3.232 \(\int \frac{\coth (x)}{a+b \sinh (x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{\log (\sinh (x))}{a}-\frac{\log (a+b \sinh (x))}{a} \]

[Out]

Log[Sinh[x]]/a - Log[a + b*Sinh[x]]/a

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Rubi [A]  time = 0.0408944, antiderivative size = 20, 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 = {2721, 36, 29, 31} \[ \frac{\log (\sinh (x))}{a}-\frac{\log (a+b \sinh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(a + b*Sinh[x]),x]

[Out]

Log[Sinh[x]]/a - Log[a + b*Sinh[x]]/a

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

Mathematica [A]  time = 0.0095571, size = 20, normalized size = 1. \[ \frac{\log (\sinh (x))}{a}-\frac{\log (a+b \sinh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(a + b*Sinh[x]),x]

[Out]

Log[Sinh[x]]/a - Log[a + b*Sinh[x]]/a

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Maple [A]  time = 0.02, size = 21, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \sinh \left ( x \right ) \right ) }{a}}-{\frac{\ln \left ( a+b\sinh \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(sinh(x))/a-ln(a+b*sinh(x))/a

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Maxima [B]  time = 1.06044, size = 62, normalized size = 3.1 \begin{align*} -\frac{\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a} + \frac{\log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac{\log \left (e^{\left (-x\right )} - 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.0267, size = 116, normalized size = 5.8 \begin{align*} -\frac{\log \left (\frac{2 \,{\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac{2 \, \sinh \left (x\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*sinh(x)),x, algorithm="fricas")

[Out]

-(log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) - log(2*sinh(x)/(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 \sinh{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)/(a + b*sinh(x)), x)

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Giac [A]  time = 1.12772, size = 53, normalized size = 2.65 \begin{align*} -\frac{\log \left ({\left | -b{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a} + \frac{\log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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