3.118 \(\int \frac{\coth (x)}{a+b \text{csch}(x)} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log (a+b \text{csch}(x))}{a}+\frac{\log (\sinh (x))}{a} \]

[Out]

Log[a + b*Csch[x]]/a + Log[Sinh[x]]/a

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Rubi [A]  time = 0.0316641, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3885, 36, 29, 31} \[ \frac{\log (a+b \text{csch}(x))}{a}+\frac{\log (\sinh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(a + b*Csch[x]),x]

[Out]

Log[a + b*Csch[x]]/a + Log[Sinh[x]]/a

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 36

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0079351, size = 11, normalized size = 0.58 \[ \frac{\log (a \sinh (x)+b)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(a + b*Csch[x]),x]

[Out]

Log[b + a*Sinh[x]]/a

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Maple [A]  time = 0.021, size = 21, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ({\rm csch} \left (x\right ) \right ) }{a}}+{\frac{\ln \left ( a+b{\rm csch} \left (x\right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*ln(csch(x))+ln(a+b*csch(x))/a

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Maxima [A]  time = 1.01121, size = 38, normalized size = 2. \begin{align*} \frac{x}{a} + \frac{\log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54861, size = 72, normalized size = 3.79 \begin{align*} -\frac{x - \log \left (\frac{2 \,{\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)/(a + b*csch(x)), x)

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Giac [A]  time = 1.10651, size = 30, normalized size = 1.58 \begin{align*} \frac{\log \left ({\left | -a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(-a*(e^(-x) - e^x) + 2*b))/a