3.596 \(\int x \tanh ^2(x) \, dx\)

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

[Out]

x^2/2 + Log[Cosh[x]] - x*Tanh[x]

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Rubi [A]  time = 0.0194905, 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} \[ \frac{x^2}{2}-x \tanh (x)+\log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/2 + Log[Cosh[x]] - x*Tanh[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 \tanh ^2(x) \, dx &=-x \tanh (x)+\int x \, dx+\int \tanh (x) \, dx\\ &=\frac{x^2}{2}+\log (\cosh (x))-x \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.020748, size = 16, normalized size = 1. \[ \frac{x^2}{2}-x \tanh (x)+\log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/2 + Log[Cosh[x]] - x*Tanh[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*tanh(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 = 1.56092, size = 66, normalized size = 4.12 \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^{\left (2 \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*tanh(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^(2*x) + 1)

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Fricas [B]  time = 2.18759, 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 \, \cosh \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*tanh(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*cosh(x)/(cosh(x) - sinh(x))))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2
+ 1)

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Sympy [A]  time = 0.171118, size = 17, normalized size = 1.06 \begin{align*} \frac{x^{2}}{2} - x \tanh{\left (x \right )} + x - \log{\left (\tanh{\left (x \right )} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*tanh(x)**2,x)

[Out]

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

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Giac [B]  time = 1.16759, size = 69, normalized size = 4.31 \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*tanh(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)