3.118 \(\int \frac{\tanh ^2(x)}{1+\tanh (x)} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0375599, 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} \[ -\frac{x}{2}-\frac{1}{2 (\tanh (x)+1)}+\log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^2/(1 + Tanh[x]),x]

[Out]

-x/2 + Log[Cosh[x]] - 1/(2*(1 + Tanh[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{\tanh ^2(x)}{1+\tanh (x)} \, dx &=-\frac{1}{2 (1+\tanh (x))}-\frac{1}{2} \int (1-2 \tanh (x)) \, dx\\ &=-\frac{x}{2}-\frac{1}{2 (1+\tanh (x))}+\int \tanh (x) \, dx\\ &=-\frac{x}{2}+\log (\cosh (x))-\frac{1}{2 (1+\tanh (x))}\\ \end{align*}

Mathematica [A]  time = 0.0276375, size = 23, normalized size = 1.21 \[ \frac{1}{4} (-2 x+\sinh (2 x)-\cosh (2 x)+4 \log (\cosh (x))) \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^2/(1 + Tanh[x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 24, normalized size = 1.3 \begin{align*} -{\frac{1}{2+2\,\tanh \left ( x \right ) }}-{\frac{3\,\ln \left ( 1+\tanh \left ( x \right ) \right ) }{4}}-{\frac{\ln \left ( \tanh \left ( x \right ) -1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^2/(1+tanh(x)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.84957, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{2} \, x - \frac{1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^2/(1+tanh(x)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.32245, 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 \, \cosh \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(tanh(x)^2/(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*cosh(x)/(cosh(x) - sinh(x))) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

________________________________________________________________________________________

Sympy [B]  time = 0.428343, size = 61, normalized size = 3.21 \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{1}{2 \tanh{\left (x \right )} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Giac [A]  time = 1.20933, size = 23, normalized size = 1.21 \begin{align*} -\frac{3}{2} \, x - \frac{1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^2/(1+tanh(x)),x, algorithm="giac")

[Out]

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