3.93 \(\int \frac{\text{csch}(x)}{1+\coth (x)} \, dx\)

Optimal. Leaf size=10 \[ -\frac{\text{csch}(x)}{\coth (x)+1} \]

[Out]

-(Csch[x]/(1 + Coth[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0221637, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3488} \[ -\frac{\text{csch}(x)}{\coth (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(1 + Coth[x]),x]

[Out]

-(Csch[x]/(1 + Coth[x]))

Rule 3488

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rubi steps

\begin{align*} \int \frac{\text{csch}(x)}{1+\coth (x)} \, dx &=-\frac{\text{csch}(x)}{1+\coth (x)}\\ \end{align*}

Mathematica [A]  time = 0.0031722, size = 7, normalized size = 0.7 \[ \sinh (x)-\cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]/(1 + Coth[x]),x]

[Out]

-Cosh[x] + Sinh[x]

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \begin{align*} -{\frac{{\rm csch} \left (x\right )}{1+{\rm coth} \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)/(1+coth(x)),x)

[Out]

-csch(x)/(1+coth(x))

________________________________________________________________________________________

Maxima [A]  time = 1.06341, size = 8, normalized size = 0.8 \begin{align*} -e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(1+coth(x)),x, algorithm="maxima")

[Out]

-e^(-x)

________________________________________________________________________________________

Fricas [A]  time = 2.51462, size = 32, normalized size = 3.2 \begin{align*} -\frac{1}{\cosh \left (x\right ) + \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(1+coth(x)),x, algorithm="fricas")

[Out]

-1/(cosh(x) + sinh(x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(1+coth(x)),x)

[Out]

Integral(csch(x)/(coth(x) + 1), x)

________________________________________________________________________________________

Giac [A]  time = 1.15261, size = 8, normalized size = 0.8 \begin{align*} -e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(1+coth(x)),x, algorithm="giac")

[Out]

-e^(-x)