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)/(a^2-b^2)^(1/2))/(a^2-b^2)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.40, size = 147, normalized size = 3.87 \[ \left [\frac {\log \left (\frac {{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + a - b}{{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - a + b}\right )}{\sqrt {a^{2} - b^{2}}}, \frac {2 \, \sqrt {-a^{2} + b^{2}} \arctan \left (\frac {\sqrt {-a^{2} + b^{2}}}{{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)}\right )}{a^{2} - b^{2}}\right ] \]

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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giac [A]  time = 0.11, size = 35, normalized size = 0.92 \[ \frac {2 \, \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} \]

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)

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maple [A]  time = 0.07, size = 39, normalized size = 1.03 \[ \frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 0.17, size = 35, normalized size = 0.92 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {b^2-a^2}}{a-b}\right )}{\sqrt {b^2-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)*(a + b*coth(x))),x)

[Out]

-(2*atan((exp(x)*(b^2 - a^2)^(1/2))/(a - b)))/(b^2 - a^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}{\relax (x )}}{a + b \coth {\relax (x )}}\, dx \]

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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