3.3 \(\int \frac{\sin ^2(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=50 \[ -\frac{3 i x}{8}-\frac{i}{8 (-\cot (x)+i)}+\frac{i}{4 (\cot (x)+i)}-\frac{1}{8 (\cot (x)+i)^2} \]

[Out]

((-3*I)/8)*x - (I/8)/(I - Cot[x]) - 1/(8*(I + Cot[x])^2) + (I/4)/(I + Cot[x])

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Rubi [A]  time = 0.050595, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3487, 44, 203} \[ -\frac{3 i x}{8}-\frac{i}{8 (-\cot (x)+i)}+\frac{i}{4 (\cot (x)+i)}-\frac{1}{8 (\cot (x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(I + Cot[x]),x]

[Out]

((-3*I)/8)*x - (I/8)/(I - Cot[x]) - 1/(8*(I + Cot[x])^2) + (I/4)/(I + Cot[x])

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sin ^2(x)}{i+\cot (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{(i-x)^2 (i+x)^3} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{i}{8 (-i+x)^2}-\frac{1}{4 (i+x)^3}+\frac{i}{4 (i+x)^2}-\frac{3 i}{8 \left (1+x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=-\frac{i}{8 (i-\cot (x))}-\frac{1}{8 (i+\cot (x))^2}+\frac{i}{4 (i+\cot (x))}+\frac{3}{8} i \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{3 i x}{8}-\frac{i}{8 (i-\cot (x))}-\frac{1}{8 (i+\cot (x))^2}+\frac{i}{4 (i+\cot (x))}\\ \end{align*}

Mathematica [A]  time = 0.0517726, size = 36, normalized size = 0.72 \[ -\frac{1}{32} i (12 x-8 \sin (2 x)+\sin (4 x)-4 i \cos (2 x)+i \cos (4 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(I + Cot[x]),x]

[Out]

(-I/32)*(12*x - (4*I)*Cos[2*x] + I*Cos[4*x] - 8*Sin[2*x] + Sin[4*x])

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Maple [A]  time = 0.062, size = 47, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{2}}}{\tan \left ( x \right ) -i}}-{\frac{1}{8\, \left ( \tan \left ( x \right ) -i \right ) ^{2}}}-{\frac{3\,\ln \left ( \tan \left ( x \right ) -i \right ) }{16}}+{\frac{{\frac{i}{8}}}{i+\tan \left ( x \right ) }}+{\frac{3\,\ln \left ( i+\tan \left ( x \right ) \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(I+cot(x)),x)

[Out]

1/2*I/(tan(x)-I)-1/8/(tan(x)-I)^2-3/16*ln(tan(x)-I)+1/8*I/(I+tan(x))+3/16*ln(I+tan(x))

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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)^2/(I+cot(x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.65179, size = 139, normalized size = 2.78 \begin{align*} \frac{1}{32} \,{\left (-4 i \, x e^{\left (4 i \, x\right )} +{\left (-8 i \, x e^{\left (2 i \, x\right )} + 2 \, e^{\left (4 i \, x\right )} - 2\right )} e^{\left (2 i \, x\right )} - 4 \, e^{\left (2 i \, x\right )} + 1\right )} e^{\left (-4 i \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(I+cot(x)),x, algorithm="fricas")

[Out]

1/32*(-4*I*x*e^(4*I*x) + (-8*I*x*e^(2*I*x) + 2*e^(4*I*x) - 2)*e^(2*I*x) - 4*e^(2*I*x) + 1)*e^(-4*I*x)

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Sympy [A]  time = 0.754485, size = 34, normalized size = 0.68 \begin{align*} - \frac{3 i x}{8} + \frac{e^{2 i x}}{16} - \frac{3 e^{- 2 i x}}{16} + \frac{e^{- 4 i x}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/(I+cot(x)),x)

[Out]

-3*I*x/8 + exp(2*I*x)/16 - 3*exp(-2*I*x)/16 + exp(-4*I*x)/32

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Giac [A]  time = 1.26947, size = 69, normalized size = 1.38 \begin{align*} -\frac{-3 i \, \tan \left (x\right ) + 1}{16 \,{\left (-i \, \tan \left (x\right ) + 1\right )}} + \frac{9 \, \tan \left (x\right )^{2} - 2 i \, \tan \left (x\right ) + 3}{32 \,{\left (\tan \left (x\right ) - i\right )}^{2}} + \frac{3}{16} \, \log \left (\tan \left (x\right ) + i\right ) - \frac{3}{16} \, \log \left (\tan \left (x\right ) - i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(I+cot(x)),x, algorithm="giac")

[Out]

-1/16*(-3*I*tan(x) + 1)/(-I*tan(x) + 1) + 1/32*(9*tan(x)^2 - 2*I*tan(x) + 3)/(tan(x) - I)^2 + 3/16*log(tan(x)
+ I) - 3/16*log(tan(x) - I)