\(\int \frac {\csc ^4(x)}{i+\cot (x)} \, dx\) [8]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=i \cot (x)-\frac {\cot ^2(x)}{2} \]

[Out]

I*cot(x)-1/2*cot(x)^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568} \[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=-\frac {\cot ^2(x)}{2}+i \cot (x) \]

[In]

Int[Csc[x]^4/(I + Cot[x]),x]

[Out]

I*Cot[x] - Cot[x]^2/2

Rule 3568

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int (i-x) \, dx,x,\cot (x)) \\ & = i \cot (x)-\frac {\cot ^2(x)}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.45 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=i \cot (x)-\frac {\csc ^2(x)}{2} \]

[In]

Integrate[Csc[x]^4/(I + Cot[x]),x]

[Out]

I*Cot[x] - Csc[x]^2/2

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {2}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\)	\(12\)
derivativedivides	\(i \cot \left (x \right )-\frac {\cot \left (x \right )^{2}}{2}\)	\(13\)
default	\(i \cot \left (x \right )-\frac {\cot \left (x \right )^{2}}{2}\)	\(13\)

[In]

int(csc(x)^4/(I+cot(x)),x,method=_RETURNVERBOSE)

[Out]

2/(exp(2*I*x)-1)^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=\frac {2}{e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1} \]

[In]

integrate(csc(x)^4/(I+cot(x)),x, algorithm="fricas")

[Out]

2/(e^(4*I*x) - 2*e^(2*I*x) + 1)

Sympy [F]

\[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=\int \frac {\csc ^{4}{\left (x \right )}}{\cot {\left (x \right )} + i}\, dx \]

[In]

integrate(csc(x)**4/(I+cot(x)),x)

[Out]

Integral(csc(x)**4/(cot(x) + I), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=\frac {2 i \, \tan \left (x\right ) - 1}{2 \, \tan \left (x\right )^{2}} \]

[In]

integrate(csc(x)^4/(I+cot(x)),x, algorithm="maxima")

[Out]

1/2*(2*I*tan(x) - 1)/tan(x)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=-\frac {-2 i \, \tan \left (x\right ) + 1}{2 \, \tan \left (x\right )^{2}} \]

[In]

integrate(csc(x)^4/(I+cot(x)),x, algorithm="giac")

[Out]

-1/2*(-2*I*tan(x) + 1)/tan(x)^2

Mupad [B] (verification not implemented)

Time = 12.55 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx=-\frac {\mathrm {cot}\left (x\right )\,\left (\mathrm {cot}\left (x\right )-2{}\mathrm {i}\right )}{2} \]

[In]

int(1/(sin(x)^4*(cot(x) + 1i)),x)

[Out]

-(cot(x)*(cot(x) - 2i))/2