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

Optimal. Leaf size=19 \[ -\frac{x}{2}+\frac{1}{2 (\tanh (x)+1)}+\log (\sinh (x)) \]

[Out]

-x/2 + Log[Sinh[x]] + 1/(2*(1 + Tanh[x]))

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Rubi [A]  time = 0.0404755, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3551, 3479, 8, 3475} \[ -\frac{x}{2}+\frac{1}{2 (\tanh (x)+1)}+\log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(1 + Tanh[x]),x]

[Out]

-x/2 + Log[Sinh[x]] + 1/(2*(1 + Tanh[x]))

Rule 3551

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

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 8

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

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0281768, size = 23, normalized size = 1.21 \[ \frac{1}{4} (-2 x-\sinh (2 x)+\cosh (2 x)+4 \log (\sinh (x))) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(1 + Tanh[x]),x]

[Out]

(-2*x + Cosh[2*x] + 4*Log[Sinh[x]] - Sinh[2*x])/4

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Maple [B]  time = 0.027, size = 43, normalized size = 2.3 \begin{align*} - \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}-{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) -{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(tanh(1/2*x)+1)+1/(tanh(1/2*x)+1)^2-3/2*ln(tanh(1/2*x)+1)+ln(tanh(1/2*x))-1/2*ln(tanh(1/2*x)-1)

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Maxima [A]  time = 1.27018, size = 32, normalized size = 1.68 \begin{align*} \frac{1}{2} \, x + \frac{1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 1/4*e^(-2*x) + log(e^(-x) + 1) + log(e^(-x) - 1)

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Fricas [B]  time = 2.37501, size = 259, normalized size = 13.63 \begin{align*} -\frac{6 \, x \cosh \left (x\right )^{2} + 12 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 6 \, x \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 1}{4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(6*x*cosh(x)^2 + 12*x*cosh(x)*sinh(x) + 6*x*sinh(x)^2 - 4*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*log
(2*sinh(x)/(cosh(x) - sinh(x))) - 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)/(tanh(x) + 1), x)

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Giac [A]  time = 1.20533, size = 24, normalized size = 1.26 \begin{align*} -\frac{3}{2} \, x + \frac{1}{4} \, e^{\left (-2 \, x\right )} + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x + 1/4*e^(-2*x) + log(abs(e^(2*x) - 1))