3.423 \(\int \frac{\coth (c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=34 \[ \frac{\log (\sinh (c+d x))}{a d}-\frac{\log (a+b \sinh (c+d x))}{a d} \]

[Out]

Log[Sinh[c + d*x]]/(a*d) - Log[a + b*Sinh[c + d*x]]/(a*d)

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Rubi [A]  time = 0.0473075, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2721, 36, 29, 31} \[ \frac{\log (\sinh (c+d x))}{a d}-\frac{\log (a+b \sinh (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]/(a + b*Sinh[c + d*x]),x]

[Out]

Log[Sinh[c + d*x]]/(a*d) - Log[a + b*Sinh[c + d*x]]/(a*d)

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

Mathematica [A]  time = 0.0213791, size = 28, normalized size = 0.82 \[ \frac{\log (\sinh (c+d x))-\log (a+b \sinh (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]/(a + b*Sinh[c + d*x]),x]

[Out]

(Log[Sinh[c + d*x]] - Log[a + b*Sinh[c + d*x]])/(a*d)

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Maple [A]  time = 0.001, size = 35, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sinh \left ( dx+c \right ) \right ) }{da}}-{\frac{\ln \left ( a+b\sinh \left ( dx+c \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.08305, size = 101, normalized size = 2.97 \begin{align*} -\frac{\log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.39192, size = 170, normalized size = 5. \begin{align*} -\frac{\log \left (\frac{2 \,{\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - \log \left (\frac{2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(c + d*x)/(a + b*sinh(c + d*x)), x)

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Giac [A]  time = 1.32442, size = 85, normalized size = 2.5 \begin{align*} \frac{\frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac{\log \left ({\left | b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b \right |}\right )}{a} + \frac{\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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