3.129 \(\int \frac{\coth ^2(x)}{1+\coth (x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0382739, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3540, 3475} \[ -\frac{x}{2}-\frac{1}{2 (\coth (x)+1)}+\log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/(1 + Coth[x]),x]

[Out]

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

Rule 3540

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> -Simp[
(b*(a*c + b*d)^2*(a + b*Tan[e + f*x])^m)/(2*a^3*f*m), x] + Dist[1/(2*a^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Si
mp[a*c^2 - 2*b*c*d + a*d^2 - 2*b*d^2*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]

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

Mathematica [A]  time = 0.0285523, 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]^2/(1 + Coth[x]),x]

[Out]

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

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Maple [A]  time = 0.017, size = 24, normalized size = 1.3 \begin{align*} -{\frac{1}{2+2\,{\rm coth} \left (x\right )}}-{\frac{3\,\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)^2/(1+coth(x)),x)

[Out]

-1/2/(1+coth(x))-3/4*ln(1+coth(x))-1/4*ln(coth(x)-1)

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Maxima [A]  time = 1.10925, 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)^2/(1+coth(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 = 3.09236, 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)^2/(1+coth(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 [B]  time = 0.905729, size = 92, normalized size = 4.84 \begin{align*} \frac{x \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{x}{2 \tanh{\left (x \right )} + 2} - \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} - \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{2 \log{\left (\tanh{\left (x \right )} \right )} \tanh{\left (x \right )}}{2 \tanh{\left (x \right )} + 2} + \frac{2 \log{\left (\tanh{\left (x \right )} \right )}}{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)**2/(1+coth(x)),x)

[Out]

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

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Giac [A]  time = 1.12442, 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)^2/(1+coth(x)),x, algorithm="giac")

[Out]

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