3.2.26 \(\int \frac {\tanh (x)}{1+\coth (x)} \, dx\) [126]

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

[Out]

-1/2*x+1/2/(1+coth(x))+ln(cosh(x))

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Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {3632, 3560, 8, 3556} \begin {gather*} -\frac {x}{2}+\frac {1}{2 (\coth (x)+1)}+\log (\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/(1 + Coth[x]),x]

[Out]

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

Rule 8

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

Rule 3556

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

Rule 3560

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 3632

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]

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*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 1.21 \begin {gather*} \frac {1}{4} (-2 x-\cosh (2 x)+4 \log (\cosh (x))+\sinh (2 x)) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(15)=30\).
time = 0.51, size = 47, normalized size = 2.47

method	result	size
risch	\(-\frac {3 x}{2}-\frac {{\mathrm e}^{-2 x}}{4}+\ln \left (1+{\mathrm e}^{2 x}\right )\)	\(18\)
default	\(-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)/(1+coth(x)),x,method=_RETURNVERBOSE)

[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 = 0.50, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (15) = 30\).
time = 0.36, size = 73, normalized size = 3.84 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}

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 = 0.42, size = 17, normalized size = 0.89 \begin {gather*} -\frac {3}{2} \, x - \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}

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)

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Mupad [B]
time = 1.17, size = 17, normalized size = 0.89 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}+1\right )-\frac {3\,x}{2}-\frac {{\mathrm {e}}^{-2\,x}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)/(coth(x) + 1),x)

[Out]

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

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