\(\int \frac {\csc ^6(x)}{i+\cot (x)} \, dx\) [10]

Optimal result

Integrand size = 13, antiderivative size = 33 \[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=i \cot (x)-\frac {\cot ^2(x)}{2}+\frac {1}{3} i \cot ^3(x)-\frac {\cot ^4(x)}{4} \]

[Out]

I*cot(x)-1/2*cot(x)^2+1/3*I*cot(x)^3-1/4*cot(x)^4

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3568, 45} \[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=-\frac {1}{4} \cot ^4(x)+\frac {1}{3} i \cot ^3(x)-\frac {\cot ^2(x)}{2}+i \cot (x) \]

[In]

Int[Csc[x]^6/(I + Cot[x]),x]

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=-\frac {1}{4} \csc ^4(x)+\frac {1}{3} i \cot (x) \left (2+\csc ^2(x)\right ) \]

[In]

Integrate[Csc[x]^6/(I + Cot[x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64

method	result	size
risch	\(-\frac {4 \left (4 \,{\mathrm e}^{2 i x}-1\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{4}}\)	\(21\)
derivativedivides	\(i \cot \left (x \right )-\frac {\cot \left (x \right )^{2}}{2}+\frac {i \cot \left (x \right )^{3}}{3}-\frac {\cot \left (x \right )^{4}}{4}\)	\(26\)
default	\(i \cot \left (x \right )-\frac {\cot \left (x \right )^{2}}{2}+\frac {i \cot \left (x \right )^{3}}{3}-\frac {\cot \left (x \right )^{4}}{4}\)	\(26\)

[In]

int(csc(x)^6/(I+cot(x)),x,method=_RETURNVERBOSE)

[Out]

-4/3*(4*exp(2*I*x)-1)/(exp(2*I*x)-1)^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=-\frac {4 \, {\left (4 \, e^{\left (2 i \, x\right )} - 1\right )}}{3 \, {\left (e^{\left (8 i \, x\right )} - 4 \, e^{\left (6 i \, x\right )} + 6 \, e^{\left (4 i \, x\right )} - 4 \, e^{\left (2 i \, x\right )} + 1\right )}} \]

[In]

integrate(csc(x)^6/(I+cot(x)),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=\int \frac {\csc ^{6}{\left (x \right )}}{\cot {\left (x \right )} + i}\, dx \]

[In]

integrate(csc(x)**6/(I+cot(x)),x)

[Out]

Integral(csc(x)**6/(cot(x) + I), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=\frac {12 i \, \tan \left (x\right )^{3} - 6 \, \tan \left (x\right )^{2} + 4 i \, \tan \left (x\right ) - 3}{12 \, \tan \left (x\right )^{4}} \]

[In]

integrate(csc(x)^6/(I+cot(x)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=-\frac {-12 i \, \tan \left (x\right )^{3} + 6 \, \tan \left (x\right )^{2} - 4 i \, \tan \left (x\right ) + 3}{12 \, \tan \left (x\right )^{4}} \]

[In]

integrate(csc(x)^6/(I+cot(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.78 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {\csc ^6(x)}{i+\cot (x)} \, dx=-\frac {{\mathrm {cot}\left (x\right )}^4}{4}+\frac {{\mathrm {cot}\left (x\right )}^3\,1{}\mathrm {i}}{3}-\frac {{\mathrm {cot}\left (x\right )}^2}{2}+\mathrm {cot}\left (x\right )\,1{}\mathrm {i} \]

[In]

int(1/(sin(x)^6*(cot(x) + 1i)),x)

[Out]

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