3.126 \(\int \frac{\tanh (x)}{1+\coth (x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0417203, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3551, 3479, 8, 3475} \[ -\frac{x}{2}+\frac{1}{2 (\coth (x)+1)}+\log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/(1 + Coth[x]),x]

[Out]

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

Rule 3551

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

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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

Mathematica [A]  time = 0.0281422, 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]/(1 + Coth[x]),x]

[Out]

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

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Maple [B]  time = 0.029, size = 47, normalized size = 2.5 \begin{align*} - \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}-{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)/(1+coth(x)),x)

[Out]

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

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Maxima [A]  time = 1.67204, 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)/(1+coth(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.95861, 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)/(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*cosh(x)/(cosh(x) - sinh(x))) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)/(1+coth(x)),x)

[Out]

Integral(tanh(x)/(coth(x) + 1), x)

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Giac [A]  time = 1.15774, 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)/(1+coth(x)),x, algorithm="giac")

[Out]

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