3.597 \(\int x \coth ^2(x) \, dx\)

Optimal. Leaf size=16 \[ \frac{x^2}{2}-x \coth (x)+\log (\sinh (x)) \]

[Out]

x^2/2 - x*Coth[x] + Log[Sinh[x]]

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Rubi [A]  time = 0.0207453, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3720, 3475, 30} \[ \frac{x^2}{2}-x \coth (x)+\log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[x*Coth[x]^2,x]

[Out]

x^2/2 - x*Coth[x] + Log[Sinh[x]]

Rule 3720

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

Rule 3475

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0190905, size = 16, normalized size = 1. \[ \frac{x^2}{2}-x \coth (x)+\log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Coth[x]^2,x]

[Out]

x^2/2 - x*Coth[x] + Log[Sinh[x]]

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Maple [A]  time = 0.013, size = 28, normalized size = 1.8 \begin{align*}{\frac{{x}^{2}}{2}}-2\,x-2\,{\frac{x}{-1+{{\rm e}^{2\,x}}}}+\ln \left ( -1+{{\rm e}^{2\,x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*coth(x)^2,x)

[Out]

1/2*x^2-2*x-2*x/(-1+exp(2*x))+ln(-1+exp(2*x))

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Maxima [B]  time = 1.1065, size = 72, normalized size = 4.5 \begin{align*} -\frac{x e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} - 1} - \frac{x^{2} -{\left (x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )}}{2 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} + \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.24242, size = 305, normalized size = 19.06 \begin{align*} \frac{{\left (x^{2} - 4 \, x\right )} \cosh \left (x\right )^{2} + 2 \,{\left (x^{2} - 4 \, x\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (x^{2} - 4 \, x\right )} \sinh \left (x\right )^{2} - x^{2} + 2 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{2 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(x)^2,x, algorithm="fricas")

[Out]

1/2*((x^2 - 4*x)*cosh(x)^2 + 2*(x^2 - 4*x)*cosh(x)*sinh(x) + (x^2 - 4*x)*sinh(x)^2 - x^2 + 2*(cosh(x)^2 + 2*co
sh(x)*sinh(x) + sinh(x)^2 - 1)*log(2*sinh(x)/(cosh(x) - sinh(x))))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2
- 1)

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Sympy [A]  time = 0.907406, size = 22, normalized size = 1.38 \begin{align*} \frac{x^{2}}{2} + x - \frac{x}{\tanh{\left (x \right )}} - \log{\left (\tanh{\left (x \right )} + 1 \right )} + \log{\left (\tanh{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(x)**2,x)

[Out]

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

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Giac [B]  time = 1.146, size = 72, normalized size = 4.5 \begin{align*} \frac{x^{2} e^{\left (2 \, x\right )} - x^{2} - 4 \, x e^{\left (2 \, x\right )} + 2 \, e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 2 \, \log \left (e^{\left (2 \, x\right )} - 1\right )}{2 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(x)^2,x, algorithm="giac")

[Out]

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