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

Optimal. Leaf size=37 \[ \frac{1}{5} i \cot ^5(x)-\frac{\cot ^4(x)}{4}+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2} \]

[Out]

-Cot[x]^2/2 + (I/3)*Cot[x]^3 - Cot[x]^4/4 + (I/5)*Cot[x]^5

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Rubi [A]  time = 0.0469125, antiderivative size = 37, 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 = {3516, 848, 75} \[ \frac{1}{5} i \cot ^5(x)-\frac{\cot ^4(x)}{4}+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^6/(I + Tan[x]),x]

[Out]

-Cot[x]^2/2 + (I/3)*Cot[x]^3 - Cot[x]^4/4 + (I/5)*Cot[x]^5

Rule 3516

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

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int \frac{\csc ^6(x)}{i+\tan (x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^6 (i+x)} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{(-i+x)^2 (i+x)}{x^6} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{i}{x^6}+\frac{1}{x^5}-\frac{i}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\tan (x)\right )\\ &=-\frac{1}{2} \cot ^2(x)+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^4(x)}{4}+\frac{1}{5} i \cot ^5(x)\\ \end{align*}

Mathematica [A]  time = 0.018521, size = 41, normalized size = 1.11 \[ -\frac{2}{15} i \cot (x)-\frac{\csc ^4(x)}{4}+\frac{1}{5} i \cot (x) \csc ^4(x)-\frac{1}{15} i \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^6/(I + Tan[x]),x]

[Out]

((-2*I)/15)*Cot[x] - (I/15)*Cot[x]*Csc[x]^2 - Csc[x]^4/4 + (I/5)*Cot[x]*Csc[x]^4

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Maple [A]  time = 0.041, size = 28, normalized size = 0.8 \begin{align*} -{\frac{1}{4\, \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{{\frac{i}{3}}}{ \left ( \tan \left ( x \right ) \right ) ^{3}}}+{\frac{{\frac{i}{5}}}{ \left ( \tan \left ( x \right ) \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.36274, size = 32, normalized size = 0.86 \begin{align*} \frac{i \,{\left (30 i \, \tan \left (x\right )^{3} + 20 \, \tan \left (x\right )^{2} + 15 i \, \tan \left (x\right ) + 12\right )}}{60 \, \tan \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*I*(30*I*tan(x)^3 + 20*tan(x)^2 + 15*I*tan(x) + 12)/tan(x)^5

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Fricas [B]  time = 1.98741, size = 176, normalized size = 4.76 \begin{align*} -\frac{4 \,{\left (30 \, e^{\left (6 i \, x\right )} - 10 \, e^{\left (4 i \, x\right )} + 5 \, e^{\left (2 i \, x\right )} - 1\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+tan(x)),x, algorithm="fricas")

[Out]

-4/15*(30*e^(6*I*x) - 10*e^(4*I*x) + 5*e^(2*I*x) - 1)/(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.38777, size = 32, normalized size = 0.86 \begin{align*} -\frac{30 \, \tan \left (x\right )^{3} - 20 i \, \tan \left (x\right )^{2} + 15 \, \tan \left (x\right ) - 12 i}{60 \, \tan \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*tan(x)^3 - 20*I*tan(x)^2 + 15*tan(x) - 12*I)/tan(x)^5