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

Optimal. Leaf size=33 \[ -\frac{1}{4} \cot ^4(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}+i \cot (x) \]

[Out]

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

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Rubi [A]  time = 0.0391877, antiderivative size = 33, 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 = {3487, 43} \[ -\frac{1}{4} \cot ^4(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}+i \cot (x) \]

Antiderivative was successfully verified.

[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 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 43

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])

Rubi steps

\begin{align*} \int \frac{\csc ^6(x)}{i+\cot (x)} \, dx &=-\operatorname{Subst}\left (\int (i-x)^2 (i+x) \, dx,x,\cot (x)\right )\\ &=-\operatorname{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]  time = 0.0523081, size = 23, normalized size = 0.7 \[ -\frac{\csc ^4(x)}{4}+\frac{1}{3} i \cot (x) \left (\csc ^2(x)+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^6/(I + Cot[x]),x]

[Out]

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

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Maple [A]  time = 0.052, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{4\, \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{i}{\tan \left ( x \right ) }}+{\frac{{\frac{i}{3}}}{ \left ( \tan \left ( x \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.20775, size = 32, normalized size = 0.97 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.63184, size = 228, normalized size = 6.91 \begin{align*} -\frac{4 \,{\left (2 \,{\left (10 \, e^{\left (4 i \, x\right )} - 5 \, e^{\left (2 i \, x\right )} + 1\right )} e^{\left (2 i \, x\right )} - 15 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{15 \,{\left (e^{\left (10 i \, x\right )} - 5 \, e^{\left (8 i \, x\right )} + 10 \, e^{\left (6 i \, x\right )} - 10 \, e^{\left (4 i \, x\right )} + 5 \, e^{\left (2 i \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(2*(10*e^(4*I*x) - 5*e^(2*I*x) + 1)*e^(2*I*x) - 15*e^(4*I*x) + 3*e^(2*I*x))*e^(-2*I*x)/(e^(10*I*x) - 5*e
^(8*I*x) + 10*e^(6*I*x) - 10*e^(4*I*x) + 5*e^(2*I*x) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.16601, size = 32, normalized size = 0.97 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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