x)^4 - 240*I*tan(1/2*x)^3
- 490*tan(1/2*x)^2 + 320*I*tan(1/2*x) + 73)/(tan(1/2*x) - I)^5