3.84 \(\int \frac{\text{csch}(x)}{a+b \tanh (x)} \, dx\)

Optimal. Leaf size=52 \[ -\frac{b \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cosh (x))}{a} \]

[Out]

-((b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])) - ArcTanh[Cosh[x]]/a

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Rubi [A]  time = 0.142876, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {3518, 3110, 3770, 3074, 206} \[ -\frac{b \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cosh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(a + b*Tanh[x]),x]

[Out]

-((b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])) - ArcTanh[Cosh[x]]/a

Rule 3518

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[(Sin[e + f*x]
^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
ILtQ[n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))

Rule 3110

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(cos[c + d*x]^m*sin[c + d*x]^n)/(a*cos[c + d*x] + b*sin[c + d
*x]), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegersQ[m, n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3074

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\text{csch}(x)}{a+b \tanh (x)} \, dx &=\int \frac{\coth (x)}{a \cosh (x)+b \sinh (x)} \, dx\\ &=i \int \left (-\frac{i \text{csch}(x)}{a}+\frac{i b}{a (a \cosh (x)+b \sinh (x))}\right ) \, dx\\ &=\frac{\int \text{csch}(x) \, dx}{a}-\frac{b \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(\cosh (x))}{a}-\frac{(i b) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{a}\\ &=-\frac{b \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cosh (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0903689, size = 59, normalized size = 1.13 \[ \frac{\log \left (\tanh \left (\frac{x}{2}\right )\right )-\frac{2 b \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]/(a + b*Tanh[x]),x]

[Out]

((-2*b*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/(Sqrt[a - b]*Sqrt[a + b]) + Log[Tanh[x/2]])/a

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Maple [A]  time = 0.033, size = 53, normalized size = 1. \begin{align*} -2\,{\frac{b}{a\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }+{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a*b/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))+1/a*ln(tanh(1/2*x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.45129, size = 675, normalized size = 12.98 \begin{align*} \left [-\frac{\sqrt{-a^{2} + b^{2}} b \log \left (\frac{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{-a^{2} + b^{2}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right ) +{\left (a^{2} - b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (a^{2} - b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{3} - a b^{2}}, \frac{2 \, \sqrt{a^{2} - b^{2}} b \arctan \left (\frac{\sqrt{a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )}\right ) -{\left (a^{2} - b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) +{\left (a^{2} - b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{3} - a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(sqrt(-a^2 + b^2)*b*log(((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(-a^2 + b
^2)*(cosh(x) + sinh(x)) - a + b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b))
+ (a^2 - b^2)*log(cosh(x) + sinh(x) + 1) - (a^2 - b^2)*log(cosh(x) + sinh(x) - 1))/(a^3 - a*b^2), (2*sqrt(a^2
- b^2)*b*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))) - (a^2 - b^2)*log(cosh(x) + sinh(x) + 1)
+ (a^2 - b^2)*log(cosh(x) + sinh(x) - 1))/(a^3 - a*b^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)/(a + b*tanh(x)), x)

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Giac [A]  time = 1.21293, size = 81, normalized size = 1.56 \begin{align*} -\frac{2 \, b \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{a^{2} - b^{2}}}\right )}{\sqrt{a^{2} - b^{2}} a} - \frac{\log \left (e^{x} + 1\right )}{a} + \frac{\log \left ({\left | e^{x} - 1 \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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