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

Optimal. Leaf size=17 \[ -\tanh ^{-1}(\cosh (x))+\frac{\coth (x)}{\text{csch}(x)+i} \]

[Out]

-ArcTanh[Cosh[x]] + Coth[x]/(I + Csch[x])

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Rubi [A]  time = 0.0539062, antiderivative size = 17, 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 = {3789, 3770, 3794} \[ -\tanh ^{-1}(\cosh (x))+\frac{\coth (x)}{\text{csch}(x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(I + Csch[x]),x]

[Out]

-ArcTanh[Cosh[x]] + Coth[x]/(I + Csch[x])

Rule 3789

Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[Csc[e + f*x],
x], x] - Dist[a/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3770

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

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [B]  time = 0.0325784, size = 37, normalized size = 2.18 \[ \log \left (\tanh \left (\frac{x}{2}\right )\right )-\frac{2 i \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^2/(I + Csch[x]),x]

[Out]

Log[Tanh[x/2]] - ((2*I)*Sinh[x/2])/(Cosh[x/2] + I*Sinh[x/2])

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Maple [A]  time = 0.017, size = 19, normalized size = 1.1 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) -{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02085, size = 39, normalized size = 2.29 \begin{align*} \frac{4}{2 \, e^{\left (-x\right )} + 2 i} - \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(csch(x)^2/(I+csch(x)),x, algorithm="maxima")

[Out]

4/(2*e^(-x) + 2*I) - log(e^(-x) + 1) + log(e^(-x) - 1)

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Fricas [B]  time = 1.58911, size = 89, normalized size = 5.24 \begin{align*} -\frac{{\left (e^{x} - i\right )} \log \left (e^{x} + 1\right ) -{\left (e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 2}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(csch(x)**2/(csch(x) + I), x)

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Giac [A]  time = 1.18024, size = 30, normalized size = 1.76 \begin{align*} \frac{2}{e^{x} - i} - \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(csch(x)^2/(I+csch(x)),x, algorithm="giac")

[Out]

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