3.128 \(\int \frac{\coth (x)}{1+\coth (x)} \, dx\)

Optimal. Leaf size=16 \[ \frac{x}{2}+\frac{1}{2 (\coth (x)+1)} \]

[Out]

x/2 + 1/(2*(1 + Coth[x]))

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Rubi [A]  time = 0.0215589, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3526, 8} \[ \frac{x}{2}+\frac{1}{2 (\coth (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(1 + Coth[x]),x]

[Out]

x/2 + 1/(2*(1 + Coth[x]))

Rule 3526

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^m)/(2*a*f*m), x] + Dist[(b*c + a*d)/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1),
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\coth (x)}{1+\coth (x)} \, dx &=\frac{1}{2 (1+\coth (x))}+\frac{\int 1 \, dx}{2}\\ &=\frac{x}{2}+\frac{1}{2 (1+\coth (x))}\\ \end{align*}

Mathematica [A]  time = 0.0152798, size = 18, normalized size = 1.12 \[ \frac{1}{4} (2 x+\sinh (2 x)-\cosh (2 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(1 + Coth[x]),x]

[Out]

(2*x - Cosh[2*x] + Sinh[2*x])/4

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Maple [A]  time = 0.018, size = 24, normalized size = 1.5 \begin{align*}{\frac{1}{2+2\,{\rm coth} \left (x\right )}}+{\frac{\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{4}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(1+coth(x))+1/4*ln(1+coth(x))-1/4*ln(coth(x)-1)

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Maxima [A]  time = 1.13635, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x - \frac{1}{4} \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/4*e^(-2*x)

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Fricas [B]  time = 2.80484, size = 88, normalized size = 5.5 \begin{align*} \frac{{\left (2 \, x - 1\right )} \cosh \left (x\right ) +{\left (2 \, x + 1\right )} \sinh \left (x\right )}{4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((2*x - 1)*cosh(x) + (2*x + 1)*sinh(x))/(cosh(x) + sinh(x))

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Sympy [B]  time = 0.59691, size = 27, normalized size = 1.69 \begin{align*} \frac{x \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{x}{2 \tanh{\left (x \right )} + 2} - \frac{1}{2 \tanh{\left (x \right )} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11193, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x - \frac{1}{4} \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/4*e^(-2*x)