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 = 0.0307238, 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 \[ \frac{x^2}{2}-x \tanh (x)+\log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]  Int[x*Tanh[x]^2,x]

[Out]

x^2/2 + Log[Cosh[x]] - x*Tanh[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - x \tanh{\left (x \right )} - \frac{\log{\left (- \tanh ^{2}{\left (x \right )} + 1 \right )}}{2} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*tanh(x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.00775351, 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 = 0.025, size = 28, normalized size = 1.8 \[{\frac{{x}^{2}}{2}}-2\,x+2\,{\frac{x}{1+{{\rm e}^{2\,x}}}}+\ln \left ( 1+{{\rm e}^{2\,x}} \right ) \]

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 [A]  time = 1.63203, size = 66, normalized size = 4.12 \[ -\frac{x e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} + 1} + \frac{x^{2} +{\left (x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )}}{2 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]

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 [A]  time = 0.214682, size = 126, normalized size = 7.88 \[ \frac{{\left (x^{2} - 4 \, x\right )} \cosh \left (x\right )^{2} + 2 \,{\left (x^{2} - 4 \, x\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (x^{2} - 4 \, x\right )} \sinh \left (x\right )^{2} + x^{2} + 2 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{2 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}} \]

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*cosh(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 = 0.207648, size = 17, normalized size = 1.06 \[ \frac{x^{2}}{2} - x \tanh{\left (x \right )} + x - \log{\left (\tanh{\left (x \right )} + 1 \right )} \]

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/XCAS [A]  time = 0.228031, size = 69, normalized size = 4.31 \[ \frac{x^{2} e^{\left (2 \, x\right )} + x^{2} - 4 \, x e^{\left (2 \, x\right )} + 2 \, e^{\left (2 \, x\right )}{\rm ln}\left (e^{\left (2 \, x\right )} + 1\right ) + 2 \,{\rm ln}\left (e^{\left (2 \, x\right )} + 1\right )}{2 \,{\left (e^{\left (2 \, x\right )} + 1\right )}} \]

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)*ln(e^(2*x) + 1) + 2*ln(e^(2*x)
+ 1))/(e^(2*x) + 1)