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))

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = 0.02, 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])

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 8, normalized size = 0.57 \[ \frac {2 i}{e^{x} - i} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 8, normalized size = 0.57 \[ \frac {2 i}{e^{x} - i} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.12, size = 12, normalized size = 0.86 \[ \frac {2}{\tanh \left (\frac {x}{2}\right )-i} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 12, normalized size = 0.86 \[ -\frac {2}{i \, e^{\left (-x\right )} - 1} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 10, normalized size = 0.71 \[ \frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)*(1/sinh(x) + 1i)),x)

[Out]

2i/(exp(x) - 1i)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________