3.109 \(\int \frac{\coth ^3(x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=12 \[ -\text{csch}(x)-i \log (\sinh (x)) \]

[Out]

-Csch[x] - I*Log[Sinh[x]]

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Rubi [A]  time = 0.0386526, antiderivative size = 12, 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 = {3879, 43} \[ -\text{csch}(x)-i \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(I + Csch[x]),x]

[Out]

-Csch[x] - I*Log[Sinh[x]]

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

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{\coth ^3(x)}{i+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{i-i x}{x^2} \, dx,x,i \sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{i}{x^2}-\frac{i}{x}\right ) \, dx,x,i \sinh (x)\right )\\ &=-\text{csch}(x)-i \log (\sinh (x))\\ \end{align*}

Mathematica [A]  time = 0.010364, size = 12, normalized size = 1. \[ -\text{csch}(x)-i \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(I + Csch[x]),x]

[Out]

-Csch[x] - I*Log[Sinh[x]]

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Maple [A]  time = 0.017, size = 12, normalized size = 1. \begin{align*} -{\rm csch} \left (x\right )+i\ln \left ({\rm csch} \left (x\right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(I+csch(x)),x)

[Out]

-csch(x)+I*ln(csch(x))

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Maxima [B]  time = 1.03452, size = 49, normalized size = 4.08 \begin{align*} -i \, x + \frac{2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(I+csch(x)),x, algorithm="maxima")

[Out]

-I*x + 2*e^(-x)/(e^(-2*x) - 1) - I*log(e^(-x) + 1) - I*log(e^(-x) - 1)

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Fricas [B]  time = 1.77861, size = 107, normalized size = 8.92 \begin{align*} \frac{i \, x e^{\left (2 \, x\right )} +{\left (-i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - i \, x - 2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(I+csch(x)),x, algorithm="fricas")

[Out]

(I*x*e^(2*x) + (-I*e^(2*x) + I)*log(e^(2*x) - 1) - I*x - 2*e^x)/(e^(2*x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{3}{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**3/(I+csch(x)),x)

[Out]

Integral(coth(x)**3/(csch(x) + I), x)

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Giac [B]  time = 1.17523, size = 51, normalized size = 4.25 \begin{align*} \frac{i \, e^{\left (-x\right )} - i \, e^{x} + 2}{e^{\left (-x\right )} - e^{x}} - i \, \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(I+csch(x)),x, algorithm="giac")

[Out]

(I*e^(-x) - I*e^x + 2)/(e^(-x) - e^x) - I*log(abs(-e^(-x) + e^x))