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

Optimal. Leaf size=14 \[ \frac{i \coth (x)}{\text{csch}(x)+i} \]

[Out]

(I*Coth[x])/(I + Csch[x])

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Rubi [A]  time = 0.0205694, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3794} \[ \frac{i \coth (x)}{\text{csch}(x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(I + Csch[x]),x]

[Out]

(I*Coth[x])/(I + Csch[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}(x)}{i+\text{csch}(x)} \, dx &=\frac{i \coth (x)}{i+\text{csch}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0187179, size = 27, normalized size = 1.93 \[ \frac{2 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]/(I + Csch[x]),x]

[Out]

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

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Maple [A]  time = 0.02, size = 12, normalized size = 0.9 \begin{align*} 2\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02864, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{i \, e^{\left (-x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(I*e^(-x) - 1)

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Fricas [A]  time = 1.35672, size = 20, normalized size = 1.43 \begin{align*} \frac{2 i}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I/(e^x - I)

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

[Out]

Integral(csch(x)/(csch(x) + I), x)

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Giac [A]  time = 1.15148, size = 11, normalized size = 0.79 \begin{align*} \frac{2 i}{e^{x} - i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I/(e^x - I)