3.129 \(\int \frac {\coth ^2(x)}{1+\coth (x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \[ -\frac {x}{2}-\frac {1}{2 (\coth (x)+1)}+\log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/(1 + Coth[x]),x]

[Out]

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

Rule 3475

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 23, normalized size = 1.21 \[ \frac {1}{4} (-2 x-\sinh (2 x)+\cosh (2 x)+4 \log (\sinh (x))) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^2/(1 + Coth[x]),x]

[Out]

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

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fricas [B]  time = 0.42, size = 73, normalized size = 3.84 \[ -\frac {6 \, x \cosh \relax (x)^{2} + 12 \, x \cosh \relax (x) \sinh \relax (x) + 6 \, x \sinh \relax (x)^{2} - 4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) - 1}{4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2/(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*sinh(x)/(cosh(x) - sinh(x))) - 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)

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giac [A]  time = 0.12, size = 18, normalized size = 0.95 \[ -\frac {3}{2} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 24, normalized size = 1.26 \[ -\frac {\ln \left (\coth \relax (x )-1\right )}{4}-\frac {1}{2 \left (1+\coth \relax (x )\right )}-\frac {3 \ln \left (1+\coth \relax (x )\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.30, size = 24, normalized size = 1.26 \[ \frac {1}{2} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 21, normalized size = 1.11 \[ \frac {x}{2}-\ln \left (\mathrm {coth}\relax (x)+1\right )-\frac {1}{2\,\left (\mathrm {coth}\relax (x)+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/(coth(x) + 1),x)

[Out]

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

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sympy [B]  time = 0.62, size = 92, normalized size = 4.84 \[ \frac {x \tanh {\relax (x )}}{2 \tanh {\relax (x )} + 2} + \frac {x}{2 \tanh {\relax (x )} + 2} - \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{2 \tanh {\relax (x )} + 2} - \frac {2 \log {\left (\tanh {\relax (x )} + 1 \right )}}{2 \tanh {\relax (x )} + 2} + \frac {2 \log {\left (\tanh {\relax (x )} \right )} \tanh {\relax (x )}}{2 \tanh {\relax (x )} + 2} + \frac {2 \log {\left (\tanh {\relax (x )} \right )}}{2 \tanh {\relax (x )} + 2} + \frac {1}{2 \tanh {\relax (x )} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**2/(1+coth(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) + 2*log(tanh(x))*tanh(x)/(2*tanh(x) + 2) + 2*log(tanh(x))/(2*tanh(x) + 2) + 1/(2*tanh(x) + 2
)

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