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 = 0.04, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 3770

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

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 1.97, size = 16, normalized size = 1.45 \[ -i \, x - \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\right ) \]

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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giac [A]  time = 0.12, size = 17, normalized size = 1.55 \[ -i \, x - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]

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))

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maple [B]  time = 0.12, size = 27, normalized size = 2.45 \[ i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]

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

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maxima [B]  time = 0.31, size = 20, normalized size = 1.82 \[ -i \, x - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]

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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mupad [B]  time = 0.18, size = 21, normalized size = 1.91 \[ \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )-x\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/(1/sinh(x) + 1i),x)

[Out]

log(2 - 2*exp(x)) - log(- 2*exp(x) - 2) - x*1i

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]

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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