3.144 \(\int \frac{1}{a+b \coth (x)} \, dx\)

Optimal. Leaf size=39 \[ \frac{a x}{a^2-b^2}-\frac{b \log (a \sinh (x)+b \cosh (x))}{a^2-b^2} \]

[Out]

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

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Rubi [A]  time = 0.0458942, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3484, 3530} \[ \frac{a x}{a^2-b^2}-\frac{b \log (a \sinh (x)+b \cosh (x))}{a^2-b^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[x])^(-1),x]

[Out]

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

Rule 3484

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

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0526497, size = 29, normalized size = 0.74 \[ \frac{a x-b \log (a \sinh (x)+b \cosh (x))}{a^2-b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Coth[x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.013, size = 55, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{2\,a-2\,b}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{2\,b+2\,a}}-{\frac{b\ln \left ( a+b{\rm coth} \left (x\right ) \right ) }{ \left ( a+b \right ) \left ( a-b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*coth(x)),x)

[Out]

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

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Maxima [A]  time = 1.26918, size = 50, normalized size = 1.28 \begin{align*} -\frac{b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{2} - b^{2}} + \frac{x}{a + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.53858, size = 108, normalized size = 2.77 \begin{align*} \frac{{\left (a + b\right )} x - b \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.22743, size = 148, normalized size = 3.79 \begin{align*} \begin{cases} \tilde{\infty } \left (x - \log{\left (\tanh{\left (x \right )} + 1 \right )}\right ) & \text{for}\: a = 0 \wedge b = 0 \\- \frac{x \tanh{\left (x \right )}}{2 b \tanh{\left (x \right )} - 2 b} + \frac{x}{2 b \tanh{\left (x \right )} - 2 b} - \frac{1}{2 b \tanh{\left (x \right )} - 2 b} & \text{for}\: a = - b \\\frac{x \tanh{\left (x \right )}}{2 b \tanh{\left (x \right )} + 2 b} + \frac{x}{2 b \tanh{\left (x \right )} + 2 b} + \frac{1}{2 b \tanh{\left (x \right )} + 2 b} & \text{for}\: a = b \\\frac{x - \log{\left (\tanh{\left (x \right )} + 1 \right )}}{b} & \text{for}\: a = 0 \\\frac{a x}{a^{2} - b^{2}} - \frac{b x}{a^{2} - b^{2}} + \frac{b \log{\left (\tanh{\left (x \right )} + 1 \right )}}{a^{2} - b^{2}} - \frac{b \log{\left (\tanh{\left (x \right )} + \frac{b}{a} \right )}}{a^{2} - b^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(x)),x)

[Out]

Piecewise((zoo*(x - log(tanh(x) + 1)), Eq(a, 0) & Eq(b, 0)), (-x*tanh(x)/(2*b*tanh(x) - 2*b) + x/(2*b*tanh(x)
- 2*b) - 1/(2*b*tanh(x) - 2*b), Eq(a, -b)), (x*tanh(x)/(2*b*tanh(x) + 2*b) + x/(2*b*tanh(x) + 2*b) + 1/(2*b*ta
nh(x) + 2*b), Eq(a, b)), ((x - log(tanh(x) + 1))/b, Eq(a, 0)), (a*x/(a**2 - b**2) - b*x/(a**2 - b**2) + b*log(
tanh(x) + 1)/(a**2 - b**2) - b*log(tanh(x) + b/a)/(a**2 - b**2), True))

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Giac [A]  time = 1.12986, size = 58, normalized size = 1.49 \begin{align*} -\frac{b \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{2} - b^{2}} + \frac{x}{a - b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(x)),x, algorithm="giac")

[Out]

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