3.9 \(\int \frac{\csc ^5(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=28 \[ -\frac{1}{3} \csc ^3(x)+\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x) \]

[Out]

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

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Rubi [A]  time = 0.0404402, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3501, 3768, 3770} \[ -\frac{1}{3} \csc ^3(x)+\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^5/(I + Cot[x]),x]

[Out]

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

Rule 3501

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d^2*
(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(d^2*(m - 2))/(a*(m + n -
1)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2
+ b^2, 0] && LtQ[n, 0] && GtQ[m, 1] &&  !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\csc ^5(x)}{i+\cot (x)} \, dx &=-\frac{1}{3} \csc ^3(x)-i \int \csc ^3(x) \, dx\\ &=\frac{1}{2} i \cot (x) \csc (x)-\frac{\csc ^3(x)}{3}-\frac{1}{2} i \int \csc (x) \, dx\\ &=\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x)-\frac{\csc ^3(x)}{3}\\ \end{align*}

Mathematica [B]  time = 0.0939927, size = 67, normalized size = 2.39 \[ \frac{1}{24} i \csc ^3(x) \left (6 \sin (2 x)+3 \sin (3 x) \log \left (\sin \left (\frac{x}{2}\right )\right )+9 \sin (x) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )-3 \sin (3 x) \log \left (\cos \left (\frac{x}{2}\right )\right )+8 i\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^5/(I + Cot[x]),x]

[Out]

(I/24)*Csc[x]^3*(8*I + 9*(Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x] + 6*Sin[2*x] - 3*Log[Cos[x/2]]*Sin[3*x] + 3*Lo
g[Sin[x/2]]*Sin[3*x])

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Maple [B]  time = 0.052, size = 58, normalized size = 2.1 \begin{align*} -{\frac{1}{8}\tan \left ({\frac{x}{2}} \right ) }-{\frac{1}{24} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{i}{8}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+{{\frac{i}{8}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{8} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{24} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{i}{2}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^5/(I+cot(x)),x)

[Out]

-1/8*tan(1/2*x)-1/24*tan(1/2*x)^3-1/8*I*tan(1/2*x)^2+1/8*I/tan(1/2*x)^2-1/8/tan(1/2*x)-1/24/tan(1/2*x)^3-1/2*I
*ln(tan(1/2*x))

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Maxima [B]  time = 1.22923, size = 112, normalized size = 4. \begin{align*} -\frac{{\left (-\frac{6 i \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{6 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 2\right )}{\left (\cos \left (x\right ) + 1\right )}^{3}}{48 \, \sin \left (x\right )^{3}} - \frac{\sin \left (x\right )}{8 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{i \, \sin \left (x\right )^{2}}{8 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{\sin \left (x\right )^{3}}{24 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{1}{2} i \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^5/(I+cot(x)),x, algorithm="maxima")

[Out]

-1/48*(-6*I*sin(x)/(cos(x) + 1) + 6*sin(x)^2/(cos(x) + 1)^2 + 2)*(cos(x) + 1)^3/sin(x)^3 - 1/8*sin(x)/(cos(x)
+ 1) - 1/8*I*sin(x)^2/(cos(x) + 1)^2 - 1/24*sin(x)^3/(cos(x) + 1)^3 - 1/2*I*log(sin(x)/(cos(x) + 1))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (24 \,{\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 )} e^{\left (2 i \, x\right )}{\rm integral}\left (\frac{{\left (5 \, e^{\left (9 i \, x\right )} + 108 \, e^{\left (7 i \, x\right )} + 30 \, e^{\left (5 i \, x\right )} - 20 \, e^{\left (3 i \, x\right )} + 5 \, e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{8 \,{\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 )}}, x\right ) - 5 i \, e^{\left (7 i \, x\right )} + 17 i \, e^{\left (5 i \, x\right )} - 75 i \, e^{\left (3 i \, x\right )} + 15 i \, e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{24 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^5/(I+cot(x)),x, algorithm="fricas")

[Out]

1/24*(24*(e^(8*I*x) - 4*e^(6*I*x) + 6*e^(4*I*x) - 4*e^(2*I*x) + 1)*e^(2*I*x)*integral(1/8*(5*e^(9*I*x) + 108*e
^(7*I*x) + 30*e^(5*I*x) - 20*e^(3*I*x) + 5*e^(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), x) - 5*I*e^(7*I*x) + 17*I*e^(5*I*x) - 75*I*e^(3*I*x) + 15*I*e^(I*x))*e^(-2*I*x)/(
e^(8*I*x) - 4*e^(6*I*x) + 6*e^(4*I*x) - 4*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)**5/(I+cot(x)),x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.18272, size = 85, normalized size = 3.04 \begin{align*} -\frac{1}{24} \, \tan \left (\frac{1}{2} \, x\right )^{3} - \frac{1}{8} i \, \tan \left (\frac{1}{2} \, x\right )^{2} - \frac{-22 i \, \tan \left (\frac{1}{2} \, x\right )^{3} + 3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 3 i \, \tan \left (\frac{1}{2} \, x\right ) + 1}{24 \, \tan \left (\frac{1}{2} \, x\right )^{3}} - \frac{1}{2} i \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) - \frac{1}{8} \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^5/(I+cot(x)),x, algorithm="giac")

[Out]

-1/24*tan(1/2*x)^3 - 1/8*I*tan(1/2*x)^2 - 1/24*(-22*I*tan(1/2*x)^3 + 3*tan(1/2*x)^2 - 3*I*tan(1/2*x) + 1)/tan(
1/2*x)^3 - 1/2*I*log(abs(tan(1/2*x))) - 1/8*tan(1/2*x)