3.183 \(\int \frac{\coth (x)}{a+b \cosh (x)} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a \log (a+b \cosh (x))}{a^2-b^2}+\frac{\log (1-\cosh (x))}{2 (a+b)}+\frac{\log (\cosh (x)+1)}{2 (a-b)} \]

[Out]

Log[1 - Cosh[x]]/(2*(a + b)) + Log[1 + Cosh[x]]/(2*(a - b)) - (a*Log[a + b*Cosh[x]])/(a^2 - b^2)

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Rubi [A]  time = 0.0704049, antiderivative size = 54, 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 = {2721, 801} \[ -\frac{a \log (a+b \cosh (x))}{a^2-b^2}+\frac{\log (1-\cosh (x))}{2 (a+b)}+\frac{\log (\cosh (x)+1)}{2 (a-b)} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(a + b*Cosh[x]),x]

[Out]

Log[1 - Cosh[x]]/(2*(a + b)) + Log[1 + Cosh[x]]/(2*(a - b)) - (a*Log[a + b*Cosh[x]])/(a^2 - b^2)

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

\begin{align*} \int \frac{\coth (x)}{a+b \cosh (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{2 (a+b) (b-x)}+\frac{a}{(a-b) (a+b) (a+x)}-\frac{1}{2 (a-b) (b+x)}\right ) \, dx,x,b \cosh (x)\right )\\ &=\frac{\log (1-\cosh (x))}{2 (a+b)}+\frac{\log (1+\cosh (x))}{2 (a-b)}-\frac{a \log (a+b \cosh (x))}{a^2-b^2}\\ \end{align*}

Mathematica [A]  time = 0.0695403, size = 38, normalized size = 0.7 \[ -\frac{a \log (a+b \cosh (x))-a \log (\sinh (x))+b \log \left (\tanh \left (\frac{x}{2}\right )\right )}{a^2-b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(a + b*Cosh[x]),x]

[Out]

-((a*Log[a + b*Cosh[x]] - a*Log[Sinh[x]] + b*Log[Tanh[x/2]])/(a^2 - b^2))

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Maple [A]  time = 0.02, size = 53, normalized size = 1. \begin{align*} -{\frac{a}{ \left ( a+b \right ) \left ( a-b \right ) }\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }+{\frac{1}{a+b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(a+b*cosh(x)),x)

[Out]

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

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Maxima [A]  time = 1.03312, size = 80, normalized size = 1.48 \begin{align*} -\frac{a \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{2} - b^{2}} + \frac{\log \left (e^{\left (-x\right )} + 1\right )}{a - b} + \frac{\log \left (e^{\left (-x\right )} - 1\right )}{a + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*cosh(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.09963, size = 182, normalized size = 3.37 \begin{align*} -\frac{a \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) -{\left (a + b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (a - b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*cosh(x)),x, algorithm="fricas")

[Out]

-(a*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) - (a + b)*log(cosh(x) + sinh(x) + 1) - (a - b)*log(cosh(x) + si
nh(x) - 1))/(a^2 - b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*cosh(x)),x)

[Out]

Integral(coth(x)/(a + b*cosh(x)), x)

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Giac [A]  time = 1.2286, size = 90, normalized size = 1.67 \begin{align*} -\frac{a b \log \left ({\left | b{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{2} b - b^{3}} + \frac{\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{2 \,{\left (a - b\right )}} + \frac{\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{2 \,{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(a+b*cosh(x)),x, algorithm="giac")

[Out]

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