3.107 \(\int \frac{\coth (x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=13 \[ -i \log (-\sinh (x)+i) \]

[Out]

(-I)*Log[I - Sinh[x]]

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Rubi [A]  time = 0.0227739, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3879, 31} \[ -i \log (-\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(I + Csch[x]),x]

[Out]

(-I)*Log[I - 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\coth (x)}{i+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{i+i x} \, dx,x,i \sinh (x)\right )\\ &=-i \log (i-\sinh (x))\\ \end{align*}

Mathematica [A]  time = 0.0072043, size = 13, normalized size = 1. \[ -i \log (-\sinh (x)+i) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(I + Csch[x]),x]

[Out]

(-I)*Log[I - Sinh[x]]

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Maple [A]  time = 0.023, size = 17, normalized size = 1.3 \begin{align*} i\ln \left ({\rm csch} \left (x\right ) \right ) -i\ln \left ( i+{\rm csch} \left (x\right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0097, size = 20, normalized size = 1.54 \begin{align*} -i \, x - 2 i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*x - 2*I*log(I*e^(-x) - 1)

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Fricas [A]  time = 1.68254, size = 32, normalized size = 2.46 \begin{align*} i \, x - 2 i \, \log \left (e^{x} - i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x - 2*I*log(e^x - I)

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

[Out]

Integral(coth(x)/(csch(x) + I), x)

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Giac [A]  time = 1.14821, size = 18, normalized size = 1.38 \begin{align*} i \, x - 2 i \, \log \left (i \, e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x - 2*I*log(I*e^x + 1)