3.7 \(\int \frac{\csc ^3(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=12 \[ -\csc (x)+i \tanh ^{-1}(\cos (x)) \]

[Out]

I*ArcTanh[Cos[x]] - Csc[x]

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Rubi [A]  time = 0.0345983, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3501, 3770} \[ -\csc (x)+i \tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(I + Cot[x]),x]

[Out]

I*ArcTanh[Cos[x]] - Csc[x]

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

Mathematica [B]  time = 0.0443024, size = 26, normalized size = 2.17 \[ -\csc (x)+i \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(I + Cot[x]),x]

[Out]

-Csc[x] + I*(Log[Cos[x/2]] - Log[Sin[x/2]])

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Maple [B]  time = 0.044, size = 24, normalized size = 2. \begin{align*} -{\frac{1}{2}\tan \left ({\frac{x}{2}} \right ) }-{\frac{1}{2} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-i\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)^3/(I+cot(x)),x)

[Out]

-1/2*tan(1/2*x)-1/2/tan(1/2*x)-I*ln(tan(1/2*x))

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Maxima [B]  time = 1.17078, size = 45, normalized size = 3.75 \begin{align*} -\frac{\cos \left (x\right ) + 1}{2 \, \sin \left (x\right )} - \frac{\sin \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} - 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)^3/(I+cot(x)),x, algorithm="maxima")

[Out]

-1/2*(cos(x) + 1)/sin(x) - 1/2*sin(x)/(cos(x) + 1) - I*log(sin(x)/(cos(x) + 1))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (2 \,{\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\right )} e^{\left (2 i \, x\right )}{\rm integral}\left (-\frac{{\left (5 \, e^{\left (5 i \, x\right )} + 6 \, e^{\left (3 i \, x\right )} - 3 \, e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{2 \,{\left (e^{\left (6 i \, x\right )} - 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} - 1\right )}}, x\right ) - i \, e^{\left (3 i \, x\right )} + 3 i \, e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{2 \,{\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(I+cot(x)),x, algorithm="fricas")

[Out]

1/2*(2*(e^(4*I*x) - 2*e^(2*I*x) + 1)*e^(2*I*x)*integral(-1/2*(5*e^(5*I*x) + 6*e^(3*I*x) - 3*e^(I*x))*e^(-2*I*x
)/(e^(6*I*x) - 3*e^(4*I*x) + 3*e^(2*I*x) - 1), x) - I*e^(3*I*x) + 3*I*e^(I*x))*e^(-2*I*x)/(e^(4*I*x) - 2*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)**3/(I+cot(x)),x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.2884, size = 42, normalized size = 3.5 \begin{align*} -\frac{-2 i \, \tan \left (\frac{1}{2} \, x\right ) + 1}{2 \, \tan \left (\frac{1}{2} \, x\right )} - i \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) - \frac{1}{2} \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(I+cot(x)),x, algorithm="giac")

[Out]

-1/2*(-2*I*tan(1/2*x) + 1)/tan(1/2*x) - I*log(abs(tan(1/2*x))) - 1/2*tan(1/2*x)