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

Optimal. Leaf size=50 \[ -\frac{b \log (\sinh (c+d x))}{a^2 d}+\frac{b \log (a+b \sinh (c+d x))}{a^2 d}-\frac{\text{csch}(c+d x)}{a d} \]

[Out]

-(Csch[c + d*x]/(a*d)) - (b*Log[Sinh[c + d*x]])/(a^2*d) + (b*Log[a + b*Sinh[c + d*x]])/(a^2*d)

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Rubi [A]  time = 0.0749684, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 44} \[ -\frac{b \log (\sinh (c+d x))}{a^2 d}+\frac{b \log (a+b \sinh (c+d x))}{a^2 d}-\frac{\text{csch}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csch[c + d*x]/(a*d)) - (b*Log[Sinh[c + d*x]])/(a^2*d) + (b*Log[a + b*Sinh[c + d*x]])/(a^2*d)

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0414655, size = 50, normalized size = 1. \[ -\frac{b \log (\sinh (c+d x))}{a^2 d}+\frac{b \log (a+b \sinh (c+d x))}{a^2 d}-\frac{\text{csch}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csch[c + d*x]/(a*d)) - (b*Log[Sinh[c + d*x]])/(a^2*d) + (b*Log[a + b*Sinh[c + d*x]])/(a^2*d)

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Maple [A]  time = 0.001, size = 35, normalized size = 0.7 \begin{align*} -{\frac{{\rm csch} \left (dx+c\right )}{da}}+{\frac{b\ln \left ( a{\rm csch} \left (dx+c\right )+b \right ) }{d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-csch(d*x+c)/a/d+1/d*b/a^2*ln(a*csch(d*x+c)+b)

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Maxima [B]  time = 1.17884, size = 149, normalized size = 2.98 \begin{align*} \frac{2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} + \frac{b \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{2} d} - \frac{b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac{b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.1464, size = 554, normalized size = 11.08 \begin{align*} -\frac{2 \, a \cosh \left (d x + c\right ) -{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \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 ) +{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \log \left (\frac{2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, a \sinh \left (d x + c\right )}{a^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\right )^{2} - a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.33678, size = 132, normalized size = 2.64 \begin{align*} -\frac{\frac{b \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} - \frac{b \log \left ({\left | b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b \right |}\right )}{a^{2}} + \frac{b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} + \frac{2 \, e^{\left (d x + c\right )}}{a{\left (e^{\left (d x + c\right )} + 1\right )}{\left (e^{\left (d x + c\right )} - 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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