3.108 \(\int \frac{\coth ^2(x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=11 \[ -\tanh ^{-1}(\cosh (x))-i x \]

[Out]

(-I)*x - ArcTanh[Cosh[x]]

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Rubi [A]  time = 0.0383253, antiderivative size = 11, 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 = {3888, 3770} \[ -\tanh ^{-1}(\cosh (x))-i x \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/(I + Csch[x]),x]

[Out]

(-I)*x - ArcTanh[Cosh[x]]

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

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{\coth ^2(x)}{i+\text{csch}(x)} \, dx &=\int (-i+\text{csch}(x)) \, dx\\ &=-i x+\int \text{csch}(x) \, dx\\ &=-i x-\tanh ^{-1}(\cosh (x))\\ \end{align*}

Mathematica [A]  time = 0.0329229, size = 13, normalized size = 1.18 \[ \log \left (\tanh \left (\frac{x}{2}\right )\right )-i x \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^2/(I + Csch[x]),x]

[Out]

(-I)*x + Log[Tanh[x/2]]

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Maple [B]  time = 0.029, size = 27, normalized size = 2.5 \begin{align*} -i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) +i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/(I+csch(x)),x)

[Out]

-I*ln(tanh(1/2*x)+1)+ln(tanh(1/2*x))+I*ln(tanh(1/2*x)-1)

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Maxima [B]  time = 1.00286, size = 27, normalized size = 2.45 \begin{align*} -i \, x - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2/(I+csch(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.61851, size = 49, normalized size = 4.45 \begin{align*} -i \, x - \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2/(I+csch(x)),x, algorithm="fricas")

[Out]

-I*x - log(e^x + 1) + log(e^x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\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)**2/(I+csch(x)),x)

[Out]

Integral(coth(x)**2/(csch(x) + I), x)

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Giac [A]  time = 1.19109, size = 23, normalized size = 2.09 \begin{align*} -i \, x - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2/(I+csch(x)),x, algorithm="giac")

[Out]

-I*x - log(e^x + 1) + log(abs(e^x - 1))