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

Optimal. Leaf size=38 \[ -\frac{\tanh ^{-1}\left (\frac{\sinh (x) (a \coth (x)+b)}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

[Out]

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

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Rubi [A]  time = 0.035892, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3509, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sinh (x) (a \coth (x)+b)}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]/(a + b*Coth[x]),x]

[Out]

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

Rule 3509

Int[sec[(e_.) + (f_.)*(x_)]/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Dist[f^(-1), Subst[Int[1/(a^
2 + b^2 - x^2), x], x, (b - a*Tan[e + f*x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, 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 \coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,i (-i b-i a \coth (x)) \sinh (x)\right )\\ &=-\frac{i \tan ^{-1}\left (\frac{(-i b-i a \coth (x)) \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}\\ \end{align*}

Mathematica [A]  time = 0.0311623, size = 46, normalized size = 1.21 \[ \frac{2 \tan ^{-1}\left (\frac{a+b \tanh \left (\frac{x}{2}\right )}{\sqrt{b-a} \sqrt{a+b}}\right )}{\sqrt{b-a} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]/(a + b*Coth[x]),x]

[Out]

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

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Maple [A]  time = 0.017, size = 39, normalized size = 1. \begin{align*} 2\,{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b+2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*coth(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.65223, size = 420, normalized size = 11.05 \begin{align*} \left [\frac{\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 )}{\sqrt{a^{2} - b^{2}}}, \frac{2 \, \sqrt{-a^{2} + b^{2}} \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 )}{a^{2} - b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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))/sqrt(a^2 - b^2), 2*sqr
t(-a^2 + b^2)*arctan(sqrt(-a^2 + b^2)/((a + b)*cosh(x) + (a + b)*sinh(x)))/(a^2 - 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 \coth{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)/(a + b*coth(x)), x)

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Giac [A]  time = 1.17974, size = 47, normalized size = 1.24 \begin{align*} \frac{2 \, \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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